Pseudo-Cartan Inclusions

David R. Pitts Department of Mathematics
University of Nebraska-Lincoln
Lincoln
NE 68588-0130 U.S.A. dpitts2@unl.edu

Abstract.

A pseudo-Cartan inclusion is a regular inclusion having a Cartan envelope. Unital pseudo-Cartan inclusions were classified in [PittsStReInII]; we extend this classification to include the non-unital case. The class of pseudo-Cartan inclusions coincides with the class of regular inclusions having the faithful unique pseudo-expectation property and can also be described using the ideal intersection property. We describe the twisted groupoid associated with the Cartan envelope of a pseudo-Cartan inclusion. These results significantly extend previous results obtained for the unital setting.

We explore properties of pseudo-Cartan inclusions and the relationship between a pseudo-Cartan inclusion and its Cartan envelope. For example, if ${\mathcal{D}}\subseteq{\mathcal{C}}$ is a pseudo-Cartan inclusion with Cartan envelope ${\mathcal{B}}\subseteq{\mathcal{A}}$ , then ${\mathcal{C}}$ is simple if and only if ${\mathcal{A}}$ is simple. Also every regular $*$ -automorphism of ${\mathcal{C}}$ uniquely extends to a $*$ -automorphism of ${\mathcal{A}}$ . We show that the inductive limit of pseudo-Cartan inclusions with suitable connecting maps is a pseudo-Cartan inclusion, and the minimal tensor product of pseudo-Cartan inclusions is a pseudo-Cartan inclusion. Further, we describe the Cartan envelope of pseudo-Cartan inclusions arising from these constructions. We conclude with some applications and a few open questions.

2020 Mathematics Subject Classification:

46L05

1. Introduction

A famous theorem of Gelfand states that an abelian $C^{*}$ -algebra ${\mathcal{C}}$ may be recognized as an algebra of continuous functions: ${\mathcal{C}}\simeq C_{0}(X)$ , where $X=\hat{\mathcal{C}}$ is the space of non-zero multiplicative linear functionals on ${\mathcal{C}}$ equipped with the weak- $*$ topology. In some cases, non-commutative $C^{*}$ -algebras can also be represented as functions on spaces. For example, Strătilă and Voiculescu [StratilaVoiculescuReAFAl], showed that for a unital AF-algebra ${\mathcal{A}}=\varinjlim{\mathcal{A}}_{n}$ (each ${\mathcal{A}}_{n}$ is a finite dimensional $C^{*}$ -algebra), there exists a maximal abelian $C^{*}$ -subalgebra ${\mathcal{B}}\subseteq{\mathcal{A}}$ such that for every $n$ , ${\mathcal{B}}\cap{\mathcal{A}}_{n}$ is maximal abelian in ${\mathcal{A}}_{n}$ . Strătilă and Voiculescu used ${\mathcal{B}}$ to construct a “coordinate system” $X$ for ${\mathcal{A}}$ which behaves in much the same way as a system of matrix units for $M_{n}({\mathbb{C}})$ ; furthermore, the AF-algebra ${\mathcal{A}}$ can be viewed as a collection of functions on $X$ . Shortly after Strătilă and Voiculescu’s work, Feldman and Moore [FeldmanMooreErEqReII] showed that if ${\mathcal{D}}$ is a $W^{*}$ -Cartan subalgebra of the (separably acting) von Neumann algebra ${\mathcal{M}}$ and ${\mathcal{D}}\simeq L^{\infty}(X,\mu)$ , then there is a “measured” equivalence relation $R$ on $X\times X$ along with a 2-cocycle $\sigma$ on $R$ such that ${\mathcal{M}}$ is, loosely speaking, isomorphic to an algebra of functions, $M(R,\sigma)$ , on $R$ with product given by a “generalized matrix multiplication”; further, the isomorphism of ${\mathcal{M}}$ onto $M(R,\sigma)$ carries ${\mathcal{D}}$ onto an algebra of functions supported on $\{(x,x):x\in X\}$ .

Efforts to adapt the ideas of Feldman and Moore to $C^{*}$ -algebras began with work of Renault in [RenaultGrApC*Al], continued with Kumjian’s work on $C^{*}$ -diagonals in [KumjianOnC*Di], and culminated with Renault’s definition of a Cartan MASA in a $C^{*}$ -algebra found in [RenaultCaSuC*Al]. Given a Cartan MASA ${\mathcal{D}}$ in the $C^{*}$ -algebra ${\mathcal{C}}$ , the Kumjian-Renault theory produces a Hausdorff, étale, and effective groupoid $G$ , along with a central groupoid extension $G^{(0)}\times{\mathbb{T}}\hookrightarrow\Sigma\twoheadrightarrow G$ ; this is the $C^{*}$ -algebraic analog of Feldman-Moore’s equivalence relation $R$ and 2-cocycle $\sigma$ . Moreover, there is a unique isomorphism of ${\mathcal{C}}$ onto a $C^{*}$ -algebra, $C^{*}_{r}(\Sigma,G)$ , which carries ${\mathcal{D}}$ onto $C_{0}(G^{(0)})$ . The algebra $C^{*}_{r}(\Sigma,G)$ can be viewed as an algebra of functions on $\Sigma$ with a product generalizing matrix multiplication, so here again, ${\mathcal{C}}$ may be viewed as an algebra of functions on a space. Conversely, every such central extension can be used to construct a Cartan MASA in a $C^{*}$ -algebra, so the theory gives a complete description of Cartan inclusions ${\mathcal{D}}\subseteq{\mathcal{C}}$ . These results are presented in [RenaultCaSuC*Al]; for a less terse treatment, see [SimsGroupoidsBook]. While Renault and Kumjian worked in the context of separable $C^{*}$ -algebras, Raad showed in [RaadGeReThCaSu] that this assumption can be removed.

Despite the success of the Kumjian-Renault theory, there are desirable settings in the $C^{*}$ -algebra context where the axioms for a Cartan MASA in the algebra ${\mathcal{C}}$ are not satisfied; examples include the virtual Cartan inclusions introduced in [PittsStReInI, Definition 1.1] and the weak Cartan inclusions defined in [ExelPittsChGrC*AlNoHaEtGr, Definition 2.11.5]. For these classes, the inclusion ${\mathcal{D}}\subseteq{\mathcal{C}}$ is regular and ${\mathcal{D}}$ is abelian. In such settings, it is possible to follow the Kumjian-Renault procedure to construct a groupoid $G$ and a central groupoid extension as above. However, when this is done, the resulting groupoid $G$ may fail to be Hausdorff, see [PittsStReInII, Theorem 4.4]. One approach to dealing with such situations is to accept the fact that non-Hausdorff groupoids may arise, and develop a theory which includes them. This is the approach taken in [ExelPittsChGrC*AlNoHaEtGr].

An alternate approach to studying regular inclusions of the form ${\mathcal{D}}\subseteq{\mathcal{C}}$ with ${\mathcal{D}}$ abelian, is to seek embeddings of ${\mathcal{D}}\subseteq{\mathcal{C}}$ into a Cartan inclusion ${\mathcal{B}}\subseteq{\mathcal{A}}$ so that the dynamics of ${\mathcal{D}}\subseteq{\mathcal{C}}$ are embedded into the dynamics of ${\mathcal{B}}\subseteq{\mathcal{A}}$ . (Such embeddings are called regular embeddings.) The idea is that the coordinates from the larger inclusion ${\mathcal{B}}\subseteq{\mathcal{A}}$ might then be used to study ${\mathcal{C}}$ . In [PittsStReInI], we gave a characterization of those unital regular inclusions which can be embedded into a $C^{*}$ -diagonal or a Cartan inclusion. However, the $C^{*}$ -diagonal ${\mathcal{B}}\subseteq{\mathcal{A}}$ obtained from the construction in [PittsStReInI] may bear little relation to the inclusion ${\mathcal{D}}\subseteq{\mathcal{C}}$ , and the embedding of ${\mathcal{D}}\subseteq{\mathcal{C}}$ into ${\mathcal{B}}\subseteq{\mathcal{A}}$ is not sufficiently rigid to transfer information from ${\mathcal{B}}\subseteq{\mathcal{A}}$ to ${\mathcal{D}}\subseteq{\mathcal{C}}$ .

Part of the motivation for [PittsStReInII] was to remedy this defect. In [PittsStReInII], the notion of the Cartan envelope is defined and two characterizations of those unital regular inclusions having a Cartan envelope are given. The Cartan envelope described in [PittsStReInII] is unique when it exists and is a minimal Cartan pair generated by the original inclusion ${\mathcal{D}}\subseteq{\mathcal{C}}$ . Also, in [PittsStReInII, Section 7], a description of the twisted groupoid associated to the Cartan envelope for ${\mathcal{D}}\subseteq{\mathcal{C}}$ is given.

Many regular inclusions are not unital. Examples include inclusions from graph algebras (or higher rank graph algebras), from crossed product constructions, or arise from tensor product constructions (such as those in Section 6.3 below). It is therefore of interest to extend results on Cartan envelopes to the non-unital setting.

Three primary goals of the present paper are: first, to establish results on Cartan envelopes which apply in the non-unital context; second, to give results showing that the Cartan envelope of a regular inclusion carries information about the original inclusion; and third, to give constructions of pseudo-Cartan inclusions along with their Cartan envelopes. Extending the characterization of regular inclusions having a Cartan envelope to include non-unital regular inclusions is unexpectedly subtle. On first glance, one might expect that passing from the unital setting to the non-unital case would simply be a matter of adjoining units and then applying the results on Cartan envelopes for unital inclusions from [PittsStReInII]. However, the unitization process need not preserve normalizing elements (see Definition 2.1(d)), nor does it preserve regular mappings (Definition 2.1(i)). For a concrete example, suppose ${\mathfrak{J}}\unlhd C_{0}({\mathbb{R}})$ is the ideal of functions in $C_{0}({\mathbb{R}})$ which vanish on $[-1,1]$ . Any $f\in C_{0}({\mathbb{R}})$ is a normalizer for the inclusion ${\mathfrak{J}}\subseteq C_{0}({\mathbb{R}})$ , yet $f$ is not a normalizer for the unitized inclusion $\tilde{\mathfrak{J}}\subseteq(C_{0}({\mathbb{R}}))^{\sim}$ unless $|f|$ is constant on $[-1,1]$ . In particular, this shows that while the identity mapping on $C_{0}({\mathbb{R}})$ is a regular map, its extension to $(C_{0}({\mathbb{R}}))^{\sim}$ (that is, $\operatorname{Id}_{(C_{0}({\mathbb{R}}))^{\sim}}$ ) is not regular. Furthermore, [PittsNoApUnInC*Al, Example 3.1] gives an example of an inclusion ${\mathcal{D}}\subseteq{\mathcal{C}}$ which is not regular, but whose unitization $\tilde{\mathcal{D}}\subseteq\tilde{\mathcal{C}}$ is a Cartan inclusion. Finally, a vexing issue we have been unable to resolve is whether regularity for an inclusion is preserved when units are adjoined. The ill behavior of regularity properties under the unitization process leads to serious technical difficulties when considering Cartan envelopes for non-unital inclusions. Using tools found in [PittsNoApUnInC*Al], we develop techniques to overcome those obstacles and are able to characterize those regular inclusions having a Cartan envelope; we shall describe the characterization in a moment.

Our work on Cartan envelopes yields an alternate set of axioms for a Cartan MASA ${\mathcal{D}}\subseteq{\mathcal{C}}$ . Recall that an inclusion ${\mathcal{B}}\subseteq{\mathcal{A}}$ of $C^{*}$ -algebras has the ideal intersection property (iip) if every non-zero ideal of ${\mathcal{A}}$ has non-trivial intersection with ${\mathcal{B}}$ . We also use ${\mathcal{B}}^{c}={\mathcal{A}}\cap{\mathcal{B}}^{\prime}$ for the relative commutant of ${\mathcal{B}}$ in ${\mathcal{A}}$ . The columns of the following table give Renault’s axioms for a Cartan MASA and (equivalent) alternate axioms for a Cartan MASA.

Cartan Inclusions:	Cartan Inclusions:
Renault’s Axioms	Alternate Axioms
${\mathcal{D}}\subseteq{\mathcal{C}}$ is regular	${\mathcal{D}}\subseteq{\mathcal{C}}$ is regular
${\mathcal{D}}$ is a MASA in ${\mathcal{C}}$	${\mathcal{D}}$ and ${\mathcal{D}}^{c}$ are abelian, and both
	the inclusions ${\mathcal{D}}\subseteq{\mathcal{D}}^{c}$ and ${\mathcal{D}}^{c}\subseteq{\mathcal{C}}$
	have the ideal intersection property
$\exists$ a faithful conditional expectation	$\exists$ a conditional expectation
$E:{\mathcal{C}}\rightarrow{\mathcal{D}}$	$E:{\mathcal{C}}\rightarrow{\mathcal{D}}$

(Renault’s original definition of Cartan MASA also assumes ${\mathcal{D}}$ contains an approximate identity for ${\mathcal{C}}$ , but [PittsNoApUnInC*Al, Theorem 2.4] shows that assumption can be removed.) By keeping the first two entries of the right-hand column but not requiring the third, we obtain a class of inclusions which we call pseudo-Cartan inclusions.

The characterization theorem, Theorem 3, extends the characterizations of the existence of a Cartan envelope found for unital inclusions in [PittsStReInII] to include the non-unital case. Theorem 3 shows the equivalence of the following: the inclusion ${\mathcal{D}}\subseteq{\mathcal{C}}$ (again with ${\mathcal{D}}$ abelian) has a Cartan envelope; ${\mathcal{D}}\subseteq{\mathcal{C}}$ is a pseudo-Cartan inclusion; and ${\mathcal{D}}\subseteq{\mathcal{C}}$ has the faithful unique pseudo-expectation property (see Definition 2.3). That an inclusion is a pseudo-Cartan inclusion if and only if it has the faithful unique pseudo-expectation property is the reason for the term pseudo-Cartan inclusion. A description of the Kumjian-Renault twist $G^{(0)}\times{\mathbb{T}}\hookrightarrow\Sigma\twoheadrightarrow G$ for the Cartan envelope of ${\mathcal{D}}\subseteq{\mathcal{C}}$ is given in Theorem 5.2. This description exhibits $\Sigma$ as a family of linear functionals on ${\mathcal{C}}$ and $G$ is the quotient of $\Sigma$ by a certain action of ${\mathbb{T}}$ .

The class of pseudo-Cartan inclusions contains the following classes: Cartan inclusions; the virtual Cartan inclusions of [PittsStReInI]; the weak Cartan inclusions introduced in [ExelPittsChGrC*AlNoHaEtGr]; and inclusions of the form $J\subseteq C_{0}(X)$ , where $X$ is a locally compact Hausdorff space and $J$ is an essential ideal in $C_{0}(X)$ .

Section 2 sets terminology, notation, and also gives a number of foundational results needed throughout the paper. These results concern weakly non-degenerate inclusions, pseudo-expectations, and the ideal intersection property. Some of the results in Section 2, such as Theorem 2.3, illustrate the technical challenges which can arise when verifying that desirable properties of an inclusion are preserved under the unitization process.

After presenting our results on existence and uniqueness of Cartan envelopes in Section 3, we explore properties of pseudo-Cartan inclusions and relationships between a pseudo-Cartan inclusion and its Cartan envelope. Section 4 contains the definition of pseudo-Cartan inclusions along with a few properties shared by ${\mathcal{C}}$ and ${\mathcal{A}}$ when ${\mathcal{D}}\subseteq{\mathcal{C}}$ is a pseudo-Cartan inclusion having Cartan envelope ${\mathcal{B}}\subseteq{\mathcal{A}}$ . For example, Proposition 4.3 shows ${\mathcal{C}}$ is simple if and only if ${\mathcal{A}}$ is simple. While a pseudo-Cartan inclusion can be (minimally) embedded into a Cartan inclusion, Proposition 4.4 gives a reverse process: it shows how to construct a family of pseudo-Cartan inclusions from a given Cartan MASA ${\mathcal{B}}\subseteq{\mathcal{A}}$ , and also examines the Cartan envelope for pseudo-Cartan inclusions arising from this construction.

Section 5 constructs the Kumjian-Renault groupoid model for the Cartan envelope of a given pseudo-Cartan inclusion ${\mathcal{D}}\subseteq{\mathcal{C}}$ starting with a family of linear functionals on ${\mathcal{C}}$ . In particular, this allows us to explicitly see the embedding of ${\mathcal{D}}\subseteq{\mathcal{C}}$ into its $C^{*}$ -envelope ${\mathcal{B}}\subseteq{\mathcal{A}}$ by realizing ${\mathcal{C}}$ as an algebra of functions (with a convolution product as multiplication).

Section 6 contains a rigidity property of the Cartan envelope and explores some permanence properties of pseudo-Cartan inclusions and their Cartan envelopes. The rigidity property is Proposition 6.1: it shows that given a pseudo-Cartan inclusion ${\mathcal{D}}\subseteq{\mathcal{C}}$ with Cartan envelope ${\mathcal{B}}\subseteq{\mathcal{A}}$ , any regular $*$ -automorphism of ${\mathcal{C}}$ extends uniquely to a $*$ -automorphism of ${\mathcal{A}}$ . We show in Theorem 6.2 that an inductive limit of pseudo-Cartan inclusions with suitable connecting maps is again a pseudo-Cartan inclusion inclusion and the Cartan envelope of such an inductive limit is the inductive limit of the Cartan envelopes of the approximating pseudo-Cartan inclusions. The proof of Theorem 6.2 uses an important mapping property of Cartan envelopes, Theorem 6.1. Theorem 6.3 shows that the minimal tensor product of two pseudo-Cartan inclusions is a pseudo-Cartan inclusion, and the Cartan envelope of their minimal tensor product is the minimal tensor product of their Cartan envelopes. Because quotients of pseudo-Cartan inclusions by regular ideals is the subject of a forthcoming paper (by Brown, Fuller, Reznikoff and the author), we do not consider quotients here.

We provide some applications of our work in Section 7. We show that when ${\mathcal{D}}\subseteq{\mathcal{C}}$ is a unital pseudo-Cartan inclusion, the $C^{*}$ -envelope of a Banach algebra ${\mathcal{D}}\subseteq{\mathcal{A}}\subseteq{\mathcal{C}}$ is the $C^{*}$ -subalgebra of ${\mathcal{C}}$ generated by ${\mathcal{A}}$ and that ${\mathcal{D}}$ norms ${\mathcal{C}}$ in the sense of [PopSinclairSmithNoC*Al]. We combine these results to show that if, for $i=1,2$ : ${\mathcal{D}}_{i}\subseteq{\mathcal{C}}_{i}$ are pseudo-Cartan inclusions such that their unitizations $\tilde{\mathcal{D}}_{i}\subseteq\tilde{\mathcal{C}}_{i}$ are also pseudo-Cartan inclusions; ${\mathcal{A}}_{i}$ are Banach algebras with ${\mathcal{D}}_{i}\subseteq{\mathcal{A}}_{i}\subseteq{\mathcal{C}}_{i}$ ; and $u:{\mathcal{A}}_{1}\rightarrow{\mathcal{A}}_{2}$ is an isometric isomorphism, then $u$ uniquely extends to a $*$ -isomorphism of $C^{*}({\mathcal{A}}_{1})$ onto $C^{*}(A_{2})$ . While these applications have antecedents in the literature, our results are significant generalizations. We include a few open questions in Section 8.

We require [PittsStReInII, Theorem 6.9] at two points in our work: in Section 5.1(d), and in the proof of Lemma 7. While [PittsStReInII, Theorem 6.9] is correctly stated in [PittsStReInII], due to an error in the statement of [PittsStReInII, Lemma 2.3], the proof of [PittsStReInII, Theorem 6.9] is insufficient. The correct statement of [PittsStReInII, Lemma 2.3] and a complete proof of [PittsStReInII, Theorem 6.9] may be found in Appendix A.

We thank Adam Fuller for several useful comments and Alex Kumjian for a helpful comment regarding the history of the adaptation of the Feldman-Moore theory to the $C^{*}$ -algebraic context.

2. Foundations

2.1. Terminology and Notation

For a normed linear space $X$ , we shall use $X^{\#}$ to denote the collection of all bounded linear functionals on $X$ .

Let ${\mathcal{A}}$ be a Banach algebra. When ${\mathcal{A}}$ is unital, we denote its unit by $I$ or $I_{\mathcal{A}}$ . Unless explicitly stated otherwise, when referring to an ideal $J$ in ${\mathcal{A}}$ , we will always assume $J$ is closed and two-sided. When $J$ is an ideal in ${\mathcal{A}}$ , we write $J\unlhd{\mathcal{A}}$ . The ideal $J$ is called an essential ideal if for any non-zero ideal $K\unlhd A$ , $J\cap K\neq\{0\}$ .

We will follow the notation and conventions regarding unitization of a $C^{*}$ -algebra found in [BlackadarOpAl, II.1.2.1]. For a $C^{*}$ -algebra ${\mathcal{A}}$ , let ${\mathcal{A}}^{\dagger}:={\mathcal{A}}\times{\mathbb{C}}$ with the usual operations and norm which make it into a unital $C^{*}$ -algebra. Define $\tilde{\mathcal{A}}$ and a $*$ -monomorphism $u_{\mathcal{A}}:{\mathcal{A}}\rightarrow\tilde{\mathcal{A}}$ by

(2.1.1)	$\displaystyle\tilde{\mathcal{A}}$	$\displaystyle:=\begin{cases}{\mathcal{A}}&\text{if ${\mathcal{A}}$ is unital}\\ {\mathcal{A}}^{\dagger}&\text{if ${\mathcal{A}}$ is not unital;}\end{cases}$
and for $x\in{\mathcal{A}}$ ,
(2.1.2)	$\displaystyle u_{\mathcal{A}}(x)$	$\displaystyle:=\begin{cases}x&\text{if ${\mathcal{A}}$ is unital}\\ (x,0)&\text{if ${\mathcal{A}}$ is not unital.}\end{cases}$

We shall consider the trivial algebra $\{0\}$ to be unital. Also, when there is no danger of confusion, we will identify ${\mathcal{A}}$ with $u_{\mathcal{A}}({\mathcal{A}})$ , and regard ${\mathcal{A}}\subseteq\tilde{\mathcal{A}}$ .

Suppose ${\mathcal{A}}_{1}$ and ${\mathcal{A}}_{2}$ are $C^{*}$ -algebras. Given a bounded linear map $\psi:{\mathcal{A}}_{1}\rightarrow{\mathcal{A}}_{2}$ , the map $\tilde{\psi}:\tilde{\mathcal{A}}_{1}\rightarrow\tilde{\mathcal{A}}_{2}$ given by

(2.1.3)

\tilde{\psi}(x)=\begin{cases}u_{{\mathcal{A}}_{2}}(\psi(x))&\text{if ${\mathcal{A}}_{1}$ is unital;}\\ u_{{\mathcal{A}}_{2}}(\psi(a))+\lambda I_{\tilde{\mathcal{A}}_{2}}&\text{if ${\mathcal{A}}_{1}$ is not unital and $x=(a,\lambda)\in\tilde{\mathcal{A}}_{1}$}\end{cases}

will be called the standard extension of $\psi$ to $\tilde{\mathcal{A}}_{1}$ . We will frequently use the following property of the standard extension without comment:

(2.1.4)

u_{{\mathcal{A}}_{2}}\circ\psi=\tilde{\psi}\circ u_{{\mathcal{A}}_{1}}.

It is not generally the case that $\tilde{\psi}(I_{\tilde{\mathcal{A}}_{1}})=I_{\tilde{A}_{2}}$ . When $\psi$ is a $*$ -homomorphism, so is $\tilde{\psi}$ .

Observation \the\numberby.

If $\psi$ is contractive and completely positive, then $\tilde{\psi}$ is also contractive and completely postive.

Proof.

When ${\mathcal{A}}_{1}$ is not unital, this follows from [BrownOzawaC*AlFiDiAp, Proposition 2.2.1] applied to $u_{{\mathcal{A}}_{2}}\circ\psi$ , and from the definition of $\tilde{\psi}$ when ${\mathcal{A}}_{1}$ is unital. ∎

Fundamental to our study are inclusions, which we usually consider as a $C^{*}$ -algebra ${\mathcal{B}}$ contained in another $C^{*}$ -algebra ${\mathcal{A}}$ . At times, particularly in this section, it will be helpful to explicitly mention the embedding of ${\mathcal{B}}$ into ${\mathcal{A}}$ . While this makes notation more cumbersome, the benefit of clarity outweighs the notational burdens.

Definition \the\numberby.

(a)

An inclusion is a triple $({\mathcal{A}},{\mathcal{B}},f)$ , where ${\mathcal{A}}$ and ${\mathcal{B}}$ are $C^{*}$ -algebras and $f:{\mathcal{B}}\rightarrow{\mathcal{A}}$ is a $*$ -monomorphism. We include the possibility that ${\mathcal{B}}=\{0\}$ . (Starting in Section 3, we will always assume ${\mathcal{B}}$ is abelian, but until then we do not make that restriction.)
(b)

Let $({\mathcal{A}}_{1},{\mathcal{B}}_{1},f_{1})$ be an inclusion. An inclusion $({\mathcal{A}}_{2},{\mathcal{B}}_{2},f_{2})$ together with a $*$ -monomorphism $\alpha:{\mathcal{A}}_{1}\rightarrow{\mathcal{A}}_{2}$ such that $\alpha(f_{1}({\mathcal{B}}_{1}))\subseteq f_{2}({\mathcal{B}}_{2})$ will be called an expansion of $({\mathcal{A}}_{1},{\mathcal{B}}_{1},f_{1})$ and will be denoted by $({\mathcal{A}}_{2},{\mathcal{B}}_{2},f_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ . Because $f_{2}$ is one-to-one, $({\mathcal{A}}_{2},{\mathcal{B}}_{2},f_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is an expansion of $({\mathcal{A}}_{1},{\mathcal{B}}_{1},f_{1})$ if and only if $\alpha:{\mathcal{A}}_{1}\rightarrow{\mathcal{A}}_{2}$ is a $*$ -monomorphism and there exists a $*$ -monomorphism $\underline{\alpha}:{\mathcal{B}}_{1}\rightarrow{\mathcal{B}}_{2}$ such that

(2.1.5) $\alpha\circ f_{1}=f_{2}\circ\underline{\alpha}.$
(c)

We will call the two inclusions $({\mathcal{A}}_{1},{\mathcal{B}}_{1},f_{1})$ and $({\mathcal{A}}_{2},{\mathcal{B}}_{2},f_{2})$ isomorphic inclusions if there is a $*$ -isomorphism $\alpha:{\mathcal{A}}_{1}\rightarrow{\mathcal{A}}_{2}$ such that $\alpha(f_{1}({\mathcal{B}}_{1}))=f_{2}({\mathcal{B}}_{2})$ .
(d)

The inclusion $({\mathcal{A}},{\mathcal{B}},f)$ is called a unital inclusion if ${\mathcal{A}}$ and ${\mathcal{B}}$ are unital and $f(I_{\mathcal{B}})=I_{\mathcal{A}}$ .
(e)

Given the inclusion $({\mathcal{A}},{\mathcal{B}},f)$ , the standard extension of $f$ to $\tilde{f}:\tilde{\mathcal{B}}\rightarrow\tilde{\mathcal{A}}$ described in (2.1.3) is a $*$ -monomorphism because ${\mathcal{B}}$ is an essential ideal in $\tilde{\mathcal{B}}$ . Thus $(\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{f})$ is an inclusion. We will call $\tilde{f}$ the the standard embedding of $\tilde{\mathcal{B}}$ into $\tilde{\mathcal{A}}$ . (Caution: $(\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{f})$ need not be a unital inclusion.)

Because

(2.1.6) $\tilde{f}\circ u_{\mathcal{B}}=u_{\mathcal{A}}\circ f,$

$(\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{f}\hbox{\,:\hskip-1.0pt:\,}u_{\mathcal{A}})$ is an expansion of $({\mathcal{A}},{\mathcal{B}},f)$ . We will call $(\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{f}\hbox{\,:\hskip-1.0pt:\,}u_{\mathcal{A}})$ the standard expansion of $({\mathcal{A}},{\mathcal{B}},f)$ .

(f)

For $1\leq i<j\leq 3$ , let $({\mathcal{B}}_{j},{\mathcal{B}}_{i},f_{ji})$ be inclusions. If $f_{31}=f_{32}\circ f_{21}$ , we will say the two inclusions $({\mathcal{B}}_{2},{\mathcal{B}}_{1},f_{21})$ and $({\mathcal{B}}_{3},{\mathcal{B}}_{2},f_{32})$ are intermediate inclusions for $({\mathcal{B}}_{3},{\mathcal{B}}_{1},f_{31})$ .

(2.1.7)

Notation \the\numberby. Let $({\mathcal{A}},{\mathcal{B}},f)$ be an inclusion.

•

When there is little danger of confusion, we will often identify ${\mathcal{B}}$ with its image $f({\mathcal{B}})$ , so that ${\mathcal{B}}\subseteq{\mathcal{A}}$ . When this identification is made, we will suppress $f$ and call $({\mathcal{A}},{\mathcal{B}})$ an inclusion.
•

When we write $(\tilde{\mathcal{A}},\tilde{\mathcal{B}})$ , we always mean $(\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{f})$ . Since $\tilde{f}$ is not necessarily a unital mapping, $(\tilde{\mathcal{A}},\tilde{\mathcal{B}})$ need not be a unital inclusion in the sense of Definition 2.1(d).
•

We will sometimes write $({\mathcal{A}},{\mathcal{B}},\subseteq)$ for inclusions when ${\mathcal{B}}$ is a $C^{*}$ -subalgebra of ${\mathcal{A}}$ .
•

If $({\mathcal{A}}_{2},{\mathcal{B}}_{2},f_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is an expansion of $({\mathcal{A}}_{1},{\mathcal{B}}_{1},f_{1})$ and ${\mathcal{B}}_{i}$ has been identified with $f_{i}({\mathcal{B}}_{i})$ as above, we will again suppress writing $f_{i}$ and say $({\mathcal{A}}_{2},{\mathcal{B}}_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is an expansion of $({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ .
•

If $({\mathcal{B}}_{2},{\mathcal{B}}_{1},f_{21})$ and $({\mathcal{B}}_{3},{\mathcal{B}}_{2},f_{32})$ are inclusions which are intermediate for $({\mathcal{B}}_{3},{\mathcal{B}}_{1},f_{31})$ , we will sometimes say that ${\mathcal{B}}_{2}$ is intermediate to $({\mathcal{B}}_{3},{\mathcal{B}}_{1})$ and may also indicate this using the notation ${\mathcal{B}}_{1}\subseteq{\mathcal{B}}_{2}\subseteq{\mathcal{B}}_{3}$ .

There are: various types of inclusions; objects associated to an inclusion; and properties an inclusion may have. We describe some of them here.

Definitions, Notations, and Comments \the\numberby. Let $({\mathcal{A}},{\mathcal{B}},f)$ be an inclusion.

(a)

We will use $\operatorname{\textsc{rCom}}({\mathcal{A}},{\mathcal{B}},f)$ for the relative commutant of $f({\mathcal{B}})$ in ${\mathcal{A}}$ , that is,

\operatorname{\textsc{rCom}}({\mathcal{A}},{\mathcal{B}},f):=\{a\in{\mathcal{A}}:af(b)=f(b)a\text{ for all }b\in{\mathcal{B}}\}.

When ${\mathcal{B}}$ is identified with $f({\mathcal{B}})$ , we will frequently use the notation ${\mathcal{B}}^{c}$ or $\operatorname{\textsc{rCom}}({\mathcal{A}},{\mathcal{B}})$ instead of $\operatorname{\textsc{rCom}}({\mathcal{A}},{\mathcal{B}},f)$ .

(b)

We say that $({\mathcal{A}},{\mathcal{B}},f)$ has the has the ideal intersection property if every non-zero ideal in ${\mathcal{A}}$ has non-trivial intersection with $f({\mathcal{B}})$ . We will also use the term essential inclusion as a synonym for an inclusion with the ideal intersection property. Likewise, when $({\mathcal{A}},{\mathcal{B}},f)$ is an essential inclusion, we will sometimes call the map $f$ an essential map.

(The literature contains several synonyms for the ideal intersection property, for example in [KwasniewskiMeyerApAlExPrUnPsEx], an inclusion with the ideal intersection property is said to detect ideals, and in [PittsZarikianUnPsExC*In], such an inclusion is called $C^{*}$ -essential.)
(c)

$({\mathcal{A}},{\mathcal{B}},f)$ has the approximate unit property (abbreviated AUP) if there is an approximate unit $(u_{\lambda})$ for ${\mathcal{B}}$ such that $f(u_{\lambda})$ is an approximate unit for ${\mathcal{A}}$ . In remarks following [ExelOnKuC*DiOpId, Definition 2.1], Exel notes that the Krein-Milman Theorem and [AkemannShultzPeC*Al, Lemma 2.32] yield the following characterization of the approximate unit property.

Fact \the\numberby (see [ExelOnKuC*DiOpId]).

An inclusion $({\mathcal{A}},{\mathcal{B}},f)$ has the approximate unit property if and only if no pure state of ${\mathcal{A}}$ annihilates $f({\mathcal{B}})$ .

(d)

A normalizer for $({\mathcal{A}},{\mathcal{B}},f)$ is an element of the set,

{\mathcal{N}}({\mathcal{A}},{\mathcal{B}},f):=\{v\in{\mathcal{A}}:vf({\mathcal{B}})v^{*}\cup v^{*}f({\mathcal{B}})v\subseteq f({\mathcal{B}})\}.

If ${\mathcal{B}}$ is identified with $f({\mathcal{B}})$ , we will write ${\mathcal{N}}({\mathcal{A}},{\mathcal{B}})$ rather than ${\mathcal{N}}({\mathcal{A}},{\mathcal{B}},f)$ .

(e)

Closely related to normalizers are intertwiners. An element $w\in{\mathcal{A}}$ is called an intertwiner if the sets $f({\mathcal{B}})w$ and $wf({\mathcal{B}})$ coincide. We will write ${\mathcal{I}}({\mathcal{A}},{\mathcal{B}},f)$ (or ${\mathcal{I}}({\mathcal{A}},{\mathcal{B}})$ when $f$ is suppressed) for the set of all intertwiners.
(f)

$({\mathcal{A}},{\mathcal{B}},f)$ is said to be a regular inclusion if $\overline{\operatorname{span}}\,{\mathcal{N}}({\mathcal{A}},{\mathcal{B}},f)={\mathcal{A}}$ .
(g)

If $f({\mathcal{B}})$ is maximal abelian in ${\mathcal{A}}$ , we will call $({\mathcal{A}},{\mathcal{B}},f)$ a MASA inclusion.
(h)

$({\mathcal{A}},{\mathcal{B}},f)$ is a Cartan inclusion, also called a Cartan pair, if it is a regular MASA inclusion and there exists a faithful conditional expectation $\Delta:{\mathcal{A}}\rightarrow f({\mathcal{B}})$ . Cartan inclusions necessarily have the AUP, [PittsNoApUnInC*Al, Theorem 2.6].

The conditional expectation $\Delta$ is unique; see [RenaultCaSuC*Al, Corollary 5.10] or [PittsStReInI, Theorem 3.5]. Also, $\Delta$ is invariant under ${\mathcal{N}}({\mathcal{A}},{\mathcal{B}},f)$ in the sense that for every $v\in{\mathcal{N}}({\mathcal{A}},{\mathcal{B}},f)$ and $x\in{\mathcal{A}}$ ,

(2.1.10) $\Delta(v^{*}xv)=v^{*}\Delta(x)v.$

While we do not have an explicit reference for (2.1.10), it is certainly known; for example, it follows from Renault’s (twisted) groupoid model for a Cartan pair.

Comment \the\numberby. We will repeatedly use the fact that the regular inclusion $({\mathcal{A}},{\mathcal{B}},f)$ is a Cartan inclusion if and only if $(\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{f})$ is a Cartan inclusion, see [PittsNoApUnInC*Al, Proposition 3.2]. Example 3.1 of [PittsNoApUnInC*Al] shows the necessity of the regularity hypothesis on $({\mathcal{A}},{\mathcal{B}},f)$ .
(i)

If $({\mathcal{A}}_{1},{\mathcal{B}}_{1},f_{1})$ is another inclusion, a $*$ -homomorphism $\alpha:{\mathcal{A}}\rightarrow{\mathcal{A}}_{1}$ is a regular homomorphism if

$\alpha({\mathcal{N}}({\mathcal{A}},{\mathcal{B}},f))\subseteq{\mathcal{N}}({\mathcal{A}}_{1},{\mathcal{B}}_{1},f_{1}).$

We will sometimes indicate this by saying $\alpha:({\mathcal{A}},{\mathcal{B}},f)\rightarrow({\mathcal{A}}_{1},{\mathcal{B}}_{1},f_{1})$ is a regular homomorphism.

When ${\mathcal{B}}_{2}$ is intermediate to $({\mathcal{B}}_{3},{\mathcal{B}}_{1})$ , it may happen that $\tilde{\mathcal{B}}_{2}$ is not intermediate to $(\tilde{\mathcal{B}}_{3},\tilde{\mathcal{B}}_{1})$ . Here is an example of this behavior (the example also provides an example of an inclusion whose “unitization” is not unital).

Example \the\numberby. Let $S_{0}$ and $S_{1}$ be isometries on a Hilbert space such that $S_{0}S_{0}^{*}+S_{1}S_{1}^{*}=I$ , let ${\mathcal{S}}$ be the inverse semigroup of partial isometries consisting of all finite products of elements of the set $\{S_{0},S_{1},S_{0}^{*},S_{1}^{*}\}$ , and let ${\mathcal{D}}$ be the $C^{*}$ -algebra generated by the projections in ${\mathcal{S}}$ . There is a unique multiplicative linear functional $\tau:{\mathcal{D}}\rightarrow{\mathbb{C}}$ on ${\mathcal{D}}$ such that for every $n\in{\mathbb{N}}$ , $\tau(S_{1}^{n}S_{1}^{n}{}^{*})=1$ . (Indeed, ${\mathcal{D}}$ is isomorphic to the continuous functions on the Cantor middle thirds set, and under this isomorphism, $\tau$ corresponds to evaluation at 1.) Let

{\mathcal{B}}_{3}={\mathcal{B}}_{1}=\ker\tau,\quad{\mathcal{B}}_{2}={\mathcal{D}},

and let $\sigma:{\mathcal{D}}\rightarrow\ker\tau$ be the map

\sigma(d)=S_{0}dS_{0}^{*}.

Put

f_{31}:=\sigma^{2}|_{{\mathcal{B}}_{1}},\quad f_{21}:=\sigma|_{{\mathcal{B}}_{1}}\quad\text{and}\quad f_{32}:=\sigma.

Since $f_{31}=f_{32}\circ f_{21}$ , $({\mathcal{B}}_{2},{\mathcal{B}}_{1},f_{21})$ and $({\mathcal{B}}_{3},{\mathcal{B}}_{2},f_{32})$ are intermediate inclusions for $({\mathcal{B}}_{3},{\mathcal{B}}_{1},f_{31})$ . Let $q$ denote the quotient map of $\tilde{\mathcal{B}}_{3}$ onto $\tilde{\mathcal{B}}_{3}/{\mathcal{B}}_{3}\simeq{\mathbb{C}}$ . Since ${\mathcal{B}}_{1}$ and ${\mathcal{B}}_{3}$ are not unital, $q\circ\tilde{f}_{31}\neq 0$ . On the other hand, since ${\mathcal{B}}_{2}$ is unital, $q\circ\tilde{f}_{32}\circ\tilde{f}_{21}=0$ . Therefore, $\tilde{f}_{31}\neq\tilde{f}_{32}\circ\tilde{f}_{21}$ , so $(\tilde{\mathcal{B}}_{2},\tilde{\mathcal{B}}_{1},\tilde{f}_{21})$ and $(\tilde{\mathcal{B}}_{3},\tilde{\mathcal{B}}_{2},\tilde{f}_{32})$ are not intermediate inclusions for $(\tilde{\mathcal{B}}_{3},\tilde{\mathcal{B}}_{1},\tilde{f}_{31})$ . Finally, note that $({\mathcal{B}}_{2},{\mathcal{B}}_{1},f_{21})$ is an example of an inclusion where $(\tilde{\mathcal{B}}_{2},\tilde{\mathcal{B}}_{1},\tilde{f}_{21})$ is not a unital inclusion.

2.2. Weakly Non-Degenerate Inclusions

The approximate unit property for an inclusion $({\mathcal{A}},{\mathcal{B}},f)$ has been used in the literature as a definition of non-degeneracy, see for example [CrytserNagySiCrEtGrC*Al, Definition 1.4] or [ExelPittsChGrC*AlNoHaEtGr, Definition 2.3.1(iii)]. However, when $J\unlhd{\mathcal{A}}$ is a proper and essential ideal, the inclusion $({\mathcal{A}},J,\subseteq)$ cannot have the AUP, so using the AUP as a non-degeneracy condition excludes such examples. For our purposes, the following weaker version of non-degeneracy is more appropriate.

Definition \the\numberby.

Let $({\mathcal{A}},{\mathcal{B}},f)$ be an inclusion.

(a)

Define the annihilator of ${\mathcal{B}}$ in ${\mathcal{A}}$ to be the set,

\operatorname{Ann}({\mathcal{A}},{\mathcal{B}},f):=\{a\in{\mathcal{A}}:af(b)=0=f(b)a\text{ for all }b\in{\mathcal{B}}\}.

Notice that $\operatorname{Ann}({\mathcal{A}},{\mathcal{B}},f)$ is an ideal of $\operatorname{\textsc{rCom}}({\mathcal{A}},{\mathcal{B}},f)$ .

(b)

An inclusion $({\mathcal{A}},{\mathcal{B}},f)$ such that $\operatorname{Ann}({\mathcal{A}},{\mathcal{B}},f)=\{0\}$ will be called a weakly non-degenerate inclusion.

As usual, when ${\mathcal{B}}$ is identified with $f({\mathcal{B}})\subseteq{\mathcal{A}}$ , we will write $\operatorname{Ann}({\mathcal{A}},{\mathcal{B}})$ instead of $\operatorname{Ann}({\mathcal{A}},{\mathcal{B}},f)$ and will say $({\mathcal{A}},{\mathcal{B}})$ is weakly non-degenerate when $({\mathcal{A}},{\mathcal{B}},f)$ is weakly non-degenerate.

Remark \the\numberby. We chose the term “weakly non-degenerate” instead of “non-degenerate” to avoid confusion with existing literature. Because of the similarity between the terms “weakly non-degenerate” and “non-degenerate” we will always use the term “approximate unit property” instead of “non-degenerate.” Finally, to avoid possible confusion, for a representation $\pi$ of a $C^{*}$ -algebra ${\mathcal{A}}$ on a Hilbert space ${\mathcal{H}}$ , we will sometimes explicitly write $\overline{\pi({\mathcal{A}}){\mathcal{H}}}={\mathcal{H}}$ instead of using the term “non-degenerate representation.”

Here are some conditions ensuring an inclusion is weakly non-degenerate.

Lemma \the\numberby.

Let $({\mathcal{A}},{\mathcal{B}})$ be an inclusion.

(a)

If $({\mathcal{A}},{\mathcal{B}})$ has the AUP, then $({\mathcal{A}},{\mathcal{B}})$ is weakly non-degenerate.
(b)

If ${\mathcal{B}}\unlhd{\mathcal{A}}$ is an essential ideal, then $({\mathcal{A}},{\mathcal{B}})$ is weakly non-degenerate. In particular, when ${\mathcal{A}}$ is not unital, $(\tilde{\mathcal{A}},{\mathcal{A}},u_{\mathcal{A}})$ is weakly non-degenerate.
(c)

If ${\mathcal{B}}$ is contained in the center of ${\mathcal{A}}$ and $({\mathcal{A}},{\mathcal{B}})$ has the ideal intersection property, then $({\mathcal{A}},{\mathcal{B}})$ is weakly non-degenerate.
(d)

Suppose $E:{\mathcal{A}}\rightarrow{\mathcal{B}}$ is a faithful conditional expectation. Then $({\mathcal{A}},{\mathcal{B}})$ is weakly non-degenerate.
(e)

Suppose $({\mathcal{C}},{\mathcal{D}})$ is an inclusion such that ${\mathcal{D}}$ is abelian and $({\mathcal{D}}^{c},{\mathcal{D}})$ has the ideal intersection property. Then $({\mathcal{C}},{\mathcal{D}})$ is weakly non-degenerate. In particular, any MASA inclusion is weakly non-degenerate.

Proof.

(a) Let $(u_{\lambda})\subseteq{\mathcal{B}}$ be an approximate unit for ${\mathcal{A}}$ . If $x\in\operatorname{Ann}({\mathcal{A}},{\mathcal{B}})$ , then $x=\lim xu_{\lambda}=0$ .

(b) Since ${\mathcal{B}}$ is an ideal of ${\mathcal{A}}$ , $\operatorname{Ann}({\mathcal{A}},{\mathcal{B}})$ is also an ideal in ${\mathcal{A}}$ . Clearly $\operatorname{Ann}({\mathcal{A}},{\mathcal{B}})\cap{\mathcal{B}}=\{0\}$ , and as ${\mathcal{B}}$ is an essential ideal, $\operatorname{Ann}({\mathcal{A}},{\mathcal{B}})=\{0\}$ .

(c) By hypothesis, ${\mathcal{B}}^{c}={\mathcal{A}}$ , so $\operatorname{Ann}({\mathcal{A}},{\mathcal{B}})$ is an ideal of ${\mathcal{A}}$ . As $\operatorname{Ann}({\mathcal{A}},{\mathcal{B}})$ has trivial intersection with ${\mathcal{B}}$ , the ideal intersection property gives $\operatorname{Ann}({\mathcal{A}},{\mathcal{B}})=\{0\}$ . Therefore, $({\mathcal{A}},{\mathcal{B}})$ is weakly non-degenerate.

(d) Let $x\in\operatorname{Ann}({\mathcal{A}},{\mathcal{B}})$ . Since $x^{*}x\in\operatorname{Ann}({\mathcal{A}},{\mathcal{B}})$ and $E(x^{*}x)\in{\mathcal{B}}$ ,

E(x^{*}x)^{2}=E(x^{*}xE(x^{*}x))=0,

with the first equality following from Tomiyama’s theorem, see [BrownOzawaC*AlFiDiAp, Theorem 1.5.10]. This gives $x=0$ by faithfulness of $E$ . So $({\mathcal{A}},{\mathcal{B}})$ is weakly non-degenerate.

(e) Applying part (c) to $({\mathcal{D}}^{c},{\mathcal{D}})$ gives $\operatorname{Ann}({\mathcal{D}}^{c},{\mathcal{D}})=\{0\}$ . But $\operatorname{Ann}({\mathcal{C}},{\mathcal{D}})$ is an ideal in ${\mathcal{D}}^{c}$ having trivial intersection with ${\mathcal{D}}$ . So $\operatorname{Ann}({\mathcal{C}},{\mathcal{D}})=\{0\}$ , showing $({\mathcal{C}},{\mathcal{D}})$ is weakly non-degenerate. ∎

In general, for an inclusion $({\mathcal{A}},{\mathcal{B}},f)$ , there are multiple embeddings of $\tilde{\mathcal{B}}$ into $\tilde{\mathcal{A}}$ which extend $u_{\mathcal{A}}\circ f$ , and it may happen that the image of $I_{\tilde{\mathcal{B}}}$ under the standard embedding is not $I_{\tilde{\mathcal{A}}}$ . Fortunately, these behaviors cannot occur when $({\mathcal{A}},{\mathcal{B}},f)$ is weakly non-degenerate, as we shall see in parts (e) and (f) of Lemma 2.2 below. Before proving Lemma 2.2, we require some preparation. The first is the following well-known fact (see [HamanaReEmCStAlMoCoCStAl, Lemma 4.8] or [EffrosAsNoOr, (2.2)]).

Lemma \the\numberby.

Suppose ${\mathcal{H}}={\mathcal{H}}_{1}\oplus{\mathcal{H}}_{2}$ is the direct sum of the Hilbert spaces ${\mathcal{H}}_{1}$ and ${\mathcal{H}}_{2}$ . Let $A\in{\mathcal{B}}({\mathcal{H}}_{1})$ be positive and invertible, $B\in{\mathcal{B}}({\mathcal{H}}_{2},{\mathcal{H}}_{1})$ , and $0\leq C\in{\mathcal{B}}({\mathcal{H}}_{2})$ . Then

T:=\begin{bmatrix}A&B\\ B^{*}&C\end{bmatrix}\in{\mathcal{B}}({\mathcal{H}}_{1}\oplus{\mathcal{H}}_{2})

is a positive operator if and only if $B^{*}A^{-1}B\leq C$ . Moreover, if $T\geq 0$ and $C=0$ , then $B=0$ .

Proof.

Both statements follow from the factorization,

\begin{bmatrix}A&B\\ B^{*}&C\end{bmatrix}=\begin{bmatrix}I_{{\mathcal{H}}_{1}}&A^{-1}B\\ 0&I_{{\mathcal{H}}_{2}}\end{bmatrix}^{*}\begin{bmatrix}A&0\\ 0&C-(A^{-1/2}B)^{*}(A^{-1/2}B)\end{bmatrix}\begin{bmatrix}I_{{\mathcal{H}}_{1}}&A^{-1}B\\ 0&I_{{\mathcal{H}}_{2}}\end{bmatrix}.

∎

Among inclusions, weakly non-degenerate inclusions are rather well-behaved. Here are a number of desirable properties which weakly non-degenerate inclusions possess.

Lemma \the\numberby.

Suppose $({\mathcal{A}},{\mathcal{B}},f)$ is a weakly non-degenerate inclusion. The following statements hold.

(a)

If ${\mathcal{B}}$ is unital, then ${\mathcal{A}}$ is unital and $f(I_{\mathcal{B}})$ is the unit for ${\mathcal{A}}$ .
(b)

Suppose $a\in{\mathcal{A}}$ satisfies $f(b)a=af(b)=f(b)$ for every $b\in{\mathcal{B}}$ . Then ${\mathcal{A}}$ is unital and $a=I_{\mathcal{A}}$ .
(c)

Let $(u_{\lambda})$ be an approximate unit for ${\mathcal{B}}$ . Suppose $a\in{\mathcal{A}}$ satisfies: $0\leq a$ , $\left\lVert a\right\rVert\leq 1$ , and $f(u_{\lambda})\leq a$ for every $\lambda$ . Then ${\mathcal{A}}$ is unital and $a=I_{\mathcal{A}}$ .
(d)

$(\tilde{\mathcal{A}},{\mathcal{B}},u_{\mathcal{A}}\circ f)$ is weakly non-degenerate.
(e)

$(\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{f})$ is a unital inclusion; in particular it is weakly non-degenerate.
(f)

If $\sigma:\tilde{\mathcal{B}}\rightarrow\tilde{\mathcal{A}}$ is a $*$ -monomorphism such that $\sigma\circ u_{\mathcal{B}}=u_{\mathcal{A}}\circ f$ , then $\sigma=\tilde{f}$ .

Proof.

(a) We may suppose ${\mathcal{A}}\subseteq{\mathcal{B}}({\mathcal{H}})$ . Let $I=I_{\mathcal{H}}$ , $p:=f(I_{\mathcal{B}})$ , and choose $a\in{\mathcal{A}}$ with $0\leq a$ . Relative to the decomposition, ${\mathcal{H}}=p{\mathcal{H}}+(I-p){\mathcal{H}}$ , $a$ and $a+p$ have the form,

a=\begin{bmatrix}pap&pa(I-p)\\ (I-p)ap&(I-p)a(I-p)\end{bmatrix}\quad\text{and}\quad a+p=\begin{bmatrix}pap+p&pa(I-p)\\ (I-p)ap&(I-p)a(I-p)\end{bmatrix}.

Let $x:=(I-p)a(I-p)=a-ap-pa+pap$ . Then $x\in\operatorname{Ann}({\mathcal{A}},{\mathcal{B}},f)$ . As $({\mathcal{A}},{\mathcal{B}},f)$ is weakly non-degenerate, $(I-p)a(I-p)=0$ . Since $0\leq a\leq a+p$ , applying Lemma 2.2 to $a+p$ gives $(I-p)ap=pa(I-p)=0$ . Therefore,

ap=pa=pap=a.

As ${\mathcal{A}}$ is the linear span of its positive elements, we conclude that $p=I_{\mathcal{A}}$ .

(b) Let $p=a^{*}a$ . Then for every $b\in{\mathcal{B}}$ , $pf(b)=f(b)p=f(b)$ because ${\mathcal{B}}$ is a self-adjoint subspace of ${\mathcal{A}}$ . Therefore, $p^{2}-p\in\operatorname{Ann}({\mathcal{A}},{\mathcal{B}},f)=\{0\}$ , so $p$ is a projection in ${\mathcal{A}}$ . Consider ${\mathcal{C}}:=f({\mathcal{B}})+{\mathbb{C}}p$ . Then ${\mathcal{C}}$ is a unital subalgebra of ${\mathcal{A}}$ , and since $\operatorname{Ann}({\mathcal{A}},{\mathcal{C}},\subseteq)\subseteq\operatorname{Ann}({\mathcal{A}},{\mathcal{B}},f)$ , $({\mathcal{A}},{\mathcal{C}},\subseteq)$ is weakly non-degenerate. By part (a), $p=I_{\mathcal{A}}$ . Noting that $p-a\in\operatorname{Ann}({\mathcal{A}},{\mathcal{B}},f)$ , we obtain $I_{\mathcal{A}}=p=a$ , as desired.

(c) Once again, we assume ${\mathcal{A}}\subseteq{\mathcal{B}}({\mathcal{H}})$ . Since $(f(u_{\lambda}))$ is a bounded increasing net of positive semi-definite operators, it converges in the strong operator topology. Let $p:=\textsc{sot}\lim f(u_{\lambda})$ . Then $p$ is a projection and for every $b\in{\mathcal{B}}$ ,

(2.2.2)

f(b)p=pf(b)=f(b).

Since $f(u_{\lambda})\leq a$ for each $\lambda$ , we obtain $p\leq a$ . Decomposing $a$ with respect to ${\mathcal{H}}=(I-p){\mathcal{H}}\oplus p{\mathcal{H}}$ we find $a=\begin{bmatrix}p^{\perp}ap^{\perp}&p^{\perp}ap\\ pap^{\perp}&pap\end{bmatrix}$ . As $a\geq p$ , we have $pap\geq p\geq 0$ . Since $\left\lVert pap\right\rVert\leq 1$ , we obtain $p=pap$ . Then

0\leq a-p+p^{\perp}=\begin{bmatrix}p^{\perp}ap^{\perp}+p^{\perp}&p^{\perp}ap\\ pap^{\perp}&pap-p\end{bmatrix}=\begin{bmatrix}p^{\perp}ap^{\perp}+p^{\perp}&p^{\perp}ap\\ pap^{\perp}&0\end{bmatrix}.

Lemma 2.2 gives $p^{\perp}ap=0$ . Therefore $a=\begin{bmatrix}p^{\perp}ap^{\perp}&0\\ 0&p\end{bmatrix}$ . It follows that for $b\in{\mathcal{B}}$ ,

af(b)=f(b)a=f(b).

An application of part (b) completes the proof.

(d) There is nothing to do when ${\mathcal{A}}$ is unital, so assume ${\mathcal{A}}$ has no unit. Let $(a,\lambda)\in\operatorname{Ann}(\tilde{\mathcal{A}},{\mathcal{B}},u_{\mathcal{A}}\circ f)$ . It suffices to show $\lambda=0$ , for once this is established, it follows that $a\in\operatorname{Ann}({\mathcal{A}},{\mathcal{B}},f)$ , whence $a=0$ . To show $\lambda=0$ , we argue by contradiction.

Suppose $\lambda\neq 0$ . Then by scaling, we may assume $\lambda=-1$ . Note that for $b\in{\mathcal{B}}$ , $u_{\mathcal{A}}(f(b))=(f(b),0)$ . Thus

(0,0)=(a,-1)(f(b),0)=(af(b)-f(b),0)=(f(b)a-f(b),0)=(f(b),0)(a,-1).

So for all $b\in{\mathcal{B}}$ , $af(b)=f(b)=f(b)a$ . Part (b) now gives $a=I_{\mathcal{A}}$ , contrary to assumption on ${\mathcal{A}}$ . Thus, $\lambda=0$ , completing the proof.

(e) Let $a=\tilde{f}(I_{\tilde{\mathcal{B}}})$ . Then $\left\lVert a\right\rVert\leq 1$ and $a\geq 0$ . For every $b\in{\mathcal{B}}$ , $au_{\mathcal{A}}(f(b))=u_{\mathcal{A}}(f(b))a=u_{\mathcal{A}}(f(b))$ . Applying part (d), then part (b), yields $a=I_{\tilde{\mathcal{A}}}$ . Therefore $(\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{f})$ is a unital inclusion and hence is weakly non-degenerate.

(f) If ${\mathcal{B}}$ is unital, so is ${\mathcal{A}}$ by part (a). Therefore, $\sigma=\sigma\circ u_{\mathcal{B}}=u_{\mathcal{A}}\circ f=f=\tilde{f}$ , and all is well.

Now assume ${\mathcal{B}}$ is not unital. We claim that $\tilde{f}(I_{\tilde{\mathcal{B}}})=\sigma(I_{\tilde{\mathcal{B}}})$ . For $b\in{\mathcal{B}}$ ,

\sigma(I_{\tilde{\mathcal{B}}})u_{\mathcal{A}}(f(b))=\sigma(I_{\tilde{\mathcal{B}}})\sigma(u_{\mathcal{B}}(b))=\sigma(u_{\mathcal{B}}(b))=u_{\mathcal{A}}(f(b));

likewise

u_{\mathcal{A}}(f(b))\sigma(I_{\tilde{\mathcal{B}}})=u_{\mathcal{A}}(f(b)).

Similarly, $\tilde{f}(I_{\tilde{\mathcal{B}}})u_{\mathcal{A}}(f(b))=u_{\mathcal{A}}(f(b))\tilde{f}(I_{\tilde{\mathcal{B}}})=u_{\mathcal{A}}(f(b))$ . Therefore,

\tilde{f}(I_{\tilde{\mathcal{B}}})-\sigma(I_{\tilde{\mathcal{B}}})\in\operatorname{Ann}(\tilde{\mathcal{A}},{\mathcal{B}},u_{\mathcal{A}}\circ f).

Part (d) gives $\tilde{f}(I_{\tilde{\mathcal{B}}})=\sigma(I_{\tilde{\mathcal{B}}})$ , so the claim holds.

For $(b,\lambda)\in\tilde{\mathcal{B}}$ ,

\sigma(b,\lambda)=\sigma(u_{\mathcal{B}}(b))+\lambda\tilde{f}(I_{\tilde{\mathcal{B}}})=u_{\mathcal{A}}(f(b))+\lambda\tilde{f}(I_{\tilde{\mathcal{B}}})=\tilde{f}(b,\lambda),

so $\sigma=\tilde{f}$ . ∎

We now show that the poor behavior for intermediate inclusions exhibited in Example 2.1 cannot occur when $({\mathcal{B}}_{3},{\mathcal{B}}_{1})$ is weakly non-degenerate: if ${\mathcal{B}}_{2}$ is intermediate to the weakly non-degenerate inclusion $({\mathcal{B}}_{3},{\mathcal{B}}_{1})$ , then $\tilde{\mathcal{B}}_{2}$ is intermediate to $(\tilde{\mathcal{B}}_{3},\tilde{\mathcal{B}}_{1})$ .

Corollary \the\numberby.

For $1\leq i<j\leq 3$ , let $({\mathcal{B}}_{j},{\mathcal{B}}_{i},f_{ji})$ be inclusions such that $f_{31}=f_{32}\circ f_{21}$ . If $({\mathcal{B}}_{3},{\mathcal{B}}_{1},f_{31})$ is weakly non-degenerate, then $\tilde{f}_{31}=\tilde{f}_{32}\circ\tilde{f}_{21}$ .

Proof.

Since $({\mathcal{B}}_{3},{\mathcal{B}}_{1},f_{31})$ is weakly non-degenerate, both $({\mathcal{B}}_{3},{\mathcal{B}}_{2},f_{32})$ and $({\mathcal{B}}_{2},{\mathcal{B}}_{1},f_{21})$ are weakly non-degenerate. Thus by Lemma 2.2(e),

\tilde{f}_{32}(\tilde{f}_{21}(I_{\tilde{\mathcal{B}}_{1}}))=\tilde{f}_{32}(I_{\tilde{\mathcal{B}}_{2}})=I_{\tilde{\mathcal{B}}_{3}}=\tilde{f}_{31}(I_{\tilde{\mathcal{B}}_{1}}).

Given $x\in\tilde{\mathcal{B}}_{1}$ , there exist $b_{1}\in{\mathcal{B}}_{1}$ and $\lambda\in{\mathbb{C}}$ so that $x=u_{{\mathcal{B}}_{1}}(b_{1})+\lambda I_{\tilde{\mathcal{B}}_{1}}$ . Using using (2.1.6),

	$\displaystyle\tilde{f}_{32}(\tilde{f}_{21}(x))$	$\displaystyle=\tilde{f}_{32}(\tilde{f}_{21}(u_{{\mathcal{B}}_{1}}(b_{1})))+\lambda I_{\tilde{\mathcal{B}}_{3}}=\tilde{f}_{32}(u_{{\mathcal{B}}_{2}}(f_{21}(b_{1})))+\lambda I_{\tilde{\mathcal{B}}_{3}}$
		$\displaystyle=u_{{\mathcal{B}}_{3}}(f_{32}((f_{21}(b_{1}))))+\lambda I_{\tilde{\mathcal{B}}_{3}}=u_{{\mathcal{B}}_{3}}(f_{31}(b_{1}))+\lambda I_{\tilde{\mathcal{B}}_{3}}$
		$\displaystyle=\tilde{f}_{31}(x),$

as desired. ∎

The following extends [PaulsenCoBoMaOpAl, Corollary 3.19] from unital inclusions to weakly non-degenerate inclusions.

Proposition \the\numberby.

Let $({\mathcal{A}}_{1},{\mathcal{B}},f_{1})$ and $({\mathcal{A}}_{2},{\mathcal{B}},f_{2})$ be weakly non-degenerate inclusions and suppose

\Phi:{\mathcal{A}}_{1}\rightarrow{\mathcal{A}}_{2}

is a contractive completely positive map such that $\Phi\circ f_{1}=f_{2}$ . The following statements hold.

(a)

The standard extension, $\tilde{\Phi}:\tilde{\mathcal{A}}_{1}\rightarrow\tilde{\mathcal{A}}_{2}$ , of $\Phi$ is a unital completely positive map.
(b)

For every $x\in{\mathcal{A}}_{1}$ and $h,k\in{\mathcal{B}}$ ,

$\Phi(f_{1}(h)xf_{1}(k))=f_{2}(h)\Phi(x)f_{2}(k).$

Proof.

For $i=1,2$ , parts (d) and (e) of Lemma 2.2 show that $(\tilde{\mathcal{A}}_{i},{\mathcal{B}},u_{{\mathcal{A}}_{i}}\circ f_{i})$ are weakly non-degenerate inclusions and $(\tilde{\mathcal{A}}_{i},\tilde{\mathcal{B}},\tilde{f}_{i})$ are unital inclusions.

(a) Observation 2.1 shows $\tilde{\Phi}$ is completely positive. To see $\tilde{\Phi}$ is a unital map, let $(h_{\lambda})$ be an approximate unit for ${\mathcal{B}}$ . We have $u_{{\mathcal{A}}_{1}}(f_{1}(h_{\lambda}))=\tilde{f}_{1}(u_{\mathcal{B}}(h_{\lambda}))\leq I_{\tilde{\mathcal{A}}_{1}}$ , so

u_{{\mathcal{A}}_{2}}(f_{2}(h_{\lambda}))=u_{{\mathcal{A}}_{2}}(\Phi(f_{1}(h(u_{\lambda}))))=\tilde{\Phi}(u_{{\mathcal{A}}_{1}}(f_{1}(h_{\lambda})))\leq\tilde{\Phi}(I_{\tilde{\mathcal{A}}_{1}}).

Applying Proposition 2.2(c) to $(\tilde{\mathcal{A}}_{2},{\mathcal{B}},u_{{\mathcal{A}}_{2}}\circ f_{2})$ , we obtain $\tilde{\Phi}(I_{\tilde{\mathcal{A}}_{1}})=I_{\tilde{\mathcal{A}}_{2}}$ , so $\tilde{\Phi}$ is a unital map.

(b) By [PaulsenCoBoMaOpAl, Corollary 3.19], $\tilde{\Phi}$ is a $\tilde{\mathcal{B}}$ -bimodule map, that is, for $\tilde{h},\tilde{k}\in\tilde{\mathcal{B}}$ and $\tilde{x}\in\tilde{\mathcal{A}}_{1}$ ,

\tilde{\Phi}(\tilde{f}_{1}(\tilde{h})\tilde{x}\tilde{f}_{1}(\tilde{k}))=\tilde{f}_{2}(\tilde{h})\tilde{\Phi}(\tilde{x})\tilde{f}_{2}(\tilde{k}).

Thus for $h,k\in{\mathcal{B}}$ and $x\in{\mathcal{A}}_{1}$ , (and using (2.1.4))

	$\displaystyle(u_{{\mathcal{A}}_{2}}\circ\Phi)(f_{1}(h)xf_{1}(k)))$	$\displaystyle=(\tilde{\Phi}\circ u_{{\mathcal{A}}_{1}})(f_{1}(h)xf_{1}(k))$
		$\displaystyle=\tilde{\Phi}(\tilde{f}_{1}(u_{\mathcal{B}}(h))u_{{\mathcal{A}}_{1}}(x)\tilde{f}_{1}(u_{\mathcal{B}}(k))))$
		$\displaystyle=\tilde{f}_{2}(u_{\mathcal{B}}(h))\tilde{\Phi}(u_{{\mathcal{A}}_{1}}(x))\tilde{f}_{2}(u_{{\mathcal{B}}}(k))$
		$\displaystyle=(u_{{\mathcal{A}}_{2}}\circ f_{2})(h)\tilde{\Phi}(u_{{\mathcal{A}}_{1}}(x))(u_{{\mathcal{A}}_{2}}\circ f_{2})(k)$
		$\displaystyle=(u_{{\mathcal{A}}_{2}}\circ f_{2})(h)(u_{{\mathcal{A}}_{2}}\circ\Phi)(x)(u_{{\mathcal{A}}_{2}}\circ f_{2})(k).$

As $u_{{\mathcal{A}}_{2}}$ is a $*$ -monomorphism, the proposition follows. ∎

2.3. Pseudo-Expectations

We shall require pseudo-expectations, and we give a brief discussion of them here. Hamana [HamanaInEnC*Al] showed that given a unital $C^{*}$ -algebra ${\mathcal{A}}$ there is a $C^{*}$ -algebra $I({\mathcal{A}})$ and a one-to-one unital $*$ -homomorphism $\iota:{\mathcal{A}}\rightarrow I(A)$ such that:

•

$I({\mathcal{A}})$ is an injective object in the category of operator systems and unital completely positive maps; and
•

the only unital completely positive map $\phi:I({\mathcal{A}})\rightarrow I({\mathcal{A}})$ satisfying $\phi\circ\iota=\iota$ is the identity mapping on $I({\mathcal{A}})$ .

Hamana calls the pair $(I({\mathcal{A}}),\iota)$ an injective envelope of ${\mathcal{A}}$ . The injective envelope of ${\mathcal{A}}$ is monotone closed and has the following uniqueness property: if $({\mathcal{I}}_{1},\iota_{1})$ and $({\mathcal{I}}_{2},\iota_{2})$ are injective envelopes for ${\mathcal{A}}$ , there exists a unique $*$ -isomorphism $\alpha:{\mathcal{I}}_{1}\rightarrow{\mathcal{I}}_{2}$ such that $\alpha\circ\iota_{1}=\iota_{2}$ .

Remark \the\numberby. Let ${\mathfrak{C}}$ be the category of unital, abelian $C^{*}$ -algebras and unital $*$ -homomorphisms. Hadwin and Paulsen note that an object ${\mathcal{A}}$ in ${\mathfrak{C}}$ is injective if and only if ${\mathcal{A}}$ is injective in the category of operator systems and completely positive unital maps, see [HadwinPaulsenInPrAnTo, Theorem 2.4].

When the $C^{*}$ -algebra ${\mathcal{A}}$ is non-unital, an injective envelope for ${\mathcal{A}}$ is defined to be an injective envelope $(I(\tilde{\mathcal{A}}),\iota)$ for $\tilde{\mathcal{A}}$ .

Remark \the\numberby.

To simplify notation, we will often say $(I({\mathcal{A}}),\iota)$ is an injective envelope of ${\mathcal{A}}$ regardless of whether ${\mathcal{A}}$ has a unit: when ${\mathcal{A}}$ is not unital, it is to be understood that $(I({\mathcal{A}}),\iota)$ means $(I(\tilde{A}),\iota)$ , where $(I(\tilde{\mathcal{A}}),\iota)$ is an injective envelope for $\tilde{A}$ . When there is a need for clarity, we will sometimes write $(I(\tilde{\mathcal{A}}),\iota)$ for an injective envelope of ${\mathcal{A}}$ .

We need the following fact about injective envelopes; it is due to Hamana. Let $(I({\mathcal{A}}),\iota)$ be an injective envelope for the $C^{*}$ -algebra ${\mathcal{A}}$ , let $J\unlhd{\mathcal{A}}$ be a closed ideal of ${\mathcal{A}}$ , let $(e_{\lambda})$ be an approximate unit for ${\mathcal{A}}$ , and let $P=\sup_{I({\mathcal{A}})_{s.a.}}\iota(e_{\lambda})$ , where $I({\mathcal{A}})_{s.a.}$ is the partially ordered set of self-adjoint elements of $I({\mathcal{A}})$ . By [HamanaCeReMoCoCStAl, Lemma 1.1, parts (i) and (iii)], $P$ is a central projection of $I({\mathcal{A}})$ and $(PI({\mathcal{A}}),\iota|_{J})$ is an injective envelope for $J$ . We will call $P$ the support projection for $J$ .

The proof of the next lemma follows from the uniqueness property of injective envelopes.

Lemma \the\numberby (c.f. [PittsStReInI, Proposition 1.11]).

Suppose ${\mathcal{A}}$ is an abelian $C^{*}$ -algebra and let $(I({\mathcal{A}}),\iota)$ be an injective envelope for ${\mathcal{A}}$ . For $i=1,2$ , let $J_{i}$ be closed ideals of ${\mathcal{A}}$ with support projections $Q_{i}$ . If $\phi:J_{1}\rightarrow J_{2}$ is a $*$ -isomorphism, then there is a unique $*$ -isomorphism $\overline{\phi}:Q_{1}I({\mathcal{A}})\rightarrow Q_{2}I({\mathcal{A}})$ such that $\overline{\phi}\circ\iota|_{J_{1}}=\iota\circ\phi$ .

The following is an interesting example of a weakly non-degenerate inclusion.

Lemma \the\numberby.

Suppose $(I(\tilde{\mathcal{B}}),\iota)$ is an injective envelope for the $C^{*}$ -algebra ${\mathcal{B}}$ . Then $(I(\tilde{\mathcal{B}}),{\mathcal{B}},\iota\circ u_{\mathcal{B}})$ is a weakly non-degenerate inclusion.

Proof.

We suppress $\iota$ and $u_{\mathcal{B}}$ , so that ${\mathcal{B}}\subseteq\tilde{\mathcal{B}}\subseteq I(\tilde{\mathcal{B}})$ . Let $(u_{\lambda})$ be an approximate unit for ${\mathcal{B}}$ , and let $p:=\sup_{I(\tilde{\mathcal{B}})}u_{\lambda}$ , that is, $p$ is the least upper bound of $\{u_{\lambda}\}$ taken in the partially ordered set $I(\tilde{\mathcal{B}})_{sa}$ . As ${\mathcal{B}}$ is a hereditary subalgebra of $\tilde{\mathcal{B}}$ , [HamanaCeReMoCoCStAl, Lemma 1.1(i)] shows $p$ is a projection and ${\mathfrak{P}}:=pI(\tilde{\mathcal{B}})p$ is an injective envelope for ${\mathcal{B}}$ .

We claim that $p=I_{I(\tilde{\mathcal{B}})}$ . As ${\mathcal{B}}\subseteq{\mathfrak{P}}\subseteq I(\tilde{\mathcal{B}})$ , $p=\sup_{I(\tilde{\mathcal{B}})}u_{\lambda}\leq\sup_{\mathfrak{P}}u_{\lambda}\leq p$ so that

\sup_{I(\tilde{\mathcal{B}})}u_{\lambda}=\sup_{\mathfrak{P}}u_{\lambda}.

By uniqueness of injective envelopes, there is a $*$ -isomorphism $\theta:{\mathfrak{P}}\rightarrow I({\mathcal{B}})$ with $\theta|_{\tilde{\mathcal{B}}}={\operatorname{id}}|_{\tilde{\mathcal{B}}}$ . Then

I_{I(\tilde{\mathcal{B}})}=\theta(p)=\theta(\sup_{{\mathfrak{P}}}u_{\lambda})=\sup_{I(\tilde{\mathcal{B}})}\theta(u_{\lambda})=\sup_{I(\tilde{\mathcal{B}})}u_{\lambda}=p.

Thus the claim holds.

For $a\in\operatorname{Ann}(I(\tilde{\mathcal{B}}),{\mathcal{B}})$ , [HamanaReEmCStAlMoCoCStAl, Corollary 4.10] shows

a^{*}a=a^{*}pa=a^{*}(\sup_{I(\tilde{\mathcal{B}})}u_{\lambda})a=\sup_{I(\tilde{\mathcal{B}})}a^{*}u_{\lambda}a=0.

∎

Definition \the\numberby (c.f. [PittsStReInI]).

Let $({\mathcal{A}},{\mathcal{B}},f)$ be an inclusion and let $(I({\mathcal{B}}),\iota)$ be an injective envelope for ${\mathcal{B}}$ . A pseudo-expectation for $({\mathcal{A}},{\mathcal{B}},f)$ relative to $(I({\mathcal{B}}),\iota)$ is a contractive and completely positive linear map $E:{\mathcal{A}}\rightarrow I({\mathcal{B}})$ such that $E\circ f=\iota\circ u_{\mathcal{B}}$ .

When ${\mathcal{B}}$ is identified with $f({\mathcal{B}})\subseteq{\mathcal{A}}$ and $u_{\mathcal{B}}({\mathcal{B}})\subseteq\tilde{\mathcal{B}}$ , we will write $E|_{\mathcal{B}}=\iota|_{\mathcal{B}}$ instead of $E\circ f=\iota\circ u_{\mathcal{B}}$ . When these identifications are made, we will simply say $E$ is a pseudo-expectation for $({\mathcal{A}},{\mathcal{B}})$ .

Notation \the\numberby. For an inclusion $({\mathcal{A}},{\mathcal{B}},f)$ , $\operatorname{\operatorname{PsExp}}({\mathcal{A}},{\mathcal{B}},f)$ will denote the collection of all pseudo-expectations for $({\mathcal{A}},{\mathcal{B}},f)$ (relative to a fixed injective envelope $(I({\mathcal{B}}),\iota)$ for ${\mathcal{B}}$ ). As usual, when $f({\mathcal{B}})$ is identified with ${\mathcal{B}}$ , we write $\operatorname{\operatorname{PsExp}}({\mathcal{A}},{\mathcal{B}})$ instead of $\operatorname{\operatorname{PsExp}}({\mathcal{A}},{\mathcal{B}},f)$ .

In general, there are many pseudo-expectations. In some cases however, there is a unique pseudo expectation, and this property plays an essential role in our study.

Definition \the\numberby.

We will say that the inclusion $({\mathcal{A}},{\mathcal{B}},f)$ has the unique pseudo-expectation property if $\operatorname{\operatorname{PsExp}}({\mathcal{A}},{\mathcal{B}},f)$ is a singleton set. When $({\mathcal{A}},{\mathcal{B}},f)$ has the unique pseudo-expectation property, and the (unique) pseudo-expectation is faithful, we say $({\mathcal{A}},{\mathcal{B}},f)$ has the faithful unique pseudo-expectation property.

Note that by Remark 2.3, if ${\mathcal{A}}$ is abelian, and $({\mathcal{A}},{\mathcal{B}},f)$ has the unique pseudo-expectation property, the pseudo-expectation is multiplicative.

That $\operatorname{\operatorname{PsExp}}({\mathcal{A}},{\mathcal{B}})$ is not empty follows from the following general fact.

Lemma \the\numberby.

Suppose ${\mathfrak{I}}$ is an injective $C^{*}$ -algebra and $({\mathcal{A}},{\mathcal{B}})$ is an inclusion. If $\phi:{\mathcal{B}}\rightarrow{\mathfrak{I}}$ is a contractive and completely positive map, then $\phi$ extends to a contractive and completely positive map $\Phi:{\mathcal{A}}\rightarrow{\mathfrak{I}}$ .

Proof.

We may assume ${\mathcal{A}}\subseteq{\mathcal{B}}({\mathcal{H}})$ satisfies $\overline{{\mathcal{A}}{\mathcal{H}}}={\mathcal{H}}$ for some Hilbert space ${\mathcal{H}}$ . Thus $\tilde{\mathcal{A}}={\mathcal{A}}+{\mathbb{C}}I_{\mathcal{H}}$ . Let $(u_{\lambda})$ be an approximate unit for ${\mathcal{B}}$ and put $Q:=\textsc{sot}\lim u_{\lambda}$ . Then $(Q\tilde{\mathcal{A}}Q,{\mathcal{B}}+{\mathbb{C}}Q)$ is a unital inclusion. As $\tilde{\mathcal{B}}\simeq{\mathcal{B}}+{\mathbb{C}}Q$ , we may apply [BrownOzawaC*AlFiDiAp, Proposition 2.2.1] to obtain a unital and completely positive map $\tilde{\phi}:{\mathcal{B}}+{\mathbb{C}}Q\rightarrow{\mathfrak{I}}$ which extends $\phi$ . Injectivity of ${\mathfrak{I}}$ shows that we may extend $\tilde{\phi}$ to a unital, completely positive map $\Delta:Q\tilde{\mathcal{A}}Q\rightarrow I({\mathcal{B}})$ . Now observe that the map $\Phi:{\mathcal{A}}\rightarrow{\mathfrak{I}}$ given by $\Phi(a)=\Delta(QaQ)$ is a contractive and completely positive map extending $\phi$ . ∎

The next several results explore properties of pseudo-expectations. We begin with the relationship between pseudo-expectations for $({\mathcal{A}},{\mathcal{B}},f)$ and pseudo-expectations for $(\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{f})$ .

Lemma \the\numberby.

Let $({\mathcal{A}},{\mathcal{B}},f)$ be an inclusion, and let $(I({\mathcal{B}}),\iota)$ be an injective envelope for ${\mathcal{B}}$ . The following statements hold.

(a)

Let $(h_{\lambda})$ be an approximate unit for ${\mathcal{B}}$ . Suppose $x\in\tilde{\mathcal{A}}$ satisfies $0\leq x$ , $\left\lVert x\right\rVert\leq 1$ , and $u_{\mathcal{A}}(f(h_{\lambda}))\leq x$ for every $\lambda$ . If $E$ is a pseudo-expectation for $({\mathcal{A}},{\mathcal{B}},f)$ , then $\tilde{E}(x)=I_{I({\mathcal{B}})}$ .
(b)

If $E$ is a pseudo-expectation for $({\mathcal{A}},{\mathcal{B}},f)$ , then

$\tilde{E}(I_{\tilde{\mathcal{A}}})=I_{I({\mathcal{B}})}=\tilde{E}(\tilde{f}(I_{\tilde{\mathcal{B}}}))$

and $\tilde{E}$ is a pseudo-expectation for $(\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{f})$ .
(c)
Suppose $\Phi:\tilde{\mathcal{A}}\rightarrow I({\mathcal{B}})$ is a pseudo-expectation for $(\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{f})$ , and let $E:=\Phi\circ u_{\mathcal{A}}$ . Then:
1. (i)
  
  $E$ is a pseudo-expectation for $({\mathcal{A}},{\mathcal{B}},f)$ ,
2. (ii)
  
  $\Phi(I_{\tilde{\mathcal{A}}})=\Phi(\tilde{f}(I_{\tilde{\mathcal{B}}}))=I_{I({\mathcal{B}})}$ ; and
3. (iii)
  
  $\tilde{E}=\Phi$ .

Proof.

(a) We have already noted in Observation 2.1 that $\tilde{E}$ is contractive and completely positive. Let $z=\tilde{E}(x)$ . Then

\iota(u_{\mathcal{B}}(h_{\lambda}))=E(f(h_{\lambda}))=\tilde{E}(u_{\mathcal{A}}(f(h_{\lambda})))\leq\tilde{E}(x)=z.

Since $(I({\mathcal{B}}),{\mathcal{B}},\iota\circ u_{\mathcal{B}})$ is weakly non-degenerate (Lemma 2.3), Proposition 2.2(c) shows $z=I_{I({\mathcal{B}})}$ .

(b) Taking $x=I_{\tilde{\mathcal{A}}}$ in part (a) shows $\tilde{E}$ is a unital completely positive map and hence is contractive. It remains to show $\tilde{E}\circ\tilde{f}=\iota$ . To do this, note that part (a) applied with $x=I_{\tilde{\mathcal{B}}}$ gives $I_{I({\mathcal{B}})}=\tilde{E}(\tilde{f}(I_{\tilde{\mathcal{B}}}))$ . Since $\tilde{\mathcal{B}}={\mathbb{C}}I_{\tilde{\mathcal{B}}}+\tilde{f}(u_{\mathcal{B}}({\mathcal{B}}))$ and $\tilde{E}\circ\tilde{f}\circ u_{\mathcal{B}}=E\circ f=\iota\circ u_{\mathcal{B}}$ , it follows that $\tilde{E}\circ\tilde{f}=\iota$ , so part (b) holds.

(c) Since $\Phi$ is contractive and completely positive, so is $E$ . As $E\circ f=\Phi\circ u_{\mathcal{A}}\circ f=\Phi\circ\tilde{f}\circ u_{\mathcal{B}}=\iota\circ u_{\mathcal{B}}$ , $E$ is a pseudo-expectation for $({\mathcal{A}},{\mathcal{B}},f)$ .

Part (a) shows $\Phi$ is a unital map. Since $\tilde{\mathcal{A}}={\mathbb{C}}I_{\tilde{\mathcal{A}}}+u_{\mathcal{A}}({\mathcal{A}})$ , it follows that $\Phi=\tilde{E}$ . Part (b) now implies item (ii), which completes the proof of part (c). ∎

It is worth noting a bijection between $\operatorname{\operatorname{PsExp}}(\tilde{\mathcal{A}},\tilde{\mathcal{B}})$ and $\operatorname{\operatorname{PsExp}}({\mathcal{A}},{\mathcal{B}})$ .

Corollary \the\numberby.

The map,

	$\displaystyle\operatorname{\operatorname{PsExp}}(\tilde{\mathcal{A}},\tilde{\mathcal{B}})\ni\Phi$	$\displaystyle\mapsto\Phi\|_{\mathcal{A}}\in\operatorname{\operatorname{PsExp}}({\mathcal{A}},{\mathcal{B}})$
is a bijection with inverse
	$\displaystyle\operatorname{\operatorname{PsExp}}({\mathcal{A}},{\mathcal{B}})\ni E$	$\displaystyle\mapsto\tilde{E}\in\operatorname{\operatorname{PsExp}}(\tilde{\mathcal{A}},\tilde{\mathcal{B}}).$

Proof.

Apply Lemma 2.3. ∎

Combining Proposition 2.2 and Lemma 2.3 we obtain the following.

Proposition \the\numberby.

Suppose $({\mathcal{A}},{\mathcal{B}})$ is a weakly non-degenerate inclusion and let $(I({\mathcal{B}}),\iota)$ be an injective envelope for ${\mathcal{B}}$ . If $E:{\mathcal{A}}\rightarrow I({\mathcal{B}})$ is a pseudo-expectation, then for every $x\in{\mathcal{A}}$ and $h,k\in{\mathcal{B}}$ ,

E(hxk)=\iota(h)E(x)\iota(k).

We do not know whether the faithfulness of a pseudo-expectation $E$ for an inclusion $({\mathcal{A}},{\mathcal{B}})$ implies $\tilde{E}$ is a faithful pseudo-expectation for $(\tilde{\mathcal{A}},\tilde{\mathcal{B}})$ , but we suspect it is not true in general. Our next few results concern the relationship between faithfulness for a pseudo-expectation $E$ and faithfulness of $\tilde{E}$ .

Lemma \the\numberby.

Let $E$ be a pseudo-expectation for $({\mathcal{A}},{\mathcal{B}})$ . If $\tilde{E}$ is faithful, then $({\mathcal{A}},{\mathcal{B}})$ is weakly non-degenerate.

Proof.

By Lemma 2.3(c) $\tilde{E}(I_{\tilde{\mathcal{A}}}-I_{\tilde{\mathcal{B}}})=0$ , so faithfulness of $\tilde{E}$ gives $I_{\tilde{\mathcal{A}}}=I_{\tilde{\mathcal{B}}}$ . Thus $(\tilde{\mathcal{A}},\tilde{\mathcal{B}})$ is a unital inclusion and hence is weakly non-degenerate.

When $x\in\operatorname{Ann}({\mathcal{A}},{\mathcal{B}})$ and $b\in{\mathcal{B}}$ , Proposition 2.3 implies

E(x^{*}x)\iota(b)=E(x^{*}xb)=0=E(bx^{*}x)=\iota(b)E(x^{*}x).

Therefore $E(x^{*}x)\in\operatorname{Ann}(I({\mathcal{B}}),{\mathcal{B}},\iota\circ u_{\mathcal{B}})$ . By Lemma 2.3, $E(x^{*}x)=0$ , so faithfulness of $E$ gives $x=0$ . Thus $({\mathcal{A}},{\mathcal{B}})$ is weakly non-degenerate. ∎

Proposition \the\numberby.

Let $({\mathcal{A}},{\mathcal{B}})$ be an inclusion and suppose $E$ is a pseudo-expectation for $({\mathcal{A}},{\mathcal{B}})$ . Then $\tilde{E}$ is a faithful pseudo-expectation for $(\tilde{\mathcal{A}},\tilde{\mathcal{B}})$ if and only if $({\mathcal{A}},{\mathcal{B}})$ is weakly non-degenerate and $E$ is faithful.

Proof.

( $\Rightarrow$ ) Suppose $\tilde{E}$ is faithful. Clearly $E$ is faithful, and Lemma 2.3 shows $({\mathcal{A}},{\mathcal{B}})$ is weakly non-degenerate.

( $\Leftarrow$ ) Suppose $E$ is faithful and $({\mathcal{A}},{\mathcal{B}})$ is weakly non-degenerate. Since $\tilde{E}=E$ when ${\mathcal{A}}$ is unital, we may as well assume ${\mathcal{A}}$ is not unital.

Suppose $(x,\lambda)\in\tilde{\mathcal{A}}$ and $\tilde{E}((x,\lambda)^{*}(x,\lambda))=0$ . Then

	$\displaystyle 0$	$\displaystyle=E(x^{}x)+\overline{\lambda}E(x)+\lambda E(x)^{}+\|\lambda\|^{2}I_{I({\mathcal{B}})}$
		$\displaystyle\geq E(x)^{}E(x)+\overline{\lambda}E(x)+\lambda E(x)^{}+\|\lambda\|^{2}I_{I({\mathcal{B}})}$
		$\displaystyle=(E(x)+\lambda I_{I({\mathcal{B}})})^{*}(E(x)+\lambda I_{I({\mathcal{B}})}),$

whence

E(x)=-\lambda I_{I({\mathcal{B}})}.

We claim that $\lambda=0$ .

We argue by contradiction. Suppose $\lambda\neq 0$ . By scaling, we may assume $\lambda=-1$ , so $E(x)=I_{I({\mathcal{B}})}$ . Then

E(x^{*}x)=I_{I({\mathcal{B}})}

because $0=\tilde{E}((x,-1)^{*}(x,-1))$ .

Given $b\in{\mathcal{B}}$ , Proposition 2.3 shows

E((xb-b)^{*}(xb-b))=\iota(b)^{*}E(x^{*}x)\iota(b)-\iota(b)^{*}E(x)\iota(b)-\iota(b)^{*}E(x^{*})\iota(b)+\iota(b^{*}b)=0.

Faithfulness of $E$ gives $xb=b$ . Similar considerations yield $bx=b$ . Lemma 2.2(b) shows that $x$ is the identity for ${\mathcal{A}}$ , contradicting the assumption that ${\mathcal{A}}$ is not unital. Hence $\lambda=0$ .

Thus $(x,\lambda)=(x,0)$ and using faithfulness of $E$ once again, we find $x=0$ . Therefore $\tilde{E}$ is faithful. ∎

For a unital inclusion $({\mathcal{A}},{\mathcal{B}})$ , [PittsZarikianUnPsExC*In, Corollary 3.14] shows that the faithful unique pseudo-expectation property implies ${\mathcal{B}}^{c}$ is abelian. We now note that this useful structural fact holds when $({\mathcal{A}},{\mathcal{B}})$ is not assumed unital.

Proposition \the\numberby.

If the inclusion $({\mathcal{A}},{\mathcal{B}})$ has the faithful unique pseudo-expectation property, then $B^{c}$ is abelian.

Proof.

The proof is an adaptation of the arguments establishing [PittsZarikianUnPsExC*In, Theorem 3.12 and Corollary 3.14].

We may suppose that for some Hilbert space ${\mathcal{H}}$ , ${\mathcal{A}}\subseteq{\mathcal{B}}({\mathcal{H}})$ and $\overline{{\mathcal{A}}{\mathcal{H}}}={\mathcal{H}}$ . Then $\tilde{\mathcal{A}}={\mathcal{A}}+{\mathbb{C}}I$ . Let $E:{\mathcal{A}}\rightarrow I({\mathcal{B}})$ be the faithful unique pseudo-expectation, and put

\Phi:=\tilde{E}.

While $\Phi$ is the unique pseudo-expectation for $(\tilde{\mathcal{A}},\tilde{\mathcal{B}})$ , we do not know whether it is faithful, so we cannot apply [PittsZarikianUnPsExC*In, Corollary 3.14]. Instead we argue by contradiction.

Assume ${\mathcal{B}}^{\prime}\cap{\mathcal{A}}$ is not abelian. Then there exists $x\in{\mathcal{B}}^{\prime}\cap{\mathcal{A}}$ so that $x^{2}=0$ and $\left\lVert x\right\rVert=1$ . Proceed as in the proof of [PittsZarikianUnPsExC*In, Theorem 3.12] to obtain a one-parameter family of unital completely positive maps $\theta_{\lambda}:{\mathcal{B}}({\mathcal{H}})\rightarrow{\mathcal{B}}({\mathcal{H}})$ ( $\lambda\in[0,1])$ ) such that

\theta_{\lambda}|_{\mathcal{B}}={\operatorname{id}}|_{\mathcal{B}},\quad\theta_{\lambda}(x^{*}x)=\lambda(x^{*}x+xx^{*}),\quad\text{and}\quad\theta_{\lambda}(xx^{*})=(1-\lambda)(x^{*}x+xx^{*}).

Let

{\mathcal{S}}:={\mathcal{B}}+{\mathbb{C}}x^{*}x+{\mathbb{C}}xx^{*}+{\mathbb{C}}I

and note that ${\mathcal{S}}\subseteq\tilde{\mathcal{A}}$ is an operator system such that $\theta_{\lambda}({\mathcal{S}})\subseteq{\mathcal{S}}$ . Let $\Phi_{\lambda}^{(0)}=\Phi\circ\theta_{\lambda}$ . Then $\Phi_{\lambda}^{(0)}$ is a unital completely positive map from ${\mathcal{S}}$ into $I({\mathcal{B}})$ . Using the injectivity of $I({\mathcal{B}})$ , we may extend $\Phi_{\lambda}^{(0)}$ to a unital completely positive map $\Phi_{\lambda}:\tilde{\mathcal{A}}\rightarrow I({\mathcal{B}})$ . Putting $E_{\lambda}:=\Phi_{\lambda}|_{\mathcal{A}}$ , we find $E_{\lambda}$ is a pseudo-expectation for $({\mathcal{A}},{\mathcal{B}})$ .

As $x^{*}x\in{\mathcal{S}}$ , for $\lambda\neq 0$ , the faithfulness of $E$ gives

E_{\lambda}(x^{*}x)=\Phi(\theta_{\lambda}(x^{*}x))=\lambda E(x^{*}x+xx^{*})\neq 0.

Therefore, when $\lambda,\mu\in(0,1]$ and $\lambda\neq\mu$ , $E_{\lambda}$ and $E_{\mu}$ are distinct pseudo-expectations for $({\mathcal{A}},{\mathcal{B}})$ . This contradicts the hypothesis that $({\mathcal{A}},{\mathcal{B}})$ has a unique pseudo-expectation, and completes the proof. ∎

Our next goal is to show that when $({\mathcal{C}},{\mathcal{D}})$ is an inclusion and ${\mathcal{D}}$ is abelian, then $({\mathcal{C}},{\mathcal{D}})$ has the faithful unique pseudo-expectation property if and only if $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ does. We begin with the commutative case.

Lemma \the\numberby.

Suppose $({\mathcal{A}},{\mathcal{B}},f)$ is an inclusion with ${\mathcal{A}}$ abelian. If $({\mathcal{A}},{\mathcal{B}},f)$ has the faithful unique pseudo-expectation property, then $(\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{f})$ has the faithful unique pseudo-expectation property.

Proof.

Let $E$ be the pseudo-expectation for $({\mathcal{A}},{\mathcal{B}},f)$ . Then $\tilde{E}$ is the unique pseudo-expectation for $(\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{f})$ and $\tilde{E}(I_{\tilde{\mathcal{A}}}-\tilde{f}(I_{\tilde{\mathcal{B}}}))=0$ (Lemma 2.3).

We wish to show $(\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{f})$ is a unital inclusion. Let $p:=I_{\tilde{\mathcal{A}}}-\tilde{f}(I_{\tilde{\mathcal{B}}})$ . Then $p$ is a projection in $\tilde{\mathcal{A}}$ . Suppose $0\leq x\in{\mathcal{A}}$ is such that $u_{\mathcal{A}}(x)\in p\tilde{\mathcal{A}}$ . Then

0\leq E(x)=\tilde{E}(u_{\mathcal{A}}(x))=\tilde{E}(pu_{\mathcal{A}}(x)p)\leq\left\lVert x\right\rVert\tilde{E}(p)=0,

so $E(x)=0$ . Faithfulness of $E$ yields $x=0$ . Since any $C^{*}$ -algebra is the span of its positive elements, $u_{\mathcal{A}}({\mathcal{A}})\cap p\tilde{\mathcal{A}}=\{0\}$ . Thus $p\tilde{\mathcal{A}}$ is an ideal of $\tilde{\mathcal{A}}$ having trivial intersection with $u_{\mathcal{A}}({\mathcal{A}})$ . Since $u_{\mathcal{A}}({\mathcal{A}})$ is an essential ideal of $\tilde{\mathcal{A}}$ , we conclude that $p\tilde{\mathcal{A}}=\{0\}$ , whence $p=0$ . Thus, $(\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{f})$ is a unital inclusion having the unique pseudo-expectation property.

By [PittsZarikianUnPsExC*In, Corollary 3.21], $\tilde{E}$ is a $*$ -homomorphism. Since $E$ is faithful, $\ker\tilde{E}$ is an ideal of $\tilde{\mathcal{A}}$ having trivial intersection with $u_{\mathcal{A}}({\mathcal{A}})$ . Using the fact that $u_{\mathcal{A}}({\mathcal{A}})$ is an essential ideal in $\tilde{\mathcal{A}}$ once again, we conclude that $\ker\tilde{E}=\{0\}$ , that is, $\tilde{E}$ is faithful. ∎

Theorem \the\numberby.

Suppose $({\mathcal{C}},{\mathcal{D}},f)$ is an inclusion with ${\mathcal{D}}$ abelian. Then $({\mathcal{C}},{\mathcal{D}},f)$ has the faithful unique pseudo-expectation property if and only if $(\tilde{\mathcal{C}},\tilde{\mathcal{D}},\tilde{f})$ has the faithful unique pseudo-expectation property.

Proof.

( $\Leftarrow$ ) Let $\Phi\in\operatorname{\operatorname{PsExp}}(\tilde{\mathcal{C}},\tilde{\mathcal{D}},\tilde{f})$ be the unique (and faithful) pseudo-expectation. Lemma 2.3 shows $\Phi\circ u_{\mathcal{C}}$ is the unique, and necessarily faithful, pseudo-expectation for $({\mathcal{C}},{\mathcal{D}},f)$ .

( $\Rightarrow$ ) Now suppose $({\mathcal{C}},{\mathcal{D}},f)$ has the faithful unique pseudo-expectation property and let $E$ be the pseudo-expectation. For notational purposes, let

g:=u_{\mathcal{C}}\circ f,\quad{\mathcal{B}}:=\operatorname{\textsc{rCom}}({\mathcal{C}},{\mathcal{D}},f),\quad\text{and}\quad{\mathcal{B}}^{+}:=\operatorname{\textsc{rCom}}(\tilde{\mathcal{C}},{\mathcal{D}},g),

that is,

	$\displaystyle{\mathcal{B}}$	$\displaystyle:=\{x\in{\mathcal{C}}:f(d)x=xf(d)\text{ for all }d\in{\mathcal{D}}\}$
and
	$\displaystyle{\mathcal{B}}^{+}$	$\displaystyle:=\{x\in\tilde{\mathcal{C}}:g(d)x=xg(d)\text{ for all }d\in{\mathcal{D}}\}.$

By Proposition 2.3, ${\mathcal{B}}$ is abelian; hence ${\mathcal{B}}^{+}$ is also abelian. As ${\mathcal{D}}$ is abelian, we obtain the two inclusions,

({\mathcal{B}},{\mathcal{D}},f)\quad\text{and}\quad({\mathcal{B}}^{+},{\mathcal{D}},g).

We establish faithfulness of $\tilde{E}:\tilde{\mathcal{C}}\rightarrow I({\mathcal{D}})$ by considering two cases: ${\mathcal{B}}$ is unital; and ${\mathcal{B}}$ is not unital.

Suppose first ${\mathcal{B}}$ is unital. By the definition of pseudo-expectation, $E\circ f=\iota\circ u_{\mathcal{D}}$ . Therefore, $\Delta:=E|_{\mathcal{B}}$ is a faithful pseudo-expectation for $({\mathcal{B}},{\mathcal{D}},f)$ . Lemma 2.3(b) shows $\tilde{\Delta}$ is a pseudo-expectation for $(\tilde{\mathcal{B}},\tilde{\mathcal{D}},\tilde{f})$ . But $\tilde{\mathcal{B}}={\mathcal{B}}$ by hypothesis, so $\tilde{\Delta}$ is a faithful pseudo-expectation for $(\tilde{\mathcal{B}},\tilde{\mathcal{D}},\tilde{f})$ . By Lemma 2.3, $({\mathcal{B}},{\mathcal{D}},f)$ is weakly non-degenerate. Recalling that $\operatorname{Ann}({\mathcal{C}},{\mathcal{D}},f)\subseteq{\mathcal{B}}$ , we find $\operatorname{Ann}({\mathcal{C}},{\mathcal{D}},f)=\operatorname{Ann}({\mathcal{B}},{\mathcal{D}},f)$ ; therefore $({\mathcal{C}},{\mathcal{D}},f)$ is weakly non-degenerate. Proposition 2.3 now shows that $\tilde{E}$ is a faithful pseudo-expectation for $(\tilde{\mathcal{C}},\tilde{\mathcal{D}},\tilde{f})$ . Thus $(\tilde{\mathcal{C}},\tilde{\mathcal{D}},\tilde{f})$ has the faithful unique pseudo-expectation property (Corollary 2.3).

Now suppose ${\mathcal{B}}$ is not unital. By definition of ${\mathcal{B}}$ , ${\mathcal{C}}$ cannot be unital. Thus $\tilde{\mathcal{C}}={\mathcal{C}}^{\dagger}$ , $\tilde{\mathcal{B}}={\mathcal{B}}^{\dagger}$ , and for $d\in{\mathcal{D}}$ and $(x,\lambda)\in\tilde{\mathcal{C}}$ ,

g(d)=(f(d),0)\quad\text{and}\quad\tilde{E}(x,\lambda)=E(x)+\lambda I_{I({\mathcal{B}})}.

Then ${\mathcal{B}}^{+}=\{(b,\lambda)\in{\mathcal{C}}^{\dagger}:b\in{\mathcal{B}}\}={\mathcal{B}}^{\dagger}$ . Therefore,

(\tilde{\mathcal{B}},{\mathcal{D}},u_{\mathcal{B}}\circ f)=({\mathcal{B}}^{+},{\mathcal{D}},g)\quad\text{and hence}\quad(\tilde{\mathcal{B}},\tilde{\mathcal{D}},\tilde{f})=({\mathcal{B}}^{+},\tilde{\mathcal{D}},\tilde{g}).

We claim that $({\mathcal{B}},{\mathcal{D}},f)$ has the faithful unique pseudo-expectation property. Lemma 2.3 implies that $\tilde{E}$ is the unique pseudo-expectation for $(\tilde{\mathcal{C}},\tilde{\mathcal{D}},\tilde{f})$ . If $\Psi:{\mathcal{B}}^{+}\rightarrow I({\mathcal{D}})$ is a pseudo-expectation for $({\mathcal{B}}^{+},\tilde{\mathcal{D}},\tilde{f})$ , then $\Psi$ is a unital map (Lemma 2.3(c)). Injectivity of $I({\mathcal{B}})$ then implies that $\Psi$ extends to a pseudo-expectation $P$ for $(\tilde{\mathcal{C}},\tilde{\mathcal{D}},\tilde{f})$ . Thus $P=\tilde{E}$ , whence $\tilde{E}|_{{\mathcal{B}}^{+}}$ is the unique pseudo-expectation for $({\mathcal{B}}^{+},\tilde{\mathcal{D}},\tilde{f})=(\tilde{\mathcal{B}},\tilde{\mathcal{D}},\tilde{f})$ . Corollary 2.3 shows $({\mathcal{B}},{\mathcal{D}},f)$ has the unique pseudo-expectation property and $E|_{\mathcal{B}}$ is the pseudo-expectation. Since $E$ is faithful on ${\mathcal{C}}$ , $E|_{\mathcal{B}}$ is faithful. Thus $({\mathcal{B}},{\mathcal{D}},f)$ has the faithful unique pseudo-expectation property and $\Delta:=E|_{{\mathcal{B}}}$ is its pseudo-expectation.

By Lemma 2.3, $(\tilde{\mathcal{B}},\tilde{\mathcal{D}},\tilde{f})$ has the faithful unique pseudo expectation property. Thus, $\tilde{\Delta}$ is faithful. Proposition 2.3 shows $({\mathcal{B}},{\mathcal{D}},f)$ is weakly non-degenerate. Because $\operatorname{Ann}({\mathcal{C}},{\mathcal{D}},f)\subseteq{\mathcal{B}}$ , $\operatorname{Ann}({\mathcal{C}},{\mathcal{D}},f)=\operatorname{Ann}({\mathcal{B}},{\mathcal{D}},f)=\{0\}$ . Therefore, $({\mathcal{C}},{\mathcal{D}},f)$ is weakly non-degenerate. Another application of Proposition 2.3 now shows $\tilde{E}$ is faithful. Corollary 2.3 shows $(\tilde{\mathcal{C}},\tilde{\mathcal{D}},\tilde{f})$ has the faithful unique pseudo-expectation property, completing the proof. ∎

Corollary \the\numberby.

Let $({\mathcal{C}},{\mathcal{D}},f)$ be an inclusion such that ${\mathcal{D}}$ is abelian. If $({\mathcal{C}},{\mathcal{D}},f)$ has the faithful unique pseudo-expectation property, then $({\mathcal{C}},{\mathcal{D}},f)$ is weakly non-degenerate.

Proof.

Combine Theorem 2.3 with Proposition 2.3. ∎

Corollary \the\numberby.

Suppose $({\mathcal{C}},{\mathcal{D}},f)$ is an inclusion such that ${\mathcal{D}}$ is abelian and let ${\mathcal{B}}$ be a $C^{*}$ -algebra such that $f({\mathcal{D}})\subseteq{\mathcal{B}}\subseteq{\mathcal{C}}$ . If $({\mathcal{C}},{\mathcal{D}},f)$ has the faithful unique pseudo-expectation property, then $({\mathcal{B}},{\mathcal{D}},f)$ has the faithful unique pseudo-expectation property.

Proof.

Corollary 2.3 shows $({\mathcal{C}},{\mathcal{D}},f)$ is weakly non-degenerate, so $(\tilde{\mathcal{C}},\tilde{\mathcal{D}},\tilde{f})$ is a unital inclusion having the faithful unique pseudo-expectation property. Corollary 2.2 shows $(\tilde{\mathcal{B}},\tilde{\mathcal{D}},\tilde{f})$ is a unital inclusion with $\tilde{f}(\tilde{\mathcal{D}})\subseteq\tilde{\mathcal{B}}\subseteq\tilde{\mathcal{C}}$ . Thus, if $E:{\mathcal{C}}\rightarrow I({\mathcal{D}})$ is the pseudo-expectation for $({\mathcal{C}},{\mathcal{D}},f)$ , then $\tilde{E}$ is the faithful and unique pseudo-expectation for $(\tilde{\mathcal{C}},\tilde{\mathcal{D}},\tilde{f})$ . Then by [PittsZarikianUnPsExC*In, Proposition 2.6], $(\tilde{\mathcal{B}},\tilde{\mathcal{D}},\tilde{f})$ has the faithful unique pseudo-expectation property and $(\tilde{E})|_{\tilde{\mathcal{B}}}$ is the unique pseudo-expectation for $(\tilde{\mathcal{B}},\tilde{\mathcal{D}},\tilde{f})$ . Another application of Theorem 2.3 shows $({\mathcal{B}},{\mathcal{D}},f)$ has the faithful unique pseudo-expectation property. ∎

We now extend [PittsStReInII, Proposition 5.5(b)] from the unital setting to include the non-unital case. The utility of the following comes from the fact that it is sometimes easier to establish the faithful unique pseudo-expectation property than to show a subalgebra is a MASA.

Proposition \the\numberby.

Let $({\mathcal{C}},{\mathcal{D}})$ be a regular inclusion with ${\mathcal{D}}$ abelian. The following statements are equivalent.

(a)

$({\mathcal{C}},{\mathcal{D}})$ is a Cartan inclusion.
(b)

$({\mathcal{C}},{\mathcal{D}})$ has the faithful unique pseudo-expectation property and there is a conditional expectation $E:{\mathcal{C}}\rightarrow{\mathcal{D}}$ .

Proof.

First notice that if $({\mathcal{C}},{\mathcal{D}})$ satisfies either condition (a) or (b), then $({\mathcal{C}},{\mathcal{D}})$ is weakly non-degenerate. Indeed, if (a) holds, then Lemma 2.2(d) implies $({\mathcal{C}},{\mathcal{D}})$ is weakly non-degenerate; when (b) holds, apply Corollary 2.3. Thus in both cases, $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is a unital inclusion (Lemma 2.2(e)).

Our arguments establishing (a) $\Leftrightarrow$ (b) differ depending on whether ${\mathcal{C}}$ is unital or non-unital, so we consider those cases separately.

Case 1: Assume ${\mathcal{C}}$ is unital.

(a) $\Rightarrow$ (b) By hypothesis, $({\mathcal{C}},{\mathcal{D}})$ is a MASA inclusion, hence $({\mathcal{C}},{\mathcal{D}})$ is a unital inclusion. Then [PittsStReInII, Proposition 5.5(b)] gives (b).

(b) $\Rightarrow$ (a) For any $d\in{\mathcal{D}}$ , we have

E(I_{\mathcal{C}})d=E(d)=d=dE(I_{\mathcal{C}}),

so $E(I_{\mathcal{C}})$ is the unit for ${\mathcal{D}}$ . As $({\mathcal{C}},{\mathcal{D}})$ is weakly non-degenerate, it is a unital inclusion. Now [PittsStReInII, Proposition 5.5(b)] shows $({\mathcal{C}},{\mathcal{D}})$ is a Cartan inclusion.

Case 2: Assume ${\mathcal{C}}$ is not unital. Then ${\mathcal{D}}$ is not unital by Lemma 2.2.

(a) $\Rightarrow$ (b) By [PittsNoApUnInC*Al, Proposition 3.2], $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is a Cartan inclusion. Case 1 shows: i) $({\mathcal{C}},{\mathcal{D}})$ has the faithful unique pseudo-expectation property (see Corollary 2.3); and ii) there exists a conditional expectation $\tilde{E}:\tilde{\mathcal{C}}\rightarrow\tilde{\mathcal{D}}$ . We wish to show $\tilde{E}|_{\mathcal{C}}$ is a conditional expectation of ${\mathcal{C}}$ onto ${\mathcal{D}}$ . Let $(u_{\lambda})$ be an approximate unit for ${\mathcal{D}}$ . Then $(u_{\lambda})$ is an approximate unit for ${\mathcal{C}}$ by [PittsNoApUnInC*Al, Theorem 2.5]. So for $x\in{\mathcal{C}}$ , the fact that ${\mathcal{D}}$ is an ideal in $\tilde{\mathcal{D}}$ gives

\tilde{E}(x)=\lim_{\lambda}\tilde{E}(xu_{\lambda})=\lim_{\lambda}\tilde{E}(x)u_{\lambda}\in{\mathcal{D}}.

Thus $\tilde{E}|_{\mathcal{C}}$ is a conditional expectation of ${\mathcal{C}}$ onto ${\mathcal{D}}$ . This establishes (b).

(b) $\Rightarrow$ (a) By Proposition 2.3 and Corollary 2.3, $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ has the faithful unique pseudo-expectation property. Noting that $\tilde{E}$ is a conditional expectation of $\tilde{\mathcal{C}}$ onto $\tilde{\mathcal{D}}$ , we conclude from Case 1 that $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is a Cartan inclusion. Applying [PittsNoApUnInC*Al, Proposition 3.2] again, we find $({\mathcal{C}},{\mathcal{D}})$ is a Cartan inclusion. This completes the proof. ∎

2.4. The Ideal Intersection Property

The purpose of this section is to show that the inclusions $({\mathcal{A}},{\mathcal{B}},f)$ and $(\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{f})$ both have the ideal intersection property or both do not, and also to explore some relationships between the ideal intersection property and the faithful unique pseudo-expectation property.

Lemma \the\numberby.

Let $({\mathcal{A}},{\mathcal{B}},f)$ be an inclusion. Then $({\mathcal{A}},{\mathcal{B}},f)$ has the ideal intersection property if and only if $(\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{f})$ has the ideal intersection property.

Proof.

Suppose $({\mathcal{A}},{\mathcal{B}},f)$ has the ideal intersection property and let $J\unlhd\tilde{\mathcal{A}}$ satisfy $J\cap\tilde{f}(\tilde{\mathcal{B}})=\{0\}$ . As

(J\cap u_{\mathcal{A}}({\mathcal{A}}))\cap u_{\mathcal{A}}(f({\mathcal{B}}))=(J\cap u_{\mathcal{A}}({\mathcal{A}}))\cap\tilde{f}(u_{\mathcal{B}}({\mathcal{B}}))\subseteq J\cap\tilde{f}(\tilde{\mathcal{B}})=\{0\},

the ideal intersection property for $({\mathcal{A}},{\mathcal{B}},f)$ gives $J\cap u_{\mathcal{A}}({\mathcal{A}})=\{0\}$ . Since $u_{\mathcal{A}}({\mathcal{A}})$ is an essential ideal in $\tilde{\mathcal{A}}$ , we obtain $J=0$ . Thus $(\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{f})$ has the ideal intersection property.

Now suppose $(\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{f})$ has the ideal intersection property and let $J\unlhd{\mathcal{A}}$ satisfy $J\cap f({\mathcal{B}})=\{0\}$ . Then $u_{\mathcal{A}}(J)\unlhd\tilde{\mathcal{A}}$ , and we claim

(2.4.1)

u_{\mathcal{A}}(J)\cap\tilde{f}(\tilde{\mathcal{B}})=\{0\}.

When ${\mathcal{B}}$ is unital, $\tilde{\mathcal{B}}={\mathcal{B}}$ and $\tilde{f}=u_{\mathcal{A}}\circ f$ , so (2.4.1) holds because $u_{\mathcal{A}}$ is one-to-one.

Suppose then that ${\mathcal{B}}$ is not unital. Recall that $\{0\}$ is a unital algebra, so in particular, ${\mathcal{B}}\neq\{0\}$ . If $x\in u_{\mathcal{A}}(J)\cap\tilde{f}(\tilde{\mathcal{B}})$ , then $x$ has the form

x=\tilde{f}(b,\lambda)=u_{\mathcal{A}}(f(b))+\lambda I_{\tilde{\mathcal{A}}}

for some $b\in{\mathcal{B}}$ and $\lambda\in{\mathbb{C}}$ . Since $J\cap f({\mathcal{B}})=\{0\}$ , (2.4.1) will follow once we show $\lambda=0$ .

Arguing by contradiction, suppose $\lambda\neq 0$ . By scaling, we may assume $\lambda=-1$ , so that $x=u_{\mathcal{A}}(f(b))-I_{\tilde{\mathcal{A}}}$ , for some $b\in{\mathcal{B}}$ . For any $h\in{\mathcal{B}}$ , we have $xu_{\mathcal{A}}(f(h))\in u_{\mathcal{A}}(J)\cap u_{\mathcal{A}}(f({\mathcal{B}}))=\{0\}$ , so $xu_{\mathcal{A}}(f(h))=0$ . Taking $h=b^{*}$ , we conclude that

bb^{*}=b^{*}.

Since ${\mathcal{B}}$ is not unital, this forces $b=0$ . Therefore $x=-I_{\tilde{\mathcal{A}}}\in u_{\mathcal{A}}(J)$ , whence $u_{\mathcal{A}}(J)=\tilde{\mathcal{A}}$ . This is impossible if ${\mathcal{A}}$ is not unital because $u_{\mathcal{A}}(J)\subseteq u_{\mathcal{A}}({\mathcal{A}})\neq\tilde{\mathcal{A}}$ . On the other hand, if ${\mathcal{A}}$ is unital, we again reach a contradiction because $\{0\}=u_{\mathcal{A}}(J\cap f({\mathcal{B}}))=\tilde{\mathcal{A}}\cap u_{\mathcal{A}}(f({\mathcal{B}}))=u_{\mathcal{A}}(f({\mathcal{B}}))$ , contrary to the fact that ${\mathcal{B}}\neq\{0\}$ . Therefore, $\lambda=0$ , completing the proof of (2.4.1).

Since $(\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{f})$ has the ideal intersection property, (2.4.1) gives $J=\{0\}$ . Thus $({\mathcal{A}},{\mathcal{B}},f)$ has the ideal intersection property. ∎

Our next result concerns the ideal intersection property for intermediate inclusions in the abelian case. It is the same as [PittsStReInII, Lemma 5.4] except the hypothesis that the $C^{*}$ -algebras involved have a common unit is dropped and we explicitly consider the inclusion mappings.

Lemma \the\numberby.

For $1\leq i<j\leq 3$ , Let $({\mathcal{D}}_{j},{\mathcal{D}}_{i},f_{ji})$ be inclusions such that $f_{31}=f_{32}\circ f_{21}$ and ${\mathcal{D}}_{3}$ is abelian. Then $({\mathcal{D}}_{3},{\mathcal{D}}_{1},f_{31})$ has the ideal intersection property if and only if both $({\mathcal{D}}_{3},{\mathcal{D}}_{2},f_{32})$ and $({\mathcal{D}}_{2},{\mathcal{D}}_{1},f_{21})$ have the ideal intersection property.

Proof.

The implication $(\Leftarrow)$ is left to the reader.

$(\Rightarrow)$ Suppose $({\mathcal{D}}_{3},{\mathcal{D}}_{1},f_{31})$ has the ideal intersection property. Lemma 2.2(c) shows $({\mathcal{D}}_{3},{\mathcal{D}}_{1},f_{31})$ is weakly non-degenerate. Therefore, both $({\mathcal{D}}_{2},{\mathcal{D}}_{1},f_{21})$ and $({\mathcal{D}}_{3},{\mathcal{D}}_{2},f_{32})$ are weakly non-degenerate. Then for $1\leq i<j\leq 3$ , each of $(\tilde{\mathcal{D}}_{j},\tilde{\mathcal{D}}_{i},\tilde{f}_{ji})$ is a unital inclusion, hence $\tilde{f}_{31}=\tilde{f}_{32}\circ\tilde{f}_{21}$ (Corollary 2.2). By Lemma 2.2(a), these inclusions have a common unit in the sense that

I_{\tilde{\mathcal{D}}_{3}}=\tilde{f}_{32}(I_{\tilde{\mathcal{D}}_{2}})\quad\text{and}\quad I_{\tilde{\mathcal{D}}_{2}}=\tilde{f}_{21}(I_{\tilde{\mathcal{D}}_{1}}).

By [PittsStReInII, Lemma 5.4], $(\tilde{\mathcal{D}}_{2},\tilde{\mathcal{D}}_{1},\tilde{f}_{21})$ and $(\tilde{\mathcal{D}}_{3},\tilde{\mathcal{D}}_{2},\tilde{f}_{32})$ have the ideal intersection property. An application of Lemma 2.4 completes the proof. ∎

Our final result of this section extends a portion of [PittsZarikianUnPsExC*In, Corollary 3.22] to include non-unital settings.

Proposition \the\numberby.

Suppose $({\mathcal{D}}_{2},{\mathcal{D}}_{1})$ is an inclusion with ${\mathcal{D}}_{2}$ abelian and let $(I({\mathcal{D}}_{1}),\iota_{1})$ be an injective envelope for ${\mathcal{D}}_{1}$ . Then $({\mathcal{D}}_{2},{\mathcal{D}}_{1})$ has the ideal intersection property if and only if $({\mathcal{D}}_{2},{\mathcal{D}}_{1})$ has the faithful unique pseudo-expectation property.

When this occurs, the unique pseudo-expectation $\iota_{2}:{\mathcal{D}}_{2}\rightarrow I({\mathcal{D}}_{1})$ is a $*$ -monomorphism and $(I({\mathcal{D}}_{1}),\iota_{2})$ is an injective envelope for ${\mathcal{D}}_{2}$ .

Proof.

( $\Rightarrow$ ) Suppose $({\mathcal{D}}_{2},{\mathcal{D}}_{1})$ has the ideal intersection property. An application of Lemma 2.2(e) shows $({\mathcal{D}}_{2},{\mathcal{D}}_{1})$ is weakly non-degenerate. By Proposition 2.2(e) and Lemma 2.4, $(\tilde{\mathcal{D}}_{2},\tilde{\mathcal{D}}_{1})$ is a unital inclusion having the ideal intersection property, so [PittsZarikianUnPsExC*In, Corollary 3.22] shows $(\tilde{\mathcal{D}}_{2},\tilde{\mathcal{D}}_{1})$ has the faithful unique pseudo-expectation property. Corollary 2.3 shows $({\mathcal{D}}_{2},{\mathcal{D}}_{1})$ has the faithful unique pseudo-expectation property.

( $\Leftarrow$ ) Suppose $({\mathcal{D}}_{2},{\mathcal{D}}_{1})$ has the faithful unique pseudo-expectation property. By Corollary 2.3, $({\mathcal{D}}_{2},{\mathcal{D}}_{1})$ is weakly non-degenerate, whence $(\tilde{\mathcal{D}}_{2},\tilde{\mathcal{D}}_{1})$ is a unital inclusion. Apply [PittsZarikianUnPsExC*In, Corollary 3.22] to see $(\tilde{\mathcal{D}}_{2},\tilde{\mathcal{D}}_{1})$ has the ideal intersection property. Lemma 2.4 shows $({\mathcal{D}}_{2},{\mathcal{D}}_{1})$ has the ideal intersection property.

Turning to the last statement, suppose $({\mathcal{D}}_{2},{\mathcal{D}}_{1})$ has the faithful unique pseudo-expectation property with pseudo-expectation $\iota_{2}$ . Applying Corollary 2.3 and Lemma 2.2(e), we see that $(\tilde{\mathcal{D}}_{2},\tilde{\mathcal{D}}_{1})$ is a unital inclusion. Corollary 2.3 shows $\tilde{\iota}_{2}:\tilde{\mathcal{D}}_{2}\rightarrow I({\mathcal{D}}_{1})$ is the unique pseudo-expectation for $(\tilde{\mathcal{D}}_{2},\tilde{\mathcal{D}}_{1})$ and by Theorem 2.3, $\tilde{\iota}_{2}$ is faithful. By [PittsZarikianUnPsExC*In, Corollary 3.22], $\tilde{\iota}_{2}$ is a $*$ -monomorphism, hence so is $\iota_{2}$ .

Finally, since $\iota_{1}({\mathcal{D}}_{1})\subseteq\iota_{2}({\mathcal{D}}_{2})\subseteq I({\mathcal{D}}_{1})$ , the minimality of injective envelopes shows $(I({\mathcal{D}}_{1}),\iota_{2})$ is an injective envelope for ${\mathcal{D}}_{2}$ . ∎

3. Cartan Envelopes

Up to this point, we have mostly considered general inclusions. We now restrict attention to inclusions where the subalgebra is abelian.

Standing Assumption \the\numberby. Unless explicitly stated otherwise, for the remainder of this work, whenever $({\mathcal{C}},{\mathcal{D}})$ is an inclusion, we shall always assume ${\mathcal{D}}$ is abelian.

In [PittsStReInII], we defined the notion of a Cartan envelope for unital and regular inclusions and characterized when such inclusions have a Cartan envelope, see [PittsStReInII, Theorem 5.2]. The purpose of this section is to extend the characterization of regular inclusions having a Cartan envelope from unital regular inclusions to all regular inclusions. This is accomplished in Theorem 3; it is among our main results. While the statement of Theorem 3 parallels that of [PittsStReInII, Theorem 5.2], discarding the hypothesis that $({\mathcal{C}},{\mathcal{D}})$ is unital presents challenges which cannot be overcome by simply adjoining a unit.

We begin by recording a few useful facts about inclusions satisfying Assumption 3.

Lemma \the\numberby ([PittsNoApUnInC*Al, Corollary 2.2]).

Let $({\mathcal{C}},{\mathcal{D}})$ be an inclusion and fix a normalizer $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ . Then $\overline{vv^{*}{\mathcal{D}}}$ and $\overline{v^{*}v{\mathcal{D}}}$ are ideals in ${\mathcal{D}}$ and the map $vv^{*}d\mapsto v^{*}dv$ uniquely extends to a $*$ -isomorphism $\theta_{v}:\overline{vv^{*}{\mathcal{D}}}\rightarrow\overline{v^{*}v{\mathcal{D}}}$ ; moreover, for every $h\in\overline{vv^{*}{\mathcal{D}}}$ , $v\theta_{v}(h)=hv$ .

Our proofs of the statements in the next lemma depend on results from [PittsNoApUnInC*Al].

Lemma \the\numberby.

Suppose $({\mathcal{C}},{\mathcal{D}})$ is an inclusion and let ${\mathcal{D}}^{c}$ be the relative commutant of ${\mathcal{D}}$ in ${\mathcal{C}}$ . The following statements hold.

(a)

The identity mapping on ${\mathcal{C}}$ is a regular map from $({\mathcal{C}},{\mathcal{D}})$ into $({\mathcal{C}},{\mathcal{D}}^{c})$ , that is,

${\mathcal{N}}({\mathcal{C}},{\mathcal{D}})\subseteq{\mathcal{N}}({\mathcal{C}},{\mathcal{D}}^{c}).$
(b)

Suppose $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ is an inclusion and $\alpha:({\mathcal{C}},{\mathcal{D}})\rightarrow({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ is a regular $*$ -monomorphism. Then

$\alpha({\mathcal{D}})\subseteq{\mathcal{D}}_{1}^{c}.$

In addition, if $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ has the AUP, then $\alpha({\mathcal{D}})\subseteq{\mathcal{D}}_{1}$ .
(c)

If $({\mathcal{C}},{\mathcal{D}})$ has the AUP, then ${\mathcal{N}}({\mathcal{C}},{\mathcal{D}})\subseteq{\mathcal{N}}(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ .
(d)

Suppose $({\mathcal{C}},{\mathcal{D}})$ is a regular and weakly non-degenerate inclusion with ${\mathcal{C}}$ not unital. If ${\mathcal{N}}({\mathcal{C}},{\mathcal{D}})\subseteq{\mathcal{N}}(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ , then $({\mathcal{C}},{\mathcal{D}})$ has the AUP.
(e)

Suppose $({\mathcal{C}},{\mathcal{D}})$ is regular and has the AUP. Then $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is regular.

Proof.

a) Let $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ . Let $d\in{\mathcal{D}}$ and $x\in{\mathcal{D}}^{c}$ . Using Lemma 3, we find that for every $h\in\overline{vv^{*}{\mathcal{D}}}$ ,

(vxv^{*})dh=vx\theta_{v}(dh)v^{*}=v\theta_{v}(dh)xv^{*}=dh(vxv^{*}).

Taking $h=u_{\lambda}$ where $(u_{\lambda})$ is an approximate unit for $\overline{vv^{*}{\mathcal{D}}}$ , and using the fact that $\left\lVert v^{*}u_{\lambda}-v^{*}\right\rVert\rightarrow 0$ , we obtain $vxv^{*}\in{\mathcal{D}}^{c}$ . Likewise $v^{*}xv\in{\mathcal{D}}^{c}$ , so (a) holds.

b) Let $0\leq d\in{\mathcal{D}}$ . Since $d^{1/2}\in{\mathcal{D}}\subseteq{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ , we have $\alpha(d^{1/2})\in{\mathcal{N}}({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ . For every $h\in{\mathcal{D}}_{1}$ , [PittsNoApUnInC*Al, Proposition 2.1] gives,

(3.2)

\alpha(d)h=\alpha(d^{1/2})^{*}\alpha(d^{1/2})h=h\alpha(d)\in{\mathcal{D}}_{1}.

Thus $\alpha(d)\in{\mathcal{D}}_{1}^{c}$ . Since ${\mathcal{D}}$ is the linear span of its positive elements, $\alpha({\mathcal{D}})\subseteq{\mathcal{D}}_{1}^{c}$ .

Let $(u_{\lambda})\subseteq{\mathcal{D}}_{1}$ be an approximate unit for ${\mathcal{C}}_{1}$ . Replacing $h$ in (3.2) with $u_{\lambda}$ and taking the limit along $\lambda$ gives $\alpha(d)\in{\mathcal{D}}_{1}$ .

c) Suppose $({\mathcal{C}},{\mathcal{D}})$ has the AUP. By Lemma 2.2(e), $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is a unital inclusion. The conclusion is obvious when ${\mathcal{C}}$ is unital. So assume ${\mathcal{C}}$ is not unital and let $(u_{\lambda})$ be a net in ${\mathcal{D}}$ which is an approximate unit for ${\mathcal{C}}$ . For $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ , [PittsNoApUnInC*Al, Proposition 2.1] shows $v^{*}vu_{\lambda}$ and $vv^{*}u_{\lambda}$ belong to ${\mathcal{D}}$ . Taking the limit shows $v^{*}v$ and $vv^{*}$ are elements of ${\mathcal{D}}$ . Thus, if $(d,\lambda)\in\tilde{\mathcal{D}}$ ,

(3.3)

(v,0)^{*}(d,\lambda)(v,0)=(v^{*}dv+\lambda v^{*}v,0)\quad\text{and}\quad(v,0)(d,\lambda)(v,0)^{*}=(vdv^{*}+\lambda vv^{*},0).

This gives ${\mathcal{N}}({\mathcal{C}},{\mathcal{D}})\subseteq{\mathcal{N}}(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ .

d) Suppose $({\mathcal{C}},{\mathcal{D}})$ is regular, weakly non-degenerate, and ${\mathcal{N}}({\mathcal{C}},{\mathcal{D}})\subseteq{\mathcal{N}}(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ . Since ${\mathcal{C}}$ is not unital, neither is ${\mathcal{D}}$ , so that $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})=({\mathcal{C}}^{\dagger},{\mathcal{D}}^{\dagger})$ . The calculation in (3.3) with $d=0$ and $\lambda=1$ shows that for every $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ , $v^{*}v\in{\mathcal{D}}$ . Then [PittsNoApUnInC*Al, Observation 1.3(2)] shows $({\mathcal{C}},{\mathcal{D}})$ has the AUP.

e) Suppose $({\mathcal{C}},{\mathcal{D}})$ is regular and has the AUP. There is nothing to do if ${\mathcal{C}}$ is unital, for then $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})=({\mathcal{C}},{\mathcal{D}})$ . Assume then, that ${\mathcal{C}}$ is not unital. Part (c) shows

\tilde{\mathcal{C}}=\{(0,\lambda):\lambda\in{\mathbb{C}}\}+\overline{\operatorname{span}}\{(v,0):v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})\},

so $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is regular. ∎

While the following corollary is immediate from parts (c) and (d) of Lemma 3, it is worth making explicit. In particular, $u_{\mathcal{C}}$ is not regular when ${\mathcal{D}}$ is an essential and proper ideal in the abelian $C^{*}$ -algebra ${\mathcal{C}}$ ; this also shows that (c) $\not\Rightarrow$ (b) in Corollary 3.

Corollary \the\numberby.

Let $({\mathcal{C}},{\mathcal{D}})$ be a regular and weakly non-degenerate inclusion with ${\mathcal{C}}$ not unital. Consider the following statements.

(a)

$({\mathcal{C}},{\mathcal{D}})$ has the AUP;
(b)

the inclusion mapping $u_{\mathcal{C}}:({\mathcal{C}},{\mathcal{D}})\rightarrow(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is a regular $*$ -monomorphism;
(c)

$(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is a regular inclusion.

The following implications hold: (a) $\Leftrightarrow$ (b) $\Rightarrow$ (c).

Proof.

(a) $\Leftrightarrow$ (b) Use 3(c) and 3(d).

(b) $\Rightarrow$ (c) By hypothesis, $\{(v,0):v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})\}\subseteq{\mathcal{N}}(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ . As ${\mathbb{C}}I_{\tilde{\mathcal{C}}}\subseteq{\mathcal{N}}(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ , it follows that $\operatorname{span}{\mathcal{N}}(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is dense in $\tilde{\mathcal{C}}$ .

∎

Example \the\numberby.

Lemma 3(b) shows $\alpha({\mathcal{D}})\subseteq{\mathcal{B}}$ whenever $({\mathcal{A}},{\mathcal{B}})$ has the AUP and $\alpha$ is a regular map. However, if $({\mathcal{A}},{\mathcal{B}})$ does not have the AUP, this can fail. For an elementary example of this behavior, let

{\mathcal{C}}={\mathcal{A}}=C_{0}({\mathbb{R}}),\quad{\mathcal{D}}={\mathcal{B}}=\{h\in C_{0}({\mathbb{R}}):h(x)=0\text{ for all }x>1\},

and define $\alpha:{\mathcal{C}}\rightarrow{\mathcal{A}}$ by

\alpha(h)(x)=h(x/2).

Since ${\mathcal{N}}({\mathcal{C}},{\mathcal{D}})={\mathcal{N}}({\mathcal{A}},{\mathcal{B}})=C_{0}({\mathbb{R}})$ , $\alpha$ is a regular map, but $\alpha({\mathcal{D}})$ is not contained in ${\mathcal{B}}$ .

The following extends parts of [PittsStReInI, Theorem 3.5] to include inclusions which may not be unital.

Theorem \the\numberby.

Suppose $({\mathcal{C}},{\mathcal{D}})$ is a regular MASA inclusion. Then $({\mathcal{C}},{\mathcal{D}})$ has the unique pseudo-expectation property. Furthermore, $({\mathcal{C}},{\mathcal{D}})$ has the faithful unique pseudo-expectation property if and only if $({\mathcal{C}},{\mathcal{D}})$ has the ideal intersection property.

Proof.

Since $({\mathcal{C}},{\mathcal{D}})$ is a regular MASA inclusion, it has the AUP ([PittsNoApUnInC*Al, Theorem 2.5]), whence $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is a unital inclusion (Lemma 2.2(e)), and regular by Lemma 3(e).

We claim that $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is a MASA inclusion. If ${\mathcal{D}}$ is unital, then so is ${\mathcal{C}}$ by Lemma 2.2(a); thus in this case $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})=({\mathcal{C}},{\mathcal{D}})$ and all is well. Suppose then that ${\mathcal{D}}$ is not unital. Since ${\mathcal{D}}$ is a MASA, ${\mathcal{C}}$ cannot be unital. Thus $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})=({\mathcal{C}}^{\dagger},{\mathcal{D}}^{\dagger})$ and a routine argument shows $\tilde{\mathcal{D}}$ is a MASA in $\tilde{\mathcal{C}}$ .

We have established that $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is a regular, unital, MASA inclusion. By [PittsStReInI, Theorem 3.5] $(\tilde{\mathcal{C}},\tilde{\mathcal{C}})$ has the unique pseudo-expectation property, so Corollary 2.3 shows $({\mathcal{C}},{\mathcal{D}})$ also has the unique pseudo-expectation property.

Turning to the second statement, let $E:{\mathcal{C}}\rightarrow I({\mathcal{D}})$ be the pseudo-expectation. Then $\tilde{E}$ is the pseudo-expectation for $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ .

Suppose $({\mathcal{C}},{\mathcal{D}})$ has the ideal intersection property. Lemma 2.4 implies $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ has the ideal intersection property. By [PittsStReInI, Theorem 3.15], the left kernel ${\mathcal{L}}:=\{x\in\tilde{\mathcal{C}}:\tilde{E}(x^{*}x)=0\}$ is an ideal of $\tilde{\mathcal{C}}$ having trivial intersection with $\tilde{\mathcal{D}}$ . Therefore, $\tilde{E}$ is faithful, whence $E$ is faithful.

Now suppose $E$ is faithful. By Proposition 2.3, $\tilde{E}$ is faithful. If $J\unlhd\tilde{\mathcal{C}}$ has trivial intersection with $\tilde{\mathcal{D}}$ , [PittsStReInI, Theorem 3.15] gives $J\subseteq{\mathcal{L}}=\{0\}$ . So $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ has the ideal intersection property. By Lemma 2.4, $({\mathcal{C}},{\mathcal{D}})$ has the ideal intersection property, completing the proof. ∎

Definition \the\numberby.

Suppose $({\mathcal{C}},{\mathcal{D}})$ is an inclusion and $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is an expansion of $({\mathcal{C}},{\mathcal{D}})$ .

(a)

We call $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ an essential expansion of $({\mathcal{C}},{\mathcal{D}})$ if $({\mathcal{B}},{\mathcal{D}},\alpha|_{{\mathcal{D}}})$ is an inclusion having the ideal intersection property.
(b)

If the $*$ -monomorphism $\alpha$ is a regular map, we will say $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a regular expansion of $({\mathcal{C}},{\mathcal{D}})$ .
(c)

If $({\mathcal{A}},{\mathcal{B}})$ is a Cartan pair, we say $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan expansion of $({\mathcal{C}},{\mathcal{D}})$ .

Remarks \the\numberby.

(a)

It may seem odd that in Definition 3(a), we only require that $({\mathcal{B}},{\mathcal{D}},\alpha|_{{\mathcal{D}}})$ has the ideal intersection property instead of also placing that requirement on $({\mathcal{A}},{\mathcal{B}},\alpha)$ . In the context of most interest to us, that is, when $({\mathcal{C}},{\mathcal{D}})$ and $({\mathcal{A}},{\mathcal{B}})$ both have the faithful unique pseudo-expectation property, we will see in Observation 4.3 that this is automatic.
(b)

In some cases, regular maps produce expansions. For example, Lemma 3(b) shows that when $\alpha:({\mathcal{C}}_{1},{\mathcal{D}}_{1})\rightarrow({\mathcal{C}}_{2},{\mathcal{D}}_{2})$ is a regular $*$ -monomorphism, and $({\mathcal{C}}_{2},{\mathcal{D}}_{2})$ is a Cartan inclusion, then $({\mathcal{C}}_{2},{\mathcal{D}}_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is automatically a regular expansion of $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ .

The next two lemmas give some useful properties of essential expansions.

Lemma \the\numberby.

Suppose $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is an essential expansion for the inclusion $({\mathcal{C}},{\mathcal{D}})$ and there is a faithful conditional expectation $\Delta:{\mathcal{A}}\rightarrow{\mathcal{B}}$ . Then $({\mathcal{A}},{\mathcal{D}},\alpha|_{\mathcal{D}})$ and $({\mathcal{A}},{\mathcal{C}},\alpha)$ are weakly non-degenerate inclusions. In particular, if ${\mathcal{C}}$ is unital, then ${\mathcal{A}}$ is unital and $\alpha(I_{\mathcal{C}})=I_{\mathcal{A}}$ .

Proof.

By hypothesis, $({\mathcal{B}},{\mathcal{D}},\alpha|_{\mathcal{D}})$ has the ideal intersection property, so it is weakly non-degenerate by Lemma 2.2(c). Suppose $0\leq a\in{\mathcal{A}}$ belongs to $\operatorname{Ann}({\mathcal{A}},{\mathcal{D}},\alpha|_{\mathcal{D}})$ . Then for every $d\in{\mathcal{D}}$ ,

0=\Delta(\alpha(d)a)=\alpha(d)\Delta(a)=\Delta(a)\alpha(d)=\Delta(a\alpha(d)).

Thus $\Delta(a)\in\operatorname{Ann}({\mathcal{B}},{\mathcal{D}},\alpha|_{\mathcal{D}})$ . Faithfulness of $\Delta$ gives $a=0$ . As $\operatorname{Ann}({\mathcal{A}},{\mathcal{D}},\alpha)$ is the span of its postive elements, we obtain $\operatorname{Ann}({\mathcal{A}},{\mathcal{D}},\alpha)=\{0\}$ , so $({\mathcal{A}},{\mathcal{D}},\alpha|_{\mathcal{D}})$ is weakly non-degenerate. That $({\mathcal{A}},{\mathcal{C}},\alpha)$ is weakly non-degenerate follows from the fact that $\operatorname{Ann}({\mathcal{A}},{\mathcal{C}},\alpha)\subseteq\operatorname{Ann}({\mathcal{A}},{\mathcal{D}},\alpha|_{\mathcal{D}})$ . If ${\mathcal{C}}$ is unital, Lemma 2.2(a) shows ${\mathcal{A}}$ is unital and $\alpha(I_{\mathcal{C}})=I_{\mathcal{A}}$ . ∎

Lemma \the\numberby.

Let $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ be an inclusion. Suppose $({\mathcal{C}}_{2},{\mathcal{D}}_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is an essential expansion of $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ and $({\mathcal{C}}_{2},{\mathcal{D}}_{2})$ has the faithful unique pseudo-expectation property. Then

(a)

$({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ has the faithful unique pseudo-expectation property; and
(b)

$\alpha({\mathcal{D}}_{1}^{c})\subseteq{\mathcal{D}}_{2}^{c}$ .

Proof.

Before giving the proofs of (a) and (b), we make a few remarks. Let $(I({\mathcal{D}}_{1}),\iota_{1})$ be an injective envelope for ${\mathcal{D}}_{1}$ . By Proposition 2.4, $({\mathcal{D}}_{2},{\mathcal{D}}_{1},\alpha|_{{\mathcal{D}}_{1}})$ has the faithful unique pseudo-expectation property; let $\iota_{2}:{\mathcal{D}}_{2}\rightarrow I({\mathcal{D}}_{1})$ be the pseudo-expectation for $({\mathcal{D}}_{2},{\mathcal{D}}_{1},\alpha|_{{\mathcal{D}}_{1}})$ relative to $(I({\mathcal{D}}_{1}),\iota_{1})$ . Then $(I({\mathcal{D}}_{1}),\iota_{2})$ is an injective envelope for ${\mathcal{D}}_{2}$ (Proposition 2.4). Let

E_{2}:{\mathcal{C}}_{2}\rightarrow I({\mathcal{D}}_{1})

be the pseudo-expectation for $({\mathcal{C}}_{2},{\mathcal{D}}_{2})$ relative to $(I({\mathcal{D}}_{1}),\iota_{2})$ .

(a) Let us first show $({\mathcal{C}}_{2},{\mathcal{D}}_{1},\alpha|_{{\mathcal{D}}_{1}})$ has the faithful unique pseudo-expectation property. Suppose $F:{\mathcal{C}}_{2}\rightarrow I({\mathcal{D}}_{1})$ is a pseudo-expectation for $({\mathcal{C}}_{2},{\mathcal{D}}_{1},\alpha|_{{\mathcal{D}}_{1}})$ relative to $(I({\mathcal{D}}_{1}),\iota_{1})$ . Then for $d\in{\mathcal{D}}_{1}$ , $F(\alpha(d))=\iota_{1}(d)=\iota_{2}(\alpha(d))$ . Thus $F|_{{\mathcal{D}}_{2}}$ is a pseudo-expectation for $({\mathcal{D}}_{2},{\mathcal{D}}_{1},\alpha|_{{\mathcal{D}}_{1}})$ relative to $(I({\mathcal{D}}_{1}),\iota_{1})$ . Therefore $F|_{{\mathcal{D}}_{2}}=\iota_{2}$ , that is, $F$ is a pseudo-expectation for $({\mathcal{C}}_{2},{\mathcal{D}}_{2})$ relative to $(I({\mathcal{D}}_{1}),\iota_{2})$ . This forces $F=E_{2}$ , showing $({\mathcal{C}}_{2},{\mathcal{D}}_{1},\alpha|_{{\mathcal{D}}_{1}})$ has the faithful unique pseudo-expectation property.

Corollary 2.3 applied to $({\mathcal{C}}_{2},{\mathcal{D}}_{1},\alpha|_{{\mathcal{D}}_{1}})$ with ${\mathcal{B}}=\alpha({\mathcal{C}}_{1})$ implies $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ has the faithful unique pseudo-expectation property (the pseudo-expectation is $E_{2}\circ\alpha$ ).

(b) By Proposition 2.3, $\operatorname{\textsc{rCom}}({\mathcal{C}}_{2},\alpha({\mathcal{D}}_{1}))$ is abelian. As $\alpha({\mathcal{D}}_{1})\subseteq{\mathcal{D}}_{2}$ , we find

\operatorname{\textsc{rCom}}({\mathcal{C}}_{2},{\mathcal{D}}_{2})\subseteq\operatorname{\textsc{rCom}}({\mathcal{C}}_{2},\alpha({\mathcal{D}}_{1})).

Since $({\mathcal{C}}_{2},{\mathcal{D}}_{2})$ has the faithful unique pseudo-expectation property, $\operatorname{\textsc{rCom}}({\mathcal{C}}_{2},{\mathcal{D}}_{2})$ is a MASA in ${\mathcal{C}}_{2}$ , so $\operatorname{\textsc{rCom}}({\mathcal{C}}_{2},{\mathcal{D}}_{2})=\operatorname{\textsc{rCom}}({\mathcal{C}}_{2},\alpha({\mathcal{D}}_{1}))$ . As

\alpha({\mathcal{D}}_{1}^{c})\subseteq\operatorname{\textsc{rCom}}({\mathcal{C}}_{2},\alpha({\mathcal{D}}_{1}))={\mathcal{D}}_{2}^{c},

part (b) holds. ∎

The definition of Cartan envelope given in [PittsStReInII] extends to any regular inclusion $({\mathcal{C}},{\mathcal{D}})$ . Here is the definition, along with the definitions of other useful expansions.

Definition \the\numberby.

(cf. [PittsStReInII, Definition 5.1]) Let $({\mathcal{C}},{\mathcal{D}})$ be a regular inclusion.

(a)
A package for $({\mathcal{C}},{\mathcal{D}})$ is a regular expansion $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ of $({\mathcal{C}},{\mathcal{D}})$ such that
1. (i)
  
  there is a faithful conditional expectation $\Delta:{\mathcal{A}}\rightarrow{\mathcal{B}}$ ;
2. (ii)
  
  $({\mathcal{C}},{\mathcal{D}})$ generates $({\mathcal{A}},{\mathcal{B}})$ in the sense that ${\mathcal{B}}=C^{*}(\Delta(\alpha({\mathcal{C}})))$ and ${\mathcal{A}}=C^{*}(\alpha({\mathcal{C}})\cup{\mathcal{B}})$ .
If in addition, $({\mathcal{A}},{\mathcal{B}})$ is a Cartan inclusion, we say $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan package for $({\mathcal{C}},{\mathcal{D}})$ .
(b)

If $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a package for $({\mathcal{C}},{\mathcal{D}})$ which is also an essential expansion for $({\mathcal{C}},{\mathcal{D}})$ (see Definition 3(a)), then we call $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ an envelope for $({\mathcal{C}},{\mathcal{D}})$ .
(c)

We say $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ if it is an envelope and a Cartan package.

We now show that when $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is an essential and regular Cartan expansion of $({\mathcal{C}},{\mathcal{D}})$ , the inclusion generated by $({\mathcal{C}},{\mathcal{D}})$ is a Cartan envelope.

Proposition \the\numberby.

Suppose $({\mathcal{C}},{\mathcal{D}})$ is a regular inclusion and $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is an essential, regular, and Cartan expansion for $({\mathcal{C}},{\mathcal{D}})$ . Let $\Delta:{\mathcal{A}}\rightarrow{\mathcal{B}}$ be the conditional expectation and put

{\mathcal{B}}_{1}:=C^{*}(\Delta(\alpha({\mathcal{C}})))\quad\text{and}\quad{\mathcal{A}}_{1}:=C^{*}(\alpha({\mathcal{C}})\cup{\mathcal{B}}_{1}).

Then $({\mathcal{A}}_{1},{\mathcal{B}}_{1}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ and $\Delta|_{{\mathcal{A}}_{1}}:{\mathcal{A}}_{1}\rightarrow{\mathcal{B}}_{1}$ is the conditional expectation.

Proof.

By construction, $\alpha({\mathcal{C}})\subseteq{\mathcal{A}}_{1}$ and $\alpha({\mathcal{D}})\subseteq{\mathcal{B}}_{1}$ , so $({\mathcal{A}}_{1},{\mathcal{B}}_{1}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is an expansion of $({\mathcal{C}},{\mathcal{D}})$ . That $\Delta|_{{\mathcal{A}}_{1}}:{\mathcal{A}}_{1}\rightarrow{\mathcal{B}}_{1}$ is a faithful conditional expectation follows from faithfulness of $\Delta$ , the definitions of ${\mathcal{A}}_{1}$ and ${\mathcal{B}}_{1}$ , and the fact that ${\mathcal{B}}_{1}\subseteq{\mathcal{B}}$ .

Claim \the\numberby.

$\alpha:({\mathcal{C}},{\mathcal{D}})\rightarrow({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ is a regular map.

Proof of Claim 3. For $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ , we must show that $\alpha(v)$ normalizes ${\mathcal{B}}_{1}$ . As $({\mathcal{A}},{\mathcal{B}})$ is a Cartan inclusion, it has the faithful unique pseudo-expectation property by Proposition 2.3. Lemma 3 shows $({\mathcal{C}},{\mathcal{D}})$ has the faithful unique pseudo-expectation property and

(3.5)

\alpha({\mathcal{D}}^{c})\subseteq{\mathcal{B}}^{c}={\mathcal{B}}.

An argument similar to that used for establishing [PittsStReInII, (5.14)] shows that for $n\in{\mathbb{N}}$ , and every collection of $n$ elements $\{x_{j}\}_{j=1}^{n}\subseteq{\mathcal{C}}$ ,

(3.6)

\alpha(v)^{*}\left(\prod_{j=1}^{n}\Delta(\alpha(x_{j}))\right)\alpha(v)\in{\mathcal{B}}_{1}.

However, the proofs of (3.6) and [PittsStReInII, (5.14)] have some technical differences, so we include the proof of (3.6) here. The argument is by induction. When $n=1$ , the invariance of $\Delta$ under ${\mathcal{N}}({\mathcal{A}},{\mathcal{B}})$ (see (2.1.10)) gives,

\alpha(v)^{*}\Delta(\alpha(x_{1}))\alpha(v)=\Delta(\alpha(v^{*})\alpha(x_{1})\alpha(v))\in\Delta(\alpha({\mathcal{C}}))\subseteq{\mathcal{B}}_{1}.

Suppose now that (3.6) holds for some $n\in{\mathbb{N}}$ and every collection of $n$ elements of ${\mathcal{C}}$ . Let $\{x_{j}\}_{j=1}^{n+1}\subseteq{\mathcal{C}}$ and set $y=\prod_{j=1}^{n}\Delta(\alpha(x_{j}))$ . For $h\in\overline{vv^{*}{\mathcal{D}}^{c}}$ , Lemma 3(b) and Lemma 3 show $\alpha(h)\in\alpha(\overline{vv^{*}{\mathcal{D}}^{c}})\subseteq\overline{\alpha(vv^{*}){\mathcal{B}}}$ ; the induction hypothesis gives $\alpha(v^{*})y\alpha(v)\in{\mathcal{B}}_{1}$ . Given $h_{0}\in vv^{*}{\mathcal{D}}^{c}$ , write $h_{0}=vv^{*}k$ where $k\in{\mathcal{D}}^{c}$ . Using Lemma 3,

	$\displaystyle\alpha(v^{*})\>y\Delta(\alpha(x_{n+1}))\alpha(h_{0})\>\alpha(v)$	$\displaystyle=\alpha(v^{*})y\alpha(v)\>\theta_{\alpha(v)}(\Delta(\alpha(x_{n+1}))\alpha(h_{0}))$
		$\displaystyle=\alpha(v^{})y\alpha(v)\>\theta_{\alpha(v)}(\alpha(vv^{})\Delta(\alpha(x_{n+1}k)))$
		$\displaystyle=(\alpha(v^{})y\alpha(v))\>\left(\alpha(v)^{}\Delta(\alpha(x_{n+1}k))\alpha(v)\right)$
		$\displaystyle=(\alpha(v^{})y\alpha(v))\>\Delta(\alpha(v)^{}\alpha(x_{n+1}k)\alpha(v))\in{\mathcal{B}}_{1}.$

Therefore, for every $h\in\overline{vv^{*}{\mathcal{D}}^{c}}$ ,

(3.7)

\alpha(v^{*})\>y\Delta(\alpha(x_{n+1}))\alpha(h)\>\alpha(v)\in{\mathcal{B}}_{1}.

Note that $v=\lim_{n\rightarrow\infty}(vv^{*})^{1/n}v$ . Thus for any $d\in{\mathcal{D}}^{c}$ , choosing $h=d(vv^{*})^{1/n}\in\overline{vv^{*}{\mathcal{D}}^{c}}$ in (3.7), we find

(3.8)

\alpha(v)^{*}y\Delta(\alpha(x_{n+1}))\alpha(d)\alpha(v)=\lim_{n}\alpha(v)^{*}y\Delta(\alpha(x_{n+1}))\alpha(d(vv^{*})^{1/n})\alpha(v)\in{\mathcal{B}}_{1}.

As ${\mathcal{D}}^{c}$ is abelian (by (3.5)), $({\mathcal{C}},{\mathcal{D}}^{c})$ is a MASA inclusion. Lemma 3(a) shows that $({\mathcal{C}},{\mathcal{D}}^{c})$ is a regular MASA inclusion, and therefore has the AUP by [PittsNoApUnInC*Al, Theorem 2.5]. Let $(u_{\lambda})\subseteq{\mathcal{D}}^{c}$ be an approximate unit for ${\mathcal{C}}$ . Taking $d=u_{\lambda}$ in (3.8), we conclude

	$\displaystyle\alpha(v)^{*}y\Delta(\alpha(x_{n+1}))\alpha(v)$	$\displaystyle=\lim_{\lambda}\alpha(v)^{*}y\Delta(\alpha(x_{n+1}u_{\lambda}))\alpha(v)$
		$\displaystyle=\lim_{\lambda}\alpha(v)^{*}y\Delta(\alpha(x_{n+1}))\alpha(u_{\lambda})\alpha(v)\in{\mathcal{B}}_{1}.$

Thus (3.6) holds.

Since ${\mathcal{B}}_{1}$ is generated by $\Delta(\alpha({\mathcal{C}}))$ , (3.6) implies $\alpha(v)^{*}{\mathcal{B}}_{1}\alpha(v)\subseteq{\mathcal{B}}_{1}$ . As this holds for every $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ , we may replace $v$ with $v^{*}$ to obtain $\alpha(v){\mathcal{B}}_{1}\alpha(v)^{*}\subseteq{\mathcal{B}}_{1}$ . Therefore, $\alpha(v)\in{\mathcal{N}}({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ . $\diamondsuit$

By Claim 3, $\alpha({\mathcal{N}}({\mathcal{C}},{\mathcal{D}}))\cup{\mathcal{B}}_{1}\subseteq{\mathcal{N}}({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ , whence $({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ is a regular inclusion. Also,

\alpha({\mathcal{D}})\subseteq{\mathcal{B}}_{1}\subseteq{\mathcal{B}}.

Applying Lemma 2.4, $({\mathcal{B}},{\mathcal{B}}_{1})$ and $({\mathcal{B}}_{1},{\mathcal{D}},\alpha|_{\mathcal{D}})$ have the ideal intersection property. Thus, $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\,\subseteq)$ is an essential expansion of $({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ . By Lemma 3, $({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ has the faithful unique pseudo-expectation property.

Proposition 2.3 shows $({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ is a Cartan inclusion. Therefore, $({\mathcal{A}}_{1},{\mathcal{B}}_{1}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan package for $({\mathcal{C}},{\mathcal{D}})$ . As $({\mathcal{B}}_{1},{\mathcal{D}},\alpha|_{\mathcal{D}})$ has the ideal intersection property, $({\mathcal{A}}_{1},{\mathcal{B}}_{1}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is also an envelope for $({\mathcal{C}},{\mathcal{D}})$ . Thus $({\mathcal{A}}_{1},{\mathcal{B}}_{1}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ . ∎

The following will be used in the proof of Theorem 3; we will establish its converse in Proposition 4.2.

Proposition \the\numberby.

Suppose $({\mathcal{C}},{\mathcal{D}})$ is a regular inclusion such that $({\mathcal{D}}^{c},{\mathcal{D}})$ has the ideal intersection property and ${\mathcal{D}}^{c}$ is abelian. If $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}}^{c})$ , then $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is also a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ .

Proof.

Let $\Delta:{\mathcal{A}}\rightarrow{\mathcal{B}}$ be the conditional expectation.

Since ${\mathcal{D}}\subseteq{\mathcal{D}}^{c}$ , $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is an expansion of $({\mathcal{C}},{\mathcal{D}})$ . Lemma 3(a) and the regularity of $\alpha:({\mathcal{C}},{\mathcal{D}}^{c})\rightarrow({\mathcal{A}},{\mathcal{B}})$ show $\alpha:({\mathcal{C}},{\mathcal{D}})\rightarrow({\mathcal{A}},{\mathcal{B}})$ is a regular mapping. Since $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}}^{c})$ , ${\mathcal{B}}=C^{*}(\Delta(\alpha({\mathcal{C}})))$ and ${\mathcal{A}}=C^{*}({\mathcal{B}}\cup\alpha({\mathcal{C}}))$ . Thus, $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan package for $({\mathcal{C}},{\mathcal{D}})$ .

We have $\alpha({\mathcal{D}})\subseteq\alpha({\mathcal{D}}^{c})\subseteq{\mathcal{B}}$ . By definition of Cartan envelope, $({\mathcal{B}},{\mathcal{D}}^{c},\alpha)$ has the ideal intersection property. Since $({\mathcal{D}}^{c},{\mathcal{D}})$ has the ideal intersection property, so does $({\mathcal{B}},{\mathcal{D}},\alpha)$ (Lemma 2.4). Hence $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ . ∎

We turn to some technical preparations for the proof of the uniqueness statement for Cartan envelopes found in Theorem 3 below. We will use Lemma 3 repeatedly, often without comment. Proposition 3 shows how to produce useful $*$ -semigroups of intertwiners from $*$ -semigroups of normalizers; these semigroups are key to the uniqueness statement in Theorem 3. Lemma 3 exhibits some intertwiners associated to a normalizer and is the foundation for the proof of Proposition 3. Recall from Definition 2.1(e) that ${\mathcal{I}}({\mathcal{C}},{\mathcal{D}})$ denotes the collection of all intertwiners for $({\mathcal{C}},{\mathcal{D}})$ .

When $({\mathcal{C}},{\mathcal{D}})$ is an inclusion and $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ , it is convenient to use the notation,

(3.9)

R(v):=\overline{vv^{*}{\mathcal{D}}}\quad\text{and}\quad S(v):=\overline{v^{*}v{\mathcal{D}}}.

For an abelian $C^{*}$ -algebra ${\mathcal{D}}$ and an ideal $J\unlhd{\mathcal{D}}$ , let

(3.10)

J_{c}:=\{d\in J:\text{ there exists }h\in J\text{ such that }dh=d\}.

Observation \the\numberby.

$J_{c}$ is an algebraic ideal of ${\mathcal{D}}$ whose closure is $J$ . (In general, $J_{c}$ is not closed.)

Sketch of Proof.

For $k_{1},k_{2}\in{\mathcal{D}}$ with $k_{i}\geq 0$ , we let $k_{1}\vee k_{2}$ be the element of ${\mathcal{D}}$ whose Gelfand transform is $\hat{\mathcal{D}}\ni\sigma\mapsto\max\{\sigma(k_{1}),\sigma(k_{2})\}$ . Also, for $h\in{\mathcal{D}}$ , let $h^{\prime}$ be the element of ${\mathcal{D}}$ whose Gelfand transform is the function

\hat{\mathcal{D}}\ni\sigma\mapsto\min\{1,|\sigma(h)|\}.

Note that if $d$ and $h$ belong to $J$ , then $dh=d$ implies $dh^{\prime}=d$ ; observe that $h^{\prime}\in J$ because $h^{\prime}=f(h)$ , where $f(z)=\min\{1,|z|\}$ . Thus if $d_{1},d_{2}\in J_{c}$ and $h_{1},h_{2}\in J$ satisfy $d_{i}h_{i}=d_{i}$ , then $(d_{1}+d_{2})(h_{1}^{\prime}\vee h_{2}^{\prime})=d_{1}+d_{2}$ , so $J_{c}$ is closed under addition. That $J_{c}$ is an ideal in ${\mathcal{D}}$ is now obvious. The fact that $J_{c}$ is dense in $J$ will follow from Lemma 3(a) below. Indeed, for $d\in J$ , $R(d)\subseteq J$ and $R(d)_{c}\subseteq J_{c}$ . Since $d\in R(d)$ , Lemma 3(a) (whose proof is independent of Observation 3) gives $d\in\overline{R(d)_{c}}\subseteq\overline{J}_{c}$ , so $J\subseteq\overline{J_{c}}$ . ∎

Lemma \the\numberby.

Suppose $({\mathcal{C}},{\mathcal{D}})$ is an inclusion. Let $v,w\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ .

(a)

Suppose $vv^{*}\in{\mathcal{D}}$ (resp. $v^{*}v\in{\mathcal{D}}$ ). Then given $\varepsilon>0$ , there exists $k\in R(v)_{c}$ , (resp. $k\in S(v)_{c}$ ) such that $\left\lVert v-kv\right\rVert<\varepsilon$ (resp. $\left\lVert v-vk\right\rVert<\varepsilon$ ).
(b)

If $k\in R(v)_{c}$ (resp. $k\in S(v)_{c}$ ), then there exists $s\in R(v)_{c}$ (resp. $s\in S(v)_{c}$ ) such that $k=vv^{*}s$ (resp. $k=v^{*}vs$ ).
(c)

If $k\in R(v)_{c}$ (resp. $k\in S(v)_{c}$ ), then $kv\in{\mathcal{I}}({\mathcal{C}},{\mathcal{D}})$ (resp. $vk\in{\mathcal{I}}({\mathcal{C}},{\mathcal{D}})$ ).
(d)

For $h\in R(v)_{c}$ (resp. $h\in S(v)_{c}$ ), $\theta_{v}(h^{*})\in R(v^{*})_{c}$ (resp. $\theta_{v}^{-1}(h^{*})\in S(v^{*})_{c}$ ).
(e)

Suppose $h\in R(v)_{c}$ and $k\in R(w)_{c}$ (resp. $h\in S(v)_{c}$ and $k\in S(w)_{c})$ . Then $\theta_{v}^{-1}(\theta_{v}(h)k)\in R(vw)_{c}$ (resp. $\theta_{w}(h\theta_{w}^{-1}(k))\in S(vw)_{c}$ ).

Proof.

We give the proofs for $vv^{*}$ ; the proofs for $v^{*}v$ are similar.

(a) Suppose $vv^{*}\in{\mathcal{D}}$ and let $\varepsilon>0$ . Since $\widehat{vv^{*}}\in C_{0}(\hat{\mathcal{D}})$ , there exists a compact set $C\subseteq\hat{\mathcal{D}}$ such that $\sigma(vv^{*})<\varepsilon^{2}/4$ whenever $\sigma\in\hat{\mathcal{D}}\setminus C$ . Let

K:=C\cap\{\sigma\in\hat{\mathcal{D}}:\sigma(vv^{*})\geq\varepsilon^{2}/4\}.

Then $K$ is a compact subset of $\hat{\mathcal{D}}$ . By local compactness of $\hat{\mathcal{D}}$ , we may find open sets $U,V\subseteq\hat{\mathcal{D}}$ such that:

$\overline{U}$ and $\overline{V}$ are compact;
$K\subseteq U\subseteq\overline{U}\subseteq\{\sigma\in\hat{\mathcal{D}}:\sigma(vv^{*})\geq\varepsilon^{2}/9\}$ ; and
$\overline{U}\subseteq V\subseteq\overline{V}\subseteq\{\sigma\in\hat{\mathcal{D}}:\sigma(vv^{*})\geq\varepsilon^{2}/16\}$ .

Urysohn’s lemma ensures there exists $h,k\in\{d\in{\mathcal{D}}:d\geq 0\text{ and }\left\lVert d\right\rVert\leq 1\}$ such that $\hat{k}\equiv 1$ on $K$ , $\hat{k}$ vanishes off $U$ , $\hat{h}\equiv 1$ on $\overline{U}$ , and $\hat{h}$ vanishes off $V$ . Then $h,k\in R(v)$ and $hk=k$ , so that $k\in R(v)_{c}$ .

Note that for $\sigma\in\hat{\mathcal{D}}\setminus K$ , $\sigma(vv^{*})<\varepsilon^{2}/4$ . Also, $(kv-v)(kv-v)^{*}=k^{2}vv^{*}-2kvv^{*}+vv^{*}\in{\mathcal{D}}$ , so

\left\lVert kv-v\right\rVert^{2}=\sup_{\sigma\in\hat{\mathcal{D}}}\sigma(({kv-v})(kv-v)^{*})=\sup_{\sigma\in\hat{\mathcal{D}}}\sigma(vv^{*})(\sigma(k)-1)^{2}\leq\varepsilon^{2}/4<\varepsilon^{2}.

(b) Let

\operatorname{range}(v)=\{\sigma\in\hat{\mathcal{D}}:\sigma(vv^{*})\neq 0\}.

Then

R(v)=\{d\in{\mathcal{D}}:\hat{d}\text{ vanishes off }\operatorname{range}(v)\}\cong C_{0}(\operatorname{range}(v)).

As $k\in R(v)_{c}$ , we may find $h\in R(v)$ so that $kh=k$ . We may therefore find a compact set $F\subseteq\operatorname{range}(v)$ such that $|\sigma(h)|<1/2$ for $\sigma\in\hat{\mathcal{D}}\setminus F$ . As $kh=k$ , $\hat{k}$ vanishes off $F$ . Define a function $\delta$ on $\hat{\mathcal{D}}$ by

\delta(\sigma)=\begin{cases}\displaystyle\frac{\sigma(k)}{\sigma(vv^{*})}&\text{for }\sigma\in F;\\ 0&\text{for }\sigma\in\hat{\mathcal{D}}\setminus F.\end{cases}

Since $\widehat{vv^{*}}$ is bounded away from $0$ on $F$ , $\delta\in C_{0}(\hat{\mathcal{D}})$ , and we define $s\in{\mathcal{D}}$ by $\hat{s}=\delta$ . By construction, $k=vv^{*}s$ , and since $\hat{s}\hat{h}=\hat{s}$ , we find $s\in R(v)_{c}$ .

(c) Let $k\in R(v)_{c}$ and choose $h\in R(v)=\overline{vv^{*}{\mathcal{D}}}$ such that $kh=k$ . Set $u:=kv$ . Then for $d\in{\mathcal{D}}$ we have

du=k(dhv)=kv\theta_{v}(dh)=u\theta_{v}(dh).

Thus, ${\mathcal{D}}u\subseteq u{\mathcal{D}}$ .

For the reverse inclusion, note that $\theta_{v}(kh)=\theta_{v}(k)\theta_{v}(h)=\theta_{v}(k)$ , so that $\theta_{v}(k)\in S(v)_{c}$ . Thus, for $d\in{\mathcal{D}}$ we have

ud=kv\,(\theta_{v}(h)d)=\theta_{v}^{-1}(\theta_{v}(h)d)kv\in{\mathcal{D}}u.

Therefore $kv\in{\mathcal{I}}({\mathcal{C}},{\mathcal{D}})$ , as claimed.

(d) By part (b), we may write $h=vv^{*}s$ for $s\in R(v)_{c}$ . Then $\theta_{v}(h^{*})=v^{*}s^{*}v\in S(v)_{c}=R(v^{*})_{c}$ .

(e) Using part (b) we may choose $s\in R(v)_{c}$ , $t\in R(w)_{c}$ so that $h=(vv^{*})^{2}s$ and $k=ww^{*}t$ . Then

	$\displaystyle\theta_{v}^{-1}(\theta_{v}(h)k)$	$\displaystyle=\theta_{v}^{-1}(\theta_{v}((vv^{})^{2}s)k)=\theta_{v}^{-1}(v^{}(vv^{*}s)vk)$
		$\displaystyle=vv^{}svkv^{}=(vv^{})sv(tww^{})v^{*}$
		$\displaystyle=s(vv^{})vtww^{}v^{*}$
		$\displaystyle=s\,(vtv^{})(vw)(vw)^{}\in(vw)(vw)^{*}{\mathcal{D}}\subseteq R(vw).$

By definition, we may find $x\in R(v)$ and $y\in R(w)$ so that $h=hx$ and $k=ky$ . Since $\overline{R(v)_{c}}=R(v)$ and $\overline{R(w)_{c}}=R(w)$ (see Observation 3), we may find sequences $(x_{k})$ in $R(v)_{c}$ and $(y_{k})$ in $R(w)_{c}$ which converge to $x$ and $y$ respectively. The previous calculations applied to $x_{k}$ and $y_{k}$ show $\theta_{v}^{-1}(\theta_{v}(x_{k})y_{k})\in R(vw)$ . Taking limits yields $\theta_{v}^{-1}(\theta_{v}(x)y)\in R(vw)$ . But then

\theta_{v}^{-1}(\theta_{v}(h)k)\theta_{v}^{-1}(\theta_{v}(x)y)=\theta_{v}^{-1}(\theta_{v}(h)k),

showing that $\theta_{v}^{-1}(\theta_{v}(h)k)\in R(vw)_{c}$ . ∎

Proposition \the\numberby.

Let $({\mathcal{C}},{\mathcal{D}})$ be an inclusion. Let ${\mathcal{P}}\subseteq{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ be a $*$ -semigroup and put

{\mathcal{Q}}:=\{vh:v\in{\mathcal{P}}\text{ and }h\in S(v)_{c}\}\quad\text{and}\quad{\mathcal{S}}:={\mathcal{Q}}\cup{\mathcal{D}}.

The following statements hold.

(a)

${\mathcal{Q}}$ and ${\mathcal{S}}$ are $*$ -semigroups and both are contained in ${\mathcal{N}}({\mathcal{C}},{\mathcal{D}})\cap{\mathcal{I}}({\mathcal{C}},{\mathcal{D}})$ .
(b)

When $({\mathcal{C}},{\mathcal{D}})$ has the approximate unit property and the $*$ -algebra generated by ${\mathcal{P}}\cup{\mathcal{D}}$ is dense in ${\mathcal{C}}$ , $\operatorname{span}{\mathcal{S}}$ is a dense $*$ -subalgebra of ${\mathcal{C}}$ .

Proof.

(a) That ${\mathcal{Q}}\subseteq{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ is clear and Lemma 3(c) shows ${\mathcal{Q}}\subseteq{\mathcal{I}}({\mathcal{C}},{\mathcal{D}})$ . For $v,w\in{\mathcal{P}}$ , $h\in S(v)_{c}$ and $k\in S(w)_{c}$ ,

vhwk=vh\theta_{w}^{-1}(k)w=vw\theta_{w}(h\theta_{w}^{-1}(k))\in{\mathcal{Q}},

by Lemma 3(e). Next, Lemma 3(d) implies ${\mathcal{Q}}$ is closed under adjoints: indeed, $(vh)^{*}=h^{*}v^{*}=v^{*}\theta_{v}^{-1}(h^{*})\in{\mathcal{Q}}$ . Thus ${\mathcal{Q}}$ is a $*$ -semigroup.

Turning our attention to ${\mathcal{S}}$ , let $v\in{\mathcal{P}}$ and $h\in S(v)_{c}$ . For $d\in{\mathcal{D}}$ , the fact that $vh\in{\mathcal{I}}({\mathcal{C}},{\mathcal{D}})$ gives $d(vh)\in{\mathcal{Q}}$ and $(vh)d\in{\mathcal{Q}}$ . Thus, ${\mathcal{Q}}{\mathcal{D}}\cup{\mathcal{D}}{\mathcal{Q}}\subseteq{\mathcal{Q}}$ . It follows that ${\mathcal{S}}$ is a $*$ -semigroup. Since ${\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ and ${\mathcal{I}}({\mathcal{C}},{\mathcal{D}})$ are $*$ -semigroups containing ${\mathcal{D}}$ , ${\mathcal{S}}\subseteq{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})\cap{\mathcal{I}}({\mathcal{C}},{\mathcal{D}})$ .

(b) Note that $({\mathcal{C}},{\mathcal{D}})$ is a regular inclusion because ${\mathcal{P}}\cup{\mathcal{D}}$ generates a $*$ -semigroup of normalizers. As $({\mathcal{C}},{\mathcal{D}})$ has the AUP, [PittsNoApUnInC*Al, Observation 1.3] shows $v^{*}v\in{\mathcal{D}}$ for every $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ . Lemma 3(a) shows $v\in\overline{{\mathcal{S}}}$ for every $v\in{\mathcal{P}}$ . Thus $\overline{\operatorname{span}{\mathcal{S}}}$ contains ${\mathcal{P}}\cup{\mathcal{D}}$ , whence $\overline{\operatorname{span}{\mathcal{S}}}={\mathcal{C}}$ . Finally, as ${\mathcal{S}}$ is a $*$ -semigroup, $\operatorname{span}{\mathcal{S}}$ is a $*$ -algebra. ∎

By taking ${\mathcal{P}}={\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ , we see that for a broad class of inclusions, the supply of intertwiners is large.

Corollary \the\numberby.

If $({\mathcal{C}},{\mathcal{D}})$ is a regular inclusion with the AUP, then ${\mathcal{I}}({\mathcal{C}},{\mathcal{D}})\cap{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ has dense span in ${\mathcal{C}}$ .

We come to a main result, which is the promised characterization of those regular inclusions which have a Cartan envelope. This characterization significantly extends [PittsStReInII, Theorem 5.2], which dealt with unital inclusions.

Theorem \the\numberby (c.f. [PittsStReInII, Theorem 5.2]).

Let $({\mathcal{C}},{\mathcal{D}})$ be a regular inclusion. The following are equivalent.

(a)

$({\mathcal{C}},{\mathcal{D}})$ has a Cartan envelope.
(b)

$({\mathcal{C}},{\mathcal{D}})$ has the faithful unique pseudo-expectation property.
(c)

${\mathcal{D}}^{c}$ is abelian and both $({\mathcal{C}},{\mathcal{D}}^{c})$ and $({\mathcal{D}}^{c},{\mathcal{D}})$ have the ideal intersection property.

When $({\mathcal{C}},{\mathcal{D}})$ satisfies any of conditions (a)–(c), $({\mathcal{C}},{\mathcal{D}})$ is weakly non-degenerate and the following statements hold.

Uniqueness:: If for $j=1,2$ , $({\mathcal{A}}_{j},{\mathcal{B}}_{j}\hbox{\,:\hskip-1.0pt:\,}\alpha_{j})$ are Cartan envelopes for $({\mathcal{C}},{\mathcal{D}})$ , there exists a unique regular $*$ -isomorphism $\psi:({\mathcal{A}}_{1},{\mathcal{B}}_{1})\rightarrow({\mathcal{A}}_{2},{\mathcal{B}}_{2})$ such that $\psi\circ\alpha_{1}=\alpha_{2}$ .
Minimality:: If $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan package for $({\mathcal{C}},{\mathcal{D}})$ , there is an ideal ${\mathfrak{J}}\subseteq{\mathcal{A}}$ such that ${\mathfrak{J}}\cap\alpha({\mathcal{C}})=\{0\}$ and, letting $q:{\mathcal{A}}\rightarrow{\mathcal{A}}/{\mathfrak{J}}$ denote the quotient map, $({\mathcal{A}}/{\mathfrak{J}},{\mathcal{B}}/({\mathfrak{J}}\cap{\mathcal{B}})\hbox{\,:\hskip-1.0pt:\,}q\circ\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ .

The proof of Theorem 3 is organized as follows. We show (a) $\Rightarrow$ (b) $\Rightarrow$ (c) $\Rightarrow$ (a) and weak non-degeneracy for $({\mathcal{C}},{\mathcal{D}})$ , then the uniqueness statement. The proof of the minimality statement may be found after the proof of Proposition 3. During all parts of the proof of Theorem 3, assume that an injective envelope $(I({\mathcal{D}}),\iota)$ for ${\mathcal{D}}$ has been fixed.

(a) $\Rightarrow$ (b).

Since $({\mathcal{C}},{\mathcal{D}})$ has a Cartan envelope, Lemma 3 shows it is weakly non-degenerate.

Suppose $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ . By Proposition 2.3, $({\mathcal{A}},{\mathcal{B}})$ has the faithful unique pseudo-expectation property. By definition, $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is an essential expansion of $({\mathcal{C}},{\mathcal{D}})$ , so Lemma 3 shows $({\mathcal{C}},{\mathcal{D}})$ has the faithful unique pseudo-expectation property. ∎

(b) $\Rightarrow$ (c).

By Proposition 2.3, ${\mathcal{D}}^{c}$ is abelian, so ${\mathcal{D}}\subseteq{\mathcal{D}}^{c}\subseteq{\mathcal{C}}$ . Corollary 2.3 shows $({\mathcal{D}}^{c},{\mathcal{D}})$ has the faithful unique pseudo-expectation property, whence $({\mathcal{D}}^{c},{\mathcal{D}})$ has the ideal intersection property by Proposition 2.4.

Let $E:{\mathcal{C}}\rightarrow I({\mathcal{D}})$ be the faithful unique pseudo-expectation. Since $\iota_{2}:=E|_{{\mathcal{D}}^{c}}$ is a pseudo-expectation for $({\mathcal{D}}^{c},{\mathcal{D}})$ , Proposition 2.4 shows $\iota_{2}:=E|_{{\mathcal{D}}^{c}}$ is the only pseudo-expectation for $({\mathcal{D}}^{c},{\mathcal{D}})$ , and $(I({\mathcal{D}}),\iota_{2})$ is an injective envelope for ${\mathcal{D}}^{c}$ .

It follows from Lemma 3(a) that $({\mathcal{C}},{\mathcal{D}}^{c})$ is regular MASA inclusion, whence $({\mathcal{C}},{\mathcal{D}}^{c})$ has the unique pseudo-expectation property by Theorem 3. Letting $F:{\mathcal{C}}\rightarrow I({\mathcal{D}})$ be the pseudo-expectation for $({\mathcal{C}},{\mathcal{D}}^{c})$ relative to $(I({\mathcal{D}}),\iota_{2})$ , we see that $F|_{{\mathcal{D}}^{c}}$ is a pseudo-expectation for $({\mathcal{D}}^{c},{\mathcal{D}})$ , so $F|_{{\mathcal{D}}^{c}}=\iota_{2}$ . But then $F|_{\mathcal{D}}=\iota_{2}|_{{\mathcal{D}}}=\iota$ , hence $F$ is a pseudo-expectatation for $({\mathcal{C}},{\mathcal{D}})$ . This forces $F=E$ . As $E$ is faithful, we conclude that $({\mathcal{C}},{\mathcal{D}}^{c})$ has the faithful unique pseudo-expectation property. Theorem 3 now shows $({\mathcal{C}},{\mathcal{D}}^{c})$ has the ideal intersection property. ∎

(c) $\Rightarrow$ (a).

For notational convenience, let ${\mathcal{M}}:={\mathcal{D}}^{c}$ . By Lemma 2.2(e), $({\mathcal{C}},{\mathcal{D}})$ is weakly non-degenerate; thus Lemma 2.2(e) shows $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is a unital inclusion, and Corollary 2.2 gives $\tilde{\mathcal{D}}\subseteq\tilde{\mathcal{M}}\subseteq\tilde{\mathcal{C}}$ . By [PittsNoApUnInC*Al, Theorem 2.5], $({\mathcal{C}},{\mathcal{M}})$ has the AUP, so

{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})\stackrel{{\scriptstyle\ref{relcom}\eqref{relcom1}}}{{\subseteq}}{\mathcal{N}}({\mathcal{C}},{\mathcal{M}})\stackrel{{\scriptstyle\ref{relcom}\eqref{relcom3}}}{{\subseteq}}{\mathcal{N}}(\tilde{\mathcal{C}},\tilde{\mathcal{M}}).

Therefore, $({\mathcal{C}},{\mathcal{M}})$ is a regular MASA inclusion; it has the ideal intersection property by hypothesis. It follows from Lemma 2.4 that $(\tilde{\mathcal{C}},\tilde{\mathcal{M}})$ is also a regular MASA inclusion with the ideal intersection property.

By [PittsStReInII, Theorem 5.2], $(\tilde{\mathcal{C}},\tilde{\mathcal{M}})$ has a Cartan envelope $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ . Since $({\mathcal{B}},\alpha(\tilde{\mathcal{M}}))$ and $(\tilde{\mathcal{M}},{\mathcal{M}})$ have the ideal intersection property, so does $({\mathcal{B}},\alpha({\mathcal{M}}))$ (Lemma 2.4). Thus $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha|_{{\mathcal{C}}})$ is an essential and regular Cartan expansion for $({\mathcal{C}},{\mathcal{M}})$ . Proposition 3 shows that $({\mathcal{C}},{\mathcal{M}})$ has a Cartan envelope. Finally, Proposition 3 implies $({\mathcal{C}},{\mathcal{D}})$ has a Cartan envelope. ∎

Proof of the Uniqueness Statement in Theorem 3.

The proof follows the same pattern as the proof of [PittsStReInII, Proposition 5.24]. Suppose for $i=1,2$ that $({\mathcal{A}}_{i},{\mathcal{B}}_{i}\hbox{\,:\hskip-1.0pt:\,}\alpha_{i})$ are Cartan envelopes for $({\mathcal{C}},{\mathcal{D}})$ . The equivalence of statements (a) and (b) in Theorem 3 shows $({\mathcal{C}},{\mathcal{D}})$ has the faithful unique pseudo-expectation property. Since $({\mathcal{A}}_{i},{\mathcal{B}}_{i}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is an essential expansion of $({\mathcal{C}},{\mathcal{D}})$ , $({\mathcal{B}}_{i},{\mathcal{D}},\alpha_{i}|_{{\mathcal{D}}})$ has the ideal intersection property, so it has the faithful unique pseudo-expectation property (Proposition 2.4). Thus there exist unique $*$ -monomorphisms $\iota_{i}:{\mathcal{B}}_{i}\rightarrow I({\mathcal{D}})$ such that $\iota_{i}\circ(\alpha_{i}|_{\mathcal{D}})=\iota$ . As $\iota_{i}\circ\Delta_{i}\circ\alpha_{i}$ is a pseudo-expectation for $({\mathcal{C}},{\mathcal{D}})$ , we obtain

(3.11)

\iota_{1}\circ\Delta_{1}\circ\alpha_{1}=E=\iota_{2}\circ\Delta_{2}\circ\alpha_{2}.

But ${\mathcal{B}}_{i}=C^{*}(\Delta_{i}(\alpha_{i}({\mathcal{C}})))$ , so $\iota_{i}({\mathcal{B}}_{i})=C^{*}(E({\mathcal{C}}))$ . As $\iota_{i}$ is a $*$ -monomorphism and is the unique pseudo-expectation for $({\mathcal{B}}_{i},{\mathcal{D}},\alpha_{i}|_{{\mathcal{D}}})$ , we conclude $\psi_{\mathcal{B}}:=\iota_{2}^{-1}\circ\iota_{1}$ is the unique $*$ -isomorphism of ${\mathcal{B}}_{1}$ onto ${\mathcal{B}}_{2}$ satisfying

\psi_{\mathcal{B}}\circ\alpha_{1}|_{{\mathcal{D}}}=\alpha_{2}|_{\mathcal{D}}.

The following diagram illustrates the maps just discussed.

(3.12)

We shall extend $\psi_{\mathcal{B}}$ to the desired isomorphism of ${\mathcal{A}}_{1}$ onto ${\mathcal{A}}_{2}$ . We do this in stages.

By the regularity of the map $\alpha_{i}$ ,

{\mathcal{P}}_{i}:=\alpha_{i}({\mathcal{N}}({\mathcal{C}},{\mathcal{D}}))

is a $*$ -semigroup contained in ${\mathcal{N}}({\mathcal{A}}_{i},{\mathcal{B}}_{i})$ . Since Cartan inclusions have the AUP, Lemma 3(b) shows that for $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ , $\alpha_{i}(v^{*}v)\in{\mathcal{B}}_{i}$ . Also, (3.11) gives

(3.13)

\psi_{\mathcal{B}}\circ\Delta_{1}\circ\alpha_{1}=\Delta_{2}\circ\alpha_{2}.

Put

{\mathcal{Q}}_{i}:=\{\alpha_{i}(v)h:v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}}),h\in(\overline{\alpha_{i}(v^{*}v){\mathcal{B}}_{i}})_{c}\}\quad\text{and}\quad{\mathcal{S}}_{i}:={\mathcal{B}}_{i}\cup{\mathcal{Q}}_{i}.

Since $({\mathcal{A}}_{i},{\mathcal{B}}_{i})$ has the AUP and ${\mathcal{A}}_{i}$ is generated by ${\mathcal{B}}_{i}\cup{\mathcal{P}}_{i}$ , Proposition 3 shows that ${\mathcal{Q}}_{i}$ and ${\mathcal{S}}_{i}$ are $*$ -semigroups contained in ${\mathcal{N}}({\mathcal{A}}_{i},{\mathcal{B}}_{i})$ and $\operatorname{span}{\mathcal{S}}_{i}$ is a dense $*$ -subalgebra of ${\mathcal{A}}_{i}$ .¹¹1The proof of [PittsStReInII, Proposition 5.24] states without proof that the set ${\mathcal{M}}=\{\alpha_{i}(v)h:v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}}),h\in\overline{\alpha_{i}(v^{*}v){\mathcal{D}}_{i}}\}$ is a $*$ -semigroup and uses ${\mathcal{M}}$ instead of ${\mathcal{Q}}$ . (Note that $\operatorname{span}{\mathcal{Q}}_{i}$ is also a $*$ -subalgebra of $\operatorname{span}{\mathcal{S}}_{i}$ .)

The first stage in extending $\psi_{\mathcal{B}}$ is to extend it to an isomorphism $\psi_{\mathcal{Q}}:\operatorname{span}{\mathcal{Q}}_{1}\rightarrow\operatorname{span}{\mathcal{Q}}_{2}$ . Let $n\in{\mathbb{N}}$ and suppose for $1\leq k\leq n$ , $v_{k}\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ and $h_{k}\in(\overline{\alpha_{1}(v_{k}^{*}v_{k}){\mathcal{B}}_{1}})_{c}$ . Then

	$\displaystyle 0=\sum_{k=1}^{n}\alpha_{1}(v_{k})h_{k}$	$\displaystyle\iff\Delta_{1}\left(\sum_{k,\ell=1}^{n}h_{k}^{}\alpha_{1}(v_{k}^{}v_{\ell})h_{\ell}\right)=0$
		$\displaystyle\iff\psi_{\mathcal{B}}\left(\sum_{k,\ell=1}^{n}h_{k}^{}\Delta_{1}(\alpha_{1}(v_{k}^{}v_{\ell}))h_{\ell}\right)=0$
		$\displaystyle\stackrel{{\scriptstyle\eqref{!pscharu1}}}{{\iff}}\sum_{k,\ell=1}^{n}\psi_{\mathcal{B}}(h_{k}^{})\Delta_{2}(\alpha_{2}(v_{k}^{}v_{\ell}))\psi_{\mathcal{B}}(h_{\ell})=0$
		$\displaystyle\iff\Delta_{2}\left(\sum_{k,\ell=1}^{n}\psi_{\mathcal{B}}(h_{k}^{})\alpha_{2}(v_{k}^{}v_{\ell})\psi_{\mathcal{B}}(h_{\ell})\right)=0$
		$\displaystyle\iff\sum_{k=1}^{n}\alpha_{2}(v_{k})\psi_{\mathcal{B}}(h_{k})=0.$

Therefore, $\psi_{\mathcal{B}}$ extends uniquely to a $*$ -isomorphism $\psi_{\mathcal{Q}}:\operatorname{span}({\mathcal{Q}}_{1})\rightarrow\operatorname{span}({\mathcal{Q}}_{2})$ given by

\sum_{k=1}^{n}\alpha_{1}(v_{k})h_{k}\mapsto\sum_{k=1}^{n}\alpha_{2}(v_{k})\psi(h_{k}).

For $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ and $h\in(\overline{\alpha_{1}(v^{*}v){\mathcal{B}}_{1}})_{c}$ ,

	$\displaystyle\psi_{\mathcal{B}}(\Delta_{1}(\alpha_{1}(v)h))$	$\displaystyle=\iota_{2}^{-1}(\iota_{1}(\Delta_{1}(\alpha_{1}(v))h))=\iota_{2}^{-1}(E(v)\iota_{1}(h))$
		$\displaystyle=\iota_{2}^{-1}(E(v))\psi_{\mathcal{B}}(h)=\iota_{2}^{-1}(\iota_{2}(\Delta_{2}(\alpha_{2}(v))))\psi_{\mathcal{B}}(h)$
		$\displaystyle=\Delta_{2}(\alpha_{2}(v)\psi_{\mathcal{B}}(h))=\Delta_{2}(\psi_{\mathcal{Q}}(\alpha_{1}(v)h)).$

It follows that

(3.14)

\psi_{\mathcal{B}}\circ\Delta_{1}|_{\operatorname{span}{\mathcal{Q}}_{1}}=\Delta_{2}\circ\psi_{\mathcal{Q}}.

Next we extend $\psi_{\mathcal{Q}}$ to a $*$ -isomorphism $\psi_{\mathcal{S}}:\operatorname{span}{\mathcal{S}}_{1}\rightarrow\operatorname{span}{\mathcal{S}}_{2}$ . For $b_{1}\in{\mathcal{B}}_{1}$ and $t_{1}\in\operatorname{span}{\mathcal{Q}}_{1}$ we have

\displaystyle\begin{split}0=b_{1}+t_{1}\iff&\Delta_{1}((b_{1}+t_{1})^{*}(b_{1}+t_{1}))=0\\ \iff&b_{1}^{*}b_{1}+b_{1}^{*}\Delta_{1}(t_{1})+\Delta_{1}(t_{1})^{*}b_{1}+\Delta_{1}(t_{1}^{*}t_{1})=0\\ \stackrel{{\scriptstyle\eqref{psiIDelta}}}{{\iff}}&\psi_{\mathcal{B}}(b_{1}^{*}b_{1})+\psi_{\mathcal{B}}(b_{1})^{*}\Delta_{2}(\psi_{\mathcal{Q}}(t_{1}))+\Delta_{2}(\psi_{\mathcal{Q}}(t_{1}^{*}))\psi_{\mathcal{B}}(b_{1})\\ &\qquad+\Delta_{2}(\psi_{\mathcal{Q}}(t_{1}^{*}t_{1}))=0\\ \iff&\Delta_{2}((\psi_{\mathcal{B}}(b_{1})+\psi_{\mathcal{Q}}(t_{1}))^{*}(\psi_{\mathcal{B}}(b_{1})+\psi_{\mathcal{Q}}(t_{1})))=0\\ \iff&\psi_{\mathcal{B}}(b_{1})+\psi_{Q}(t_{1})=0.\end{split}

Therefore, the map $\operatorname{span}{\mathcal{S}}_{1}={\mathcal{B}}_{1}+\operatorname{span}{\mathcal{Q}}_{1}\ni b_{1}+t_{1}\mapsto\psi_{\mathcal{B}}(b_{1})+\psi_{Q}(t_{1})\in\operatorname{span}{\mathcal{S}}_{2}$ is well-defined, so we obtain a $*$ -isomorphism $\psi_{\mathcal{S}}:\operatorname{span}{\mathcal{S}}_{1}\rightarrow\operatorname{span}{\mathcal{S}}_{2}$ .

Now let $\tilde{\mathcal{S}}_{i}:=u_{{\mathcal{A}}_{i}}({\mathcal{S}}_{i})\cup{\mathbb{C}}I_{\tilde{\mathcal{A}}_{i}}\subseteq\tilde{\mathcal{A}}_{i}$ . Then $\tilde{\mathcal{S}}_{i}$ is a MASA skeleton for $(\tilde{\mathcal{A}}_{i},\tilde{\mathcal{B}}_{i})$ in the sense of [PittsStReInI, Definition 3.1] (see also [PittsStReInI, Definition 1.7]). Note that $\psi_{\mathcal{S}}$ extends uniquely to a $*$ -isomorphism $\tilde{\psi}_{\mathcal{S}}:\operatorname{span}\tilde{\mathcal{S}}_{1}\rightarrow\operatorname{span}\tilde{\mathcal{S}}_{2}$ . Since $(\tilde{\mathcal{A}}_{i},\tilde{\mathcal{B}}_{i})$ is a Cartan inclusion, we may argue as in the proof of [PittsStReInII, Proposition 5.24] to conclude that $\tilde{\psi}_{\mathcal{S}}$ extends to a regular $*$ -isomorphism $\tilde{\psi}$ of $(\tilde{\mathcal{A}}_{1},\tilde{\mathcal{B}}_{1})$ onto $(\tilde{\mathcal{A}}_{2},\tilde{\mathcal{B}}_{2})$ . Thus $\psi:=\tilde{\psi}|_{{\mathcal{A}}_{1}}$ is an isomorphism of ${\mathcal{A}}_{1}$ onto ${\mathcal{A}}_{2}$ . The constructions show $\psi\circ\alpha_{1}=\alpha_{2}$ .

Suppose $\psi^{\prime}:{\mathcal{A}}_{1}\rightarrow{\mathcal{A}}_{2}$ is a $*$ -isomorphism such that $\psi^{\prime}\circ\alpha_{1}=\alpha_{2}$ . The uniqueness assertion for $\psi_{\mathcal{B}}$ shows $\psi^{\prime}|_{{\mathcal{B}}_{1}}=\psi|_{{\mathcal{B}}_{1}}$ . Examining each stage of the construction of $\psi|_{\mathcal{S}}$ , we obtain $\psi^{\prime}|_{{\mathcal{S}}_{1}}=\psi|_{{\mathcal{S}}_{1}}$ . As $\operatorname{span}{\mathcal{S}}_{i}$ is dense in ${\mathcal{A}}_{i}$ , this forces $\psi^{\prime}=\psi$ and completes the proof of the uniqueness assertion. ∎

Before turning to the proof of minimality, we need a bit of terminology and a fact about abelian $C^{*}$ -algebras. Suppose ${\mathcal{A}}$ is an arbitrary $C^{*}$ -algebra and $J\unlhd{\mathcal{A}}$ . Set

J^{\perp}:=\{a\in{\mathcal{A}}:aj=ja=0\text{ for all }j\in J\}

and recall that $J$ is called a regular ideal if $J^{\perp\perp}=J$ , where $J^{\perp\perp}$ means $(J^{\perp})^{\perp}$ . (Because $J$ is an ideal, so is $J^{\perp}$ .)

Next, let $X$ be a locally compact Hausdorff space and suppose $J\unlhd C_{0}(X)$ . For an open set $G\subseteq X$ , we write $G^{\perp}$ for the interior of $X\setminus G$ and write $\operatorname{supp}(J):=\{x\in X:f(x)\neq 0\text{ for some }f\in J\}$ . The proof of the following is left to the reader.

Fact \the\numberby.

Let $J\unlhd C_{0}(X)$ and put $G:=\operatorname{supp}(J)$ . Then $\operatorname{supp}(J^{\perp\perp})=G^{\perp\perp}$ , $G$ is a dense open subset of $G^{\perp\perp}$ , and $(J^{\perp\perp},J)$ has the ideal intersection property.

We require the following invariance type result; it is the extension of [PittsStReInI, Proposition 3.14] to our context.

Proposition \the\numberby (c.f. [PittsStReInI, Proposition 3.14]).

Suppose $({\mathcal{C}},{\mathcal{D}})$ is a regular inclusion having the unique faithful pseudo-expectation property, and let $E:{\mathcal{C}}\rightarrow I({\mathcal{D}})$ be the pseudo-expectation. Fix $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ , and let $\overline{\theta}_{v}$ be the result of applying Lemma 2.3 to $\theta_{v}:\overline{vv^{*}{\mathcal{D}}}\rightarrow\overline{v^{*}v{\mathcal{D}}}$ . Then for every $x\in{\mathcal{C}}$ ,

(3.15)

E(v^{*}xv)=\overline{\theta}_{v}(E(vv^{*}x)).

Proof.

To simplify notation, throughout the proof we will identify ${\mathcal{A}}$ with its image under $u_{\mathcal{A}}$ in $\tilde{\mathcal{A}}$ , and we will write

{\mathcal{M}}:={\mathcal{D}}^{c}.

We wish to apply [PittsStReInI, Proposition 3.14] to $(\tilde{\mathcal{C}},\tilde{\mathcal{M}})$ . To do so, we show that $(\tilde{\mathcal{C}},\tilde{\mathcal{M}})$ is a regular and unital MASA inclusion with the faithful unique pseudo-expectation property, that $(I({\mathcal{D}}),\tilde{E}|_{\tilde{\mathcal{M}}})$ is an injective envelope for $\tilde{\mathcal{M}}$ , and that $\tilde{E}:\tilde{\mathcal{C}}\rightarrow I({\mathcal{D}})$ is the pseudo-expectation for $(\tilde{\mathcal{C}},\tilde{\mathcal{M}})$ .

By Corollary 2.3, $({\mathcal{C}},{\mathcal{D}})$ is weakly non-degenerate, so Lemma 2.2(e) shows $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is a unital inclusion. Also, Corollary 2.2 gives $\tilde{\mathcal{D}}\subseteq\tilde{\mathcal{M}}\subseteq\tilde{\mathcal{C}}$ . In particular, $(\tilde{\mathcal{C}},\tilde{\mathcal{M}})$ is a unital inclusion. Proposition 2.3 shows ${\mathcal{M}}$ is abelian, Corollary 2.3 shows $({\mathcal{M}},{\mathcal{D}})$ has the faithful unique pseudo-expectation property, and by Proposition 2.4, $(I({\mathcal{D}}),E|_{{\mathcal{M}}})$ is an injective envelope for ${\mathcal{M}}$ . Thus, $(I({\mathcal{D}}),\tilde{E}|_{\tilde{\mathcal{M}}})$ is an injective envelope for $\tilde{\mathcal{M}}$ . Next, $({\mathcal{C}},{\mathcal{M}})$ is a regular MASA inclusion by Lemma 3(a). Therefore $({\mathcal{C}},{\mathcal{M}})$ has the AUP by [PittsNoApUnInC*Al, Corollary 2.6], so $(\tilde{\mathcal{C}},\tilde{\mathcal{M}})$ is a unital, regular MASA inclusion (Lemma 3(e)). Corollary 2.3 shows $\tilde{E}$ is the pseudo-expectation for $(\tilde{\mathcal{C}},\tilde{\mathcal{M}})$ and $\tilde{E}$ is faithful by Theorem 2.3.

Using parts (a) and (e) of Lemma 3, we obtain

{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})\subseteq{\mathcal{N}}({\mathcal{C}},{\mathcal{M}})\subseteq{\mathcal{N}}(\tilde{\mathcal{C}},\tilde{{\mathcal{M}}}).

Thus by [PittsStReInI, Proposition 3.14], for $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ and $x\in{\mathcal{C}}$ ,

\tilde{E}(v^{*}xv)=\overline{\theta}_{v}(\tilde{E}(vv^{*}x)),

that is, $E(v^{*}xv)=\overline{\theta}_{v}(E(vv^{*}x))$ . ∎

Proof of the Minimality Statement in Theorem 3.

Suppose that $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan package for $({\mathcal{C}},{\mathcal{D}})$ and $\Delta:{\mathcal{A}}\rightarrow{\mathcal{B}}$ is the conditional expectation. By injectivity of $I({\mathcal{D}})$ , there exists a $*$ -homomorphism $\tilde{\gamma}:\tilde{\mathcal{B}}\rightarrow I({\mathcal{D}})$ such that

(3.16)

\iota=\tilde{\gamma}\circ\tilde{\alpha}|_{\tilde{\mathcal{D}}}.

To show $\tilde{\gamma}$ is the unique such $*$ -homomorphism, suppose $\tilde{\tau}:\tilde{\mathcal{B}}\rightarrow I({\mathcal{D}})$ is another $*$ -homomorphism such that $\iota=\tilde{\tau}\circ\tilde{\alpha}|_{\tilde{\mathcal{D}}}$ . Note that $\tilde{\gamma}\circ\tilde{\Delta}\circ\tilde{\alpha}|_{\tilde{\mathcal{D}}}=\iota$ , so $\tilde{\gamma}\circ\tilde{\Delta}\circ\tilde{\alpha}$ is a pseudo-expectation for $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ . Likewise $\tilde{\tau}\circ\tilde{\Delta}\circ\tilde{\alpha}$ is a pseudo-expectation for $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ . Since $({\mathcal{C}},{\mathcal{D}})$ has the unique pseudo-expectation property, so does $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ (Corollary 2.3). Therefore,

\tilde{\gamma}\circ\tilde{\Delta}\circ\tilde{\alpha}=\tilde{\tau}\circ\tilde{\Delta}\circ\tilde{\alpha}.

The definition of Cartan package shows ${\mathcal{B}}=C^{*}(\Delta(\alpha({\mathcal{C}})))$ , hence $\tilde{\mathcal{B}}=C^{*}(\tilde{\Delta}(\tilde{\alpha}(\tilde{\mathcal{C}}))$ . Thus $\tilde{\tau}=\tilde{\gamma}$ , which shows $\tilde{\gamma}$ is the unique $*$ -homomorphism satisfying (3.16).

Let $E:{\mathcal{C}}\rightarrow I({\mathcal{D}})$ be the (unique and faithful) pseudo-expectation. Since $\tilde{\gamma}\circ\tilde{\Delta}\circ\tilde{\alpha}$ is the pseudo-expectation for $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ , Lemma 2.3 shows $\tilde{E}=\tilde{\gamma}\circ\tilde{\Delta}\circ\tilde{\alpha}$ and $E=\tilde{\gamma}\circ\tilde{\Delta}\circ\tilde{\alpha}\circ u_{\mathcal{C}}$ . Define

\gamma:=\tilde{\gamma}\circ u_{\mathcal{B}}.

Using (2.1.4), we find

(3.17)

E=\tilde{\gamma}\circ u_{\mathcal{B}}\circ\Delta\circ\alpha=\gamma\circ\Delta\circ\alpha.

Claim \the\numberby.

Put

{\mathfrak{J}}:=\{a\in{\mathcal{A}}:\Delta(a^{*}a)\in\ker\gamma\}.

Then ${\mathfrak{J}}$ is a regular ideal in ${\mathcal{A}}$ such that ${\mathfrak{J}}\cap{\mathcal{B}}=\ker\gamma$ .

Proof of Claim 3.

To verify Claim 3, we use [BrownFullerPittsReznikoffReIdIdInQu, Proposition 3.19(ii)]. To do this, we require the following facts:

(a)

$\ker\gamma$ is a regular ideal of ${\mathcal{B}}$ ; and
(b)

$\ker\gamma$ is $\alpha({\mathcal{N}}({\mathcal{C}},{\mathcal{D}}))$ invariant, that is, for every $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ ,

(3.18) $\alpha(v^{*})(\ker\gamma)\alpha(v)\subseteq\ker\gamma.$

Our first step in verifying (a) is to show $\ker\gamma$ is the unique ideal in ${\mathcal{B}}$ maximal with respect to having trivial intersection with $\alpha({\mathcal{D}})$ . By [PittsZarikianUnPsExC*In, Corollary 3.21], $\ker\tilde{\gamma}$ is the unique maximal $\tilde{\alpha}(\tilde{\mathcal{D}})$ -disjoint ideal of $\tilde{\mathcal{B}}$ . Suppose $J\unlhd{\mathcal{B}}$ satisfies $J\cap\alpha({\mathcal{D}})=\{0\}$ . Then $u_{\mathcal{B}}(J)\unlhd\tilde{\mathcal{B}}$ . Let us show

(3.19)

u_{\mathcal{B}}(J)\cap\tilde{\alpha}(\tilde{\mathcal{D}})=\{0\}.

We do this by considering two cases.

Case 1: ${\mathcal{D}}$ is unital. Since $({\mathcal{C}},{\mathcal{D}})$ is weakly non-degenerate, ${\mathcal{C}}$ is unital. Let us show ${\mathcal{B}}$ is unital and that $e:=\alpha(I_{\mathcal{C}})$ is its unit. Since $\alpha$ is a regular $*$ -monomorphism, $e\in{\mathcal{N}}({\mathcal{A}},{\mathcal{B}})$ , hence $e=e^{*}e$ commutes with ${\mathcal{B}}$ by [PittsNoApUnInC*Al, Proposition 2.1]. Thus $e\in{\mathcal{B}}$ because ${\mathcal{B}}$ is a MASA in ${\mathcal{A}}$ . Let $n\geq 1$ and $x_{1},\dots,x_{n}\in{\mathcal{C}}$ . Then

e\Delta(\alpha(x_{1}))=\Delta(e\alpha(x_{1}))=\Delta(\alpha(x_{1}));\text{ likewise }\Delta(\alpha(x_{n}))e=\Delta(\alpha(x_{n})).

Hence

e\left(\prod_{k=1}^{n}\Delta(\alpha(x_{k}))\right)=\left(\prod_{k=1}^{n}\Delta(\alpha(x_{k}))\right)e=\prod_{k=1}^{n}\Delta(\alpha(x_{k})).

By the definition of a package, the span of finite products from $\Delta({\mathcal{C}})$ is dense in ${\mathcal{B}}$ . Thus $e=I_{\mathcal{B}}$ .

As $({\mathcal{A}},{\mathcal{B}})$ is a Cartan inclusion, it has the AUP. Therefore $I_{\mathcal{B}}$ is the unit for ${\mathcal{A}}$ . The equality (3.19) now follows because $u_{\mathcal{B}}$ is the identity map on ${\mathcal{B}}$ , $\tilde{\alpha}=\alpha$ and $\tilde{\mathcal{D}}={\mathcal{D}}$ .

Case 2: ${\mathcal{D}}$ is not unital. Let $y\in u_{\mathcal{B}}(J)\cap\tilde{\alpha}(\tilde{\mathcal{D}})$ . Then for some $(d,\lambda)\in\tilde{\mathcal{D}}$ and $j\in J$ ,

y=\alpha(d)+\lambda I_{\tilde{\mathcal{B}}}=u_{\mathcal{B}}(j).

For any $h\in{\mathcal{D}}$ , multiplying each side of the second equality by $\tilde{\alpha}(h,0)$ gives

\alpha(dh)+\lambda\alpha(h)=u_{\mathcal{B}}(j)\tilde{\alpha}(h,0).

Since the left side belongs to $u_{\mathcal{B}}(\alpha({\mathcal{D}}))$ and the right belongs to $u_{\mathcal{B}}(J)$ , the hypothesis that $\alpha({\mathcal{D}})\cap J=\{0\}$ gives $\alpha(dh)+\lambda\alpha(h)=0$ . Thus for every $h\in{\mathcal{D}}$ ,

dh=-\lambda h.

Let $(e_{\lambda})$ be an approximate unit for ${\mathcal{D}}$ . Choosing $h=e_{\lambda}$ and taking the limit, we obtain $\lim\lambda e_{\lambda}$ exists in ${\mathcal{D}}$ and $d=-\lim\lambda e_{\lambda}$ . As ${\mathcal{D}}$ is non-unital, this forces $y=0$ . This completes the proof of (3.19).

Equality 3.19 and the fact that $\ker\tilde{\gamma}$ is the maximal ideal of $\tilde{\mathcal{B}}$ disjoint from $\tilde{\alpha}(\tilde{\mathcal{D}})$ gives $u_{\mathcal{B}}(J)\subseteq\ker\tilde{\gamma}$ . Therefore $u_{\mathcal{B}}(J)=u_{\mathcal{B}}(J)\cap u_{\mathcal{B}}({\mathcal{B}})\subseteq\ker\tilde{\gamma}\cap u_{\mathcal{B}}({\mathcal{B}})=u_{\mathcal{B}}(\ker\gamma)$ , whence $J\subseteq\ker\gamma$ . Thus $\ker\gamma$ is the unique maximal $\alpha({\mathcal{D}})$ -disjoint ideal of ${\mathcal{B}}$ .

We are now prepared to show $\ker\gamma$ is a regular ideal of ${\mathcal{B}}$ . Suppose $d\in{\mathcal{D}}$ satisfies $\alpha(d)\in(\ker\gamma)^{\perp\perp}$ . Let ${\mathcal{I}}_{d}$ be the closed ideal of ${\mathcal{B}}$ generated by $\alpha(d)$ . Then ${\mathcal{I}}_{d}$ is contained in $(\ker\gamma)^{\perp\perp}$ , so $\alpha(d)\in{\mathcal{I}}_{d}\cap(\ker\gamma)^{\perp\perp}$ . But ${\mathcal{I}}_{d}\cap\ker\gamma\subseteq\alpha({\mathcal{D}})\cap\ker\gamma=\{0\}$ . Fact 3 shows $((\ker\gamma)^{\perp\perp},\ker\gamma)$ has the ideal intersection property, so ${\mathcal{I}}_{d}=\{0\}$ . That $d=0$ follows. Therefore $(\ker\gamma)^{\perp\perp}$ is an ideal of ${\mathcal{B}}$ having trivial intersection with $\alpha({\mathcal{D}})$ . As $\ker\gamma\subseteq(\ker\gamma)^{\perp\perp}$ , the maximality property of $\ker\gamma$ gives $\ker\gamma=(\ker\gamma)^{\perp\perp}$ , so $\ker\gamma$ is a regular ideal of ${\mathcal{B}}$ .

Turning now to (b), we will use the notation from (3.9). We claim that for $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ ,

(3.20)

\gamma\circ\theta_{\alpha(v)}=\overline{\theta}_{v}\circ\gamma|_{R(\alpha(v))}.

To see this, let $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ . Regularity of the map $\alpha$ and the fact that Cartan inclusions have the AUP, gives $\alpha(vv^{*})\in{\mathcal{B}}$ .

Let ${\mathbb{P}}_{0}$ be the collection of all polynomials $p$ with complex coefficients such that $p(0)=0$ . Given $p\in{\mathbb{P}}_{0}$ , factor $p(t)=tq(t)$ , where $q$ is another polynomial. For $x\in{\mathcal{C}}$ , let

y:=\alpha(p(vv^{*}))\Delta(\alpha(x))=\alpha(vv^{*})\alpha(q(vv^{*}))\Delta(\alpha(x)).

Using Lemma 3, (2.1.10), and the fact that $\Delta$ is a conditional expectation,

	$\displaystyle\gamma(\theta_{\alpha(v)}(y))$	$\displaystyle=\gamma(\alpha(v)^{}\Delta(\alpha(q(vv^{})x))\alpha(v))=\gamma(\Delta(\alpha(v^{}q(vv^{})xv)))$
		$\displaystyle\stackrel{{\scriptstyle\eqref{ush1}}}{{=}}E(v^{}q(vv^{})xv)=\overline{\theta}_{v}(E(vv^{}q(vv^{})x))=\overline{\theta}_{v}(\gamma(\Delta(\alpha(vv^{}q(vv^{})x))))$
		$\displaystyle=\overline{\theta}_{v}(\gamma(y)).$

The maps $\gamma$ , $\theta_{\alpha(v)}$ , and $\overline{\theta}_{v}$ are multiplicative on their domains. Let

{\mathbb{P}}_{v}:=\{p(vv^{*}):p\in{\mathbb{P}}_{0}\}.

Since ${\mathcal{B}}=C^{*}(\Delta(\alpha({\mathcal{C}}))$ , we find that for $b\in\alpha({\mathbb{P}}_{v}){\mathcal{B}}$ ,

\gamma(\theta_{\alpha(v)}(b))=\overline{\theta}_{v}(\gamma(b)).

Since $\alpha({\mathbb{P}}_{v}){\mathcal{B}}$ is dense in $R(\alpha(v))$ , continuity gives (3.20).

With (3.20) in hand, we now complete the proof of the invariance of $\ker\gamma$ . Suppose $t\in\ker\gamma$ , and let $b\in{\mathcal{B}}$ . Then

\gamma(\alpha(v)^{*}tb\alpha(v))=\gamma(\theta_{\alpha(v)}(tb\alpha(vv^{*})))\stackrel{{\scriptstyle\eqref{fJreg2a}}}{{=}}\overline{\theta}_{v}(\gamma(tb\alpha(vv^{*})))=0.

By replacing $b$ with elements from an approximate unit for ${\mathcal{B}}$ , we find $\alpha(v^{*})t\alpha(v)\in\ker\gamma$ , so (3.18) holds. This completes the proof of (b).

Let $N$ be the $*$ -semigroup generated by ${\mathcal{B}}\cup\alpha({\mathcal{N}}({\mathcal{C}},{\mathcal{D}}))$ . For every $w\in N$ , using (b), we find that $w(\ker\gamma)w^{*}\subseteq\ker\gamma$ , that is $\ker\gamma$ is an $N$ -invariant regular ideal in ${\mathcal{B}}$ . While the statement of [BrownFullerPittsReznikoffReIdIdInQu, Proposition 3.19(ii)], concerns a regular invariant ideals, the proof applies for $N$ -invariant regular ideals (in fact, the statement should have been for $N$ -invariant regular ideals). Thus, [BrownFullerPittsReznikoffReIdIdInQu, Proposition 3.19(ii)] shows ${\mathfrak{J}}$ is a regular ideal in ${\mathcal{A}}$ such that ${\mathfrak{J}}\cap{\mathcal{B}}=\ker\gamma$ . The proof of Claim 3 is now complete. $\diamondsuit$

Let $q:{\mathcal{A}}\rightarrow{\mathcal{A}}/{\mathfrak{J}}$ be the quotient map. We must show $({\mathcal{A}}/{\mathfrak{J}},{\mathcal{B}}/({\mathfrak{J}}\cap{\mathcal{B}})\hbox{\,:\hskip-1.0pt:\,}q\circ\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ .

First we show $q\circ\alpha:({\mathcal{C}},{\mathcal{D}})\rightarrow({\mathcal{A}}/{\mathfrak{J}},{\mathcal{B}}/({\mathcal{J}}\cap{\mathcal{B}}))$ is a regular $*$ -monomorphism. To show $q\circ\alpha$ is one-to-one, suppose $x\in{\mathfrak{J}}\cap\alpha({\mathcal{C}})$ . Then there exists $c\in{\mathcal{C}}$ so that $x=\alpha(c)$ . Since $x^{*}x\in{\mathfrak{J}}\cap\alpha({\mathcal{C}})$ , (3.17) shows

0=\gamma(\Delta(\alpha(c^{*}c)))=E(c^{*}c).

Since the pseudo-expectation for $({\mathcal{C}},{\mathcal{D}})$ is faithful, we find $c=0$ , hence $x=0$ . Therefore $q\circ\alpha$ is faithful. For $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ and $h\in{\mathcal{B}}$ , (3.18) gives

\alpha(v)^{*}(h+\ker\gamma)\alpha(v)\subseteq\alpha(v^{*})h\alpha(v)+\ker\gamma\in{\mathcal{B}}/({\mathcal{B}}\cap{\mathfrak{J}}),

whence $q\circ\alpha$ is a regular $*$ -monomorphism.

By [BrownFullerPittsReznikoffReIdIdInQu, Theorem 4.8], $({\mathcal{A}}/{\mathfrak{J}},{\mathcal{B}}/({\mathcal{B}}\cap{\mathfrak{J}}))$ is a Cartan inclusion. Clearly $({\mathcal{C}},{\mathcal{D}})$ generates $({\mathcal{A}}/{\mathfrak{J}},{\mathcal{B}}/({\mathfrak{J}}\cap{\mathcal{B}}))$ in the sense of part (ii) of Definition 3(a).

It remains to establish that $({\mathcal{B}}/({\mathfrak{J}}\cap{\mathcal{B}}),{\mathcal{D}},q\circ\alpha)$ has the ideal intersection property. Suppose $K\unlhd{\mathcal{B}}/({\mathcal{B}}\cap{\mathfrak{J}})$ satisfies $K\cap(q(\alpha({\mathcal{D}})))=\{0\}$ . Since $q\circ\alpha$ is faithful, ${\mathcal{B}}\cap q^{-1}(K)$ is an ideal of ${\mathcal{B}}$ having trivial intersection with $\alpha({\mathcal{D}})$ . By the maximality property of $\ker\gamma$ , ${\mathcal{B}}\cap q^{-1}(K)\subseteq\ker\gamma$ , so $K=\{0\}$ . Therefore $({\mathcal{B}}/({\mathcal{B}}\cap{\mathfrak{J}}),{\mathcal{D}},q\circ\alpha)$ has the ideal intersection property. Thus $({\mathcal{A}}/{\mathfrak{J}},{\mathcal{B}}/({\mathcal{B}}\cap{\mathfrak{J}})\hbox{\,:\hskip-1.0pt:\,}q\circ\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ . This finishes the proof of the minimality statement and also concludes the proof of Theorem 3. ∎

4. Pseudo-Cartan Inclusions and their Cartan Envelopes

In this section we give the formal definition of a pseudo-Cartan inclusion. The key results of this section are: Theorem 4.2, which shows that in the presence of the AUP, the unitization process commutes with taking Cartan envelopes, and Theorem 4.3 which gives some properties shared by a pseudo-Cartan inclusion and its Cartan envelope. Proposition 4.4 gives a construction which can be used to produce a family of pseudo-Cartan inclusions starting with a Cartan inclusion.

4.1. Definition of Pseudo-Cartan Inclusions and Examples

We are now prepared to define the notion of pseudo-Cartan inclusion. Standing Assumption 3 remains in force throughout, that is, we assume that ${\mathcal{D}}$ is abelian for every inclusion $({\mathcal{C}},{\mathcal{D}})$ .

Definition \the\numberby.

A regular inclusion $({\mathcal{C}},{\mathcal{D}})$ is a pseudo-Cartan inclusion if it satisfies the following:

(a)

${\mathcal{D}}^{c}$ is abelian; and
(b)

both $({\mathcal{C}},{\mathcal{D}}^{c})$ and $({\mathcal{D}}^{c},{\mathcal{D}})$ have the ideal intersection property.

We will frequently use the alternate descriptions of a pseudo-Cartan inclusion found in Theorem 3, often without comment.

The following is immediate from Definition 4.1.

Observation \the\numberby.

Let $({\mathcal{C}},{\mathcal{D}})$ be an inclusion with ${\mathcal{C}}$ abelian. Then ${\mathcal{D}}^{c}={\mathcal{C}}$ , so $({\mathcal{C}},{\mathcal{D}})$ is a pseudo-Cartan inclusion if and only if $({\mathcal{C}},{\mathcal{D}})$ is regular and has the ideal intersection property.

Let us compare the notion of pseudo-Cartan inclusion with other related classes of inclusions.

Cartan Inclusions:: Proposition 2.3 shows that every Cartan inclusion is a pseudo-Cartan inclusion. Furthermore, Proposition 2.3 may be interpreted as stating that a pseudo-Cartan inclusion $({\mathcal{C}},{\mathcal{D}})$ is a Cartan inclusion if and only if the pseudo-expectation $E:{\mathcal{C}}\rightarrow I({\mathcal{D}})$ “is” a conditional expectation in the sense that $E({\mathcal{C}})=\iota({\mathcal{D}})$ . This characterization is the reason we chose the term “pseudo-Cartan inclusion” in Definition 4.1.
Virtual Cartan Inclusions:: We will use the term virtual Cartan inclusion for a regular MASA inclusion $({\mathcal{C}},{\mathcal{D}})$ having the ideal intersection property. Theorem 3 shows that every virtual Cartan inclusion is a pseudo-Cartan inclusion; furthermore, the pseudo-Cartan inclusion $({\mathcal{C}},{\mathcal{D}})$ is a virtual Cartan inclusion if and only if ${\mathcal{D}}$ is a MASA in ${\mathcal{C}}$ . By [PittsNoApUnInC*Al, Theorem 2.6], every virtual Cartan inclusion has the AUP.
Weak Cartan Inclusions:: Weak Cartan inclusions were defined in [ExelPittsChGrC*AlNoHaEtGr, Definition 2.11.5]. We shall show in Proposition 4.1 below that that every weak Cartan inclusion is a pseudo-Cartan inclusion. Since the axioms for weak Cartan inclusions include the AUP, the class of pseudo-Cartan inclusions properly contains the class of weak Cartan inclusions.

The example given in [ExelPittsChGrC*AlNoHaEtGr, Remark 2.11.6] shows that there exists a Cartan inclusion $({\mathcal{C}},{\mathcal{D}})$ (as in Definition 2.1(h) above) which is not a weak Cartan inclusion, so the classes of pseudo-Cartan inclusions with the AUP and weak Cartan inclusions are also distinct. Also, an example of a unital pseudo-Cartan inclusion $({\mathcal{C}},{\mathcal{D}})$ with ${\mathcal{C}}$ abelian which is not a weak Cartan inclusion is given in the paragraph just prior to [ExelPittsChGrC*AlNoHaEtGr, 2.11.16]. However, in both these examples, the algebra ${\mathcal{C}}$ is not separable. We do not know whether every pseudo-Cartan inclusion $({\mathcal{C}},{\mathcal{D}})$ with the AUP and such that ${\mathcal{C}}$ is separable is a weak Cartan inclusion.²²2During his lecture at the 2025 Canadian Operator Theory Symposium, Dan Ursu stated that he found an example of a separable pseudo-Cartan inclusion with the AUP which is not a weak Cartan inclusion.
Pseudo-Diagonals and Abelian Cores:: These are classes of regular inclusions which are not assumed to be regular. Nagy and Reznikoff define the notion of a pseudo-diagonal, see [NagyReznikoffPsDiUnTh, p. 268]. One of the requirements for the inclusion $({\mathcal{C}},{\mathcal{D}})$ to be a pseudo-diagonal is that there exists a faithful conditional expectation $E:{\mathcal{C}}\rightarrow{\mathcal{D}}$ , however regularity of $({\mathcal{C}},{\mathcal{D}})$ is not required. Nagy and Reznikoff observe that when $({\mathcal{C}},{\mathcal{D}})$ is a pseudo-diagonal, ${\mathcal{D}}$ is necessarily a MASA in ${\mathcal{C}}$ , [NagyReznikoffPsDiUnTh, Corollary 3.2]. Thus, when $({\mathcal{C}},{\mathcal{D}})$ is a regular pseudo-diagonal, $({\mathcal{C}},{\mathcal{D}})$ is a Cartan inclusion and so is a pseudo-Cartan inclusion. Likewise, every regular abelian core (see [NagyReznikoffPsDiUnTh, p. 272]) is a Cartan inclusion and hence a pseudo-Cartan inclusion.

For a unital pseudo-Cartan inclusion $({\mathcal{C}},{\mathcal{D}})$ , the relative commutant ${\mathcal{D}}^{c}$ is a MASA in ${\mathcal{C}}$ and [PittsStReInI, Lemma 2.10] shows $({\mathcal{C}},{\mathcal{D}}^{c})$ is a regular inclusion. As $({\mathcal{C}},{\mathcal{D}}^{c})$ has the ideal intersection property (Definition 4.1(b)), it is a virtual Cartan inclusion. We now observe the same is true for possibly non-unital pseudo-Cartan inclusions.

Observation \the\numberby.

If $({\mathcal{C}},{\mathcal{D}})$ is a pseudo-Cartan inclusion, then $({\mathcal{C}},{\mathcal{D}}^{c})$ is a virtual Cartan inclusion.

Proof.

Lemma 3(a) shows that $({\mathcal{C}},{\mathcal{D}}^{c})$ is a regular inclusion, so it is a regular MASA inclusion with the ideal intersection property. ∎

Our next goal is to show that every weak Cartan inclusion is a pseudo-Cartan inclusion. Before proceeding, we recall some terminology from [ExelPittsChGrC*AlNoHaEtGr].

Definition \the\numberby ([ExelPittsChGrC*AlNoHaEtGr, Definitions 2.7.2, 2.8.1 and 2.11.5]).

Let $({\mathcal{C}},{\mathcal{D}})$ be an inclusion and $\sigma\in\hat{\mathcal{D}}$ .

(a)

For $x\in{\mathcal{C}}$ , we say $\sigma$ is free relative to $x$ if whenever $f_{1},f_{2}$ are states on ${\mathcal{C}}$ such that $f_{1}|_{\mathcal{D}}=\sigma=f_{2}|_{\mathcal{D}}$ , we have $f_{1}(x)=f_{2}(x)$ .
(b)

We say $\sigma\in\hat{\mathcal{D}}$ is free for $({\mathcal{C}},{\mathcal{D}})$ , or a free point when the context is clear, if $\sigma$ uniquely extends to a state $f$ on ${\mathcal{C}}$ .
(c)

The inclusion $({\mathcal{C}},{\mathcal{D}})$ is called topologically free if the set of free points for $({\mathcal{C}},{\mathcal{D}})$ is dense in $\hat{\mathcal{D}}$ . Other terminology for topologically free inclusions may be found in the literature, for example, the almost extension property is used in [KwasniewskiMeyerApAlExPrUnPsEx, NagyReznikoffPsDiUnTh, ZarikianPuExPrDiCrPr] instead of ‘topologically free inclusion’.

Proposition \the\numberby.

Let $({\mathcal{C}},{\mathcal{D}})$ be a weak Cartan inclusion. Then $({\mathcal{C}},{\mathcal{D}})$ is a pseudo-Cartan inclusion.

Proof.

First note that the weak Cartan inclusions considered in [ExelPittsChGrC*AlNoHaEtGr] have the AUP and are regular (see [ExelPittsChGrC*AlNoHaEtGr, Definition 2.3.1, Hypothesis 2.3.2, and Definition 2.11.5]). Next, [ExelPittsChGrC*AlNoHaEtGr, Proposition 2.11.7] shows that $({\mathcal{C}},{\mathcal{D}})$ is topologically free. Combining [KwasniewskiMeyerApAlExPrUnPsEx, Theorems 5.5 and 3.6], we find that $({\mathcal{C}},{\mathcal{D}})$ has the unique pseudo-expectation property; let $E:{\mathcal{C}}\rightarrow I({\mathcal{D}})$ be its pseudo-expectation. We must show $E$ is faithful.

Since $({\mathcal{C}},{\mathcal{D}})$ has the AUP, $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is a regular inclusion by Lemma 3(e). Corollary 2.3 shows $\tilde{E}$ is the unique pseudo-expectation for $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ . Let ${\mathcal{L}}(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ be the left kernel of $\tilde{E}$ . It follows from [PittsStReInII, Theorem 6.5] that ${\mathcal{L}}(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is the largest ideal of $\tilde{\mathcal{C}}$ having trivial intersection with $\tilde{\mathcal{D}}$ .

If $J\unlhd{\mathcal{C}}$ has trivial intersection with ${\mathcal{D}}$ , then $J$ is also an ideal of $\tilde{\mathcal{C}}$ which has trivial intersection with $\tilde{\mathcal{D}}$ . Thus $J\subseteq{\mathcal{L}}(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ . Hence ${\mathcal{L}}_{1}:={\mathcal{C}}\cap{\mathcal{L}}(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is the largest ideal of ${\mathcal{C}}$ having trivial intersection with ${\mathcal{D}}$ . Therefore, [ExelPittsChGrC*AlNoHaEtGr, Proposition 2.11.3] shows that ${\mathcal{L}}_{1}$ is the grey ideal for $({\mathcal{C}},{\mathcal{D}})$ . The grey ideal in a weak Cartan inclusion is $\{0\}$ (see [ExelPittsChGrC*AlNoHaEtGr, Definition 2.11.5]), so ${\mathcal{L}}_{1}=\{0\}$ . But ${\mathcal{L}}_{1}$ is the left kernel of $E$ , whence $({\mathcal{C}},{\mathcal{D}})$ has the faithful unique pseudo-expectation property. Theorem 3 then shows $({\mathcal{C}},{\mathcal{D}})$ is a pseudo-Cartan inclusion. ∎

4.2. The AUP and Unitization of Cartan Envelopes

Given a weakly non-degenerate regular inclusion $({\mathcal{C}},{\mathcal{D}})$ , $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is a unital inclusion by Lemma 2.2(e), but we have been unable to establish whether $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is regular. Two conditions which ensure that regularity of $({\mathcal{C}},{\mathcal{D}})$ implies $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is also regular are: (i) $({\mathcal{C}},{\mathcal{D}})$ has the AUP (Corollary 3); and (ii) ${\mathcal{C}}$ is abelian (because every unitary in $\tilde{\mathcal{C}}$ normalizes $\tilde{\mathcal{D}}$ and $\tilde{\mathcal{C}}$ is spanned by its unitaries). On the other hand, [PittsNoApUnInC*Al, Example 3.1] shows that regularity of $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ does not imply regularity of $({\mathcal{C}},{\mathcal{D}})$ . Theorem 2.3 (and Theorem 3) give the following.

Observation \the\numberby.

Suppose both $({\mathcal{C}},{\mathcal{D}})$ and $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ are regular inclusions. Then $({\mathcal{C}},{\mathcal{D}})$ is a pseudo-Cartan inclusion if and only if $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is a pseudo-Cartan inclusion.

A pseudo-Cartan inclusion $({\mathcal{C}},{\mathcal{D}})$ may not have the AUP. Our next two results show that it is possible to enlarge ${\mathcal{D}}$ to obtain a pseudo-Cartan inclusion $({\mathcal{C}},{\mathcal{E}})$ having the AUP such that the Cartan envelopes of $({\mathcal{C}},{\mathcal{D}})$ and $({\mathcal{C}},{\mathcal{E}})$ are the same. The significance of these results is that it is often possible to replace $({\mathcal{C}},{\mathcal{D}})$ with $({\mathcal{C}},{\mathcal{E}})$ when the AUP is required. The first of these results shows that ${\mathcal{E}}$ may be taken to be ${\mathcal{D}}^{c}$ ; it extends [PittsStReInII, Proposition 5.29(b)] to include not-necessarily unital inclusions.

Proposition \the\numberby.

Suppose $({\mathcal{C}},{\mathcal{D}})$ is a pseudo-Cartan inclusion and consider the virtual Cartan inclusion $({\mathcal{C}},{\mathcal{D}}^{c})$ . Then $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ if and only if $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is Cartan envelope for $({\mathcal{C}},{\mathcal{D}}^{c})$ .

Proof.

Proposition 3 shows that if $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}}^{c})$ , then $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ .

Suppose $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ . Then we may find a Cartan envelope $({\mathcal{A}}^{\prime},{\mathcal{B}}^{\prime}\hbox{\,:\hskip-1.0pt:\,}\alpha^{\prime})$ for $({\mathcal{C}},{\mathcal{D}}^{c})$ , which by Proposition 3, is also a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ . By uniqueness of Cartan envelopes for $({\mathcal{C}},{\mathcal{D}})$ , there exists a unique regular $*$ -isomorphism $\psi:({\mathcal{A}}^{\prime},{\mathcal{B}}^{\prime})\rightarrow({\mathcal{A}},{\mathcal{B}})$ such that $\psi\circ\alpha^{\prime}=\alpha$ . Since $\alpha^{\prime}:({\mathcal{C}},{\mathcal{D}}^{c})\rightarrow({\mathcal{A}}^{\prime},{\mathcal{B}}^{\prime})$ is regular, we conclude that $\alpha:({\mathcal{C}},{\mathcal{D}}^{c})\rightarrow({\mathcal{A}},{\mathcal{B}})$ is regular. Therefore, $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}}^{c})$ . ∎

Given the pseudo-Cartan inclusion $({\mathcal{C}},{\mathcal{D}})$ , examples show that ${\mathcal{D}}^{c}$ need not be contained in ${\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ . Nevertheless, by suitably enlarging ${\mathcal{D}}$ , we can use $({\mathcal{C}},{\mathcal{D}})$ to produce pseudo-Cartan inclusions $({\mathcal{C}},{\mathcal{E}})$ having the AUP such that ${\mathcal{E}}\subseteq{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ and which have the same Cartan envelope as $({\mathcal{C}},{\mathcal{D}})$ . However, unlike ${\mathcal{D}}^{c}$ , ${\mathcal{E}}$ may not be a MASA in ${\mathcal{C}}$ .

Proposition \the\numberby.

Let $({\mathcal{C}},{\mathcal{D}})$ be a pseudo-Cartan inclusion. Suppose ${\mathcal{P}}\subseteq{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ is a $*$ -semigroup such that ${\mathcal{D}}\subseteq{\mathcal{P}}$ , and $\operatorname{span}{\mathcal{P}}$ is dense in ${\mathcal{C}}$ . Let

{\mathcal{E}}:=C^{*}(\{p\in{\mathcal{P}}:p\geq 0\}).

Then $({\mathcal{C}},{\mathcal{E}})$ is a pseudo-Cartan inclusion having the AUP. Furthermore,

(a)

${\mathcal{D}}\subseteq{\mathcal{E}}\subseteq{\mathcal{D}}^{c}$ ;
(b)

${\mathcal{E}}\subseteq{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ ; and
(c)

$({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ if and only if $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{E}})$ .

Proof.

We first establish parts (a) and (b), then show $({\mathcal{C}},{\mathcal{E}})$ is a pseudo-Cartan inclusion with the AUP, and finally show part (c).

The first inclusion in (a) follows from the fact that ${\mathcal{D}}\subseteq{\mathcal{P}}$ and ${\mathcal{D}}$ is the span of its positive elements. For the second, [PittsNoApUnInC*Al, Proposition 2.1] shows that for $0\leq p\in{\mathcal{P}}$ , $p^{2}\in{\mathcal{D}}^{c}$ , so $p\in{\mathcal{D}}^{c}$ ; thus ${\mathcal{E}}\subseteq{\mathcal{D}}^{c}$ . This gives (a).

Before moving to part (b), we introduce some notation. Let us write,

{\mathcal{P}}^{+}:=\{p\in{\mathcal{P}}:p\geq 0\}.

Since ${\mathcal{D}}^{c}$ is abelian, ${\mathcal{P}}^{+}$ is a $*$ -semigroup, whence ${\mathcal{E}}=\overline{\operatorname{span}}\,{\mathcal{P}}^{+}$ .

We now turn to part (b). For $p\in{\mathcal{P}}^{+}$ and and $d\in{\mathcal{D}}$ , [PittsNoApUnInC*Al, Proposition 2.1] gives $p^{2}d^{*}d=d^{*}dp^{2}\in{\mathcal{D}}$ . Taking square roots, we obtain $p|d|=|d|p\in{\mathcal{D}}$ . Since ${\mathcal{D}}$ is the span of its positive elements, we see that for every $d\in{\mathcal{D}}$ and $p\in{\mathcal{P}}^{+}$ ,

pd=dp\in{\mathcal{D}}.

Thus, if $x\in\operatorname{span}{\mathcal{P}}^{+}$ , $xd\in{\mathcal{D}}$ , so $x^{*}dx=xdx^{*}=x^{*}xd\in{\mathcal{D}}$ also. Therefore, $\operatorname{span}{\mathcal{P}}^{+}\subseteq{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ . As ${\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ is closed, part (b) holds.

We next show $({\mathcal{C}},{\mathcal{E}})$ is a regular inclusion. For any $v\in{\mathcal{P}}$ , $v^{*}{\mathcal{P}}^{+}v\cup v{\mathcal{P}}^{+}v^{*}\subseteq{\mathcal{P}}^{+}$ , whence $v{\mathcal{E}}v^{*}\cup v^{*}{\mathcal{E}}v\subseteq{\mathcal{E}}$ . Therefore, ${\mathcal{P}}\subseteq{\mathcal{N}}({\mathcal{C}},{\mathcal{E}})$ , and because ${\mathcal{C}}=\overline{\operatorname{span}}\,{\mathcal{P}}$ , $({\mathcal{C}},{\mathcal{E}})$ is a regular inclusion.

Note that ${\mathcal{E}}^{c}={\mathcal{D}}^{c}$ . Thus, part (a) and Lemma 2.4 show $({\mathcal{E}}^{c},{\mathcal{E}})$ has the ideal intersection property. As $({\mathcal{C}},{\mathcal{D}})$ is a pseudo-Cartan inclusion, $({\mathcal{C}},{\mathcal{E}}^{c})$ has the ideal intersection property, so $({\mathcal{C}},{\mathcal{E}})$ is a pseudo-Cartan inclusion.

To show $({\mathcal{C}},{\mathcal{E}})$ has the AUP, let $(u_{\lambda})$ be an approximate unit for ${\mathcal{E}}$ . Let $\varepsilon>0$ . For $v\in{\mathcal{P}}$ , we may choose $n\in{\mathbb{N}}$ so that $\left\lVert v(v^{*}v)^{1/n}-v\right\rVert<\varepsilon/3$ . Since $(v^{*}v)^{1/n}\in{\mathcal{E}}$ , there exists $\lambda_{0}$ so that for $\lambda\geq\lambda_{0}$ , $\left\lVert v\right\rVert\left\lVert(v^{*}v)^{1/n}u_{\lambda}-(v^{*}v)^{1/n}\right\rVert<\varepsilon/3.$ For $\lambda\geq\lambda_{0}$ ,

	$\displaystyle\left\lVert vu_{\lambda}-v\right\rVert$	$\displaystyle\leq\left\lVert vu_{\lambda}-v(v^{}v)^{1/n}u_{\lambda}\right\rVert+\left\lVert v(v^{}v)^{1/n}u_{\lambda}-v(vv^{})^{1/n}\right\rVert+\left\lVert v(v^{}v)^{1/n}-v\right\rVert$
		$\displaystyle\leq 2\left\lVert v-v(v^{}v)^{1/n}\right\rVert+\left\lVert v\right\rVert\left\lVert(v^{}v)^{1/n}u_{\lambda}-(v^{*}v)^{1/n}\right\rVert<\varepsilon.$

Therefore $vu_{\lambda}\rightarrow v$ ; similarly $u_{\lambda}v\rightarrow v$ . Since $\operatorname{span}{\mathcal{P}}$ is dense in ${\mathcal{C}}$ , it follows that $({\mathcal{C}},{\mathcal{E}})$ has the AUP.

Finally, using Proposition 4.2 and the equality ${\mathcal{E}}^{c}={\mathcal{D}}^{c}$ ,

	$\displaystyle({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)\text{ is a Cartan envelope for }({\mathcal{C}},{\mathcal{D}})\iff$	$\displaystyle({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)\text{ is a Cartan envelope for }$
		$\displaystyle({\mathcal{C}},{\mathcal{D}}^{c})=({\mathcal{C}},{\mathcal{E}}^{c})$
	$\displaystyle\iff$	$\displaystyle({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)\text{ is a Cartan envelope for }$
		$\displaystyle({\mathcal{C}},{\mathcal{E}}).\hfill\qed$

As noted in Corollary 3, for the weakly non-degenerate and non-unital inclusion $({\mathcal{C}},{\mathcal{D}})$ , the inclusion mapping $u_{\mathcal{C}}$ of $({\mathcal{C}},{\mathcal{D}})$ into $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is regular if and only if $({\mathcal{C}},{\mathcal{D}})$ has the AUP. We now study the relationships between the Cartan envelopes of $({\mathcal{C}},{\mathcal{D}})$ and $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ when $({\mathcal{C}},{\mathcal{D}})$ has the AUP. We begin with a lemma concerning the image under $\alpha$ of an approximate unit for ${\mathcal{D}}$ in a Cartan package $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ .

Lemma \the\numberby.

Suppose $({\mathcal{C}},{\mathcal{D}})$ is an inclusion with the AUP and $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan package for $({\mathcal{C}},{\mathcal{D}})$ . If $(u_{\lambda})\subseteq{\mathcal{D}}$ is an approximate unit for ${\mathcal{C}}$ , then $(\alpha(u_{\lambda}))\subseteq{\mathcal{B}}$ is an approximate unit for ${\mathcal{A}}$ .

Proof.

Let $\Delta:{\mathcal{A}}\rightarrow{\mathcal{B}}$ be the conditional expectation. For $x\in{\mathcal{C}}$ , we have

\lim_{\lambda}\Delta(\alpha(x))\alpha(u_{\lambda})=\lim_{\lambda}\Delta(\alpha(xu_{\lambda}))=\Delta(\alpha(x)).

Since ${\mathcal{B}}$ is generated by $\Delta(\alpha({\mathcal{C}}))$ , it follows that $(\alpha(u_{\lambda}))$ is an approximate unit for ${\mathcal{B}}$ . As $({\mathcal{A}},{\mathcal{B}})$ is a regular MASA inclusion, an application of [PittsNoApUnInC*Al, Theorem 2.6] shows $(\alpha(u_{\lambda}))$ is an approximate unit for ${\mathcal{A}}$ . ∎

Lemma \the\numberby.

Let $({\mathcal{C}},{\mathcal{D}})$ be a regular inclusion with the AUP such that ${\mathcal{C}}$ is not unital. Suppose $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan package (resp. Cartan envelope) for $({\mathcal{C}},{\mathcal{D}})$ . Then $(\tilde{\mathcal{A}},\tilde{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\tilde{\alpha})$ is a Cartan package (resp. Cartan envelope) for $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ .

Proof.

We will use $\Delta:{\mathcal{A}}\rightarrow{\mathcal{B}}$ for the conditional expectation. Since $({\mathcal{A}},{\mathcal{B}})$ is a Cartan pair, so is $(\tilde{\mathcal{A}},\tilde{\mathcal{B}})$ ; also, $\tilde{\Delta}:\tilde{\mathcal{A}}\rightarrow\tilde{\mathcal{B}}$ is the conditional expectation of $\tilde{\mathcal{A}}$ onto $\tilde{\mathcal{B}}$ .

That $\tilde{\mathcal{A}}$ is generated by $\tilde{\alpha}(\tilde{\mathcal{C}})\cup\tilde{\Delta}(\tilde{\alpha}(\tilde{\mathcal{C}}))$ follows from the fact that ${\mathcal{A}}$ is generated by $\alpha({\mathcal{C}})\cup\Delta(\alpha({\mathcal{C}}))$ . Similarly, $\tilde{\mathcal{B}}$ is generated by $\tilde{\Delta}(\pi(\tilde{\mathcal{C}}))$ . It remains to show that $\tilde{\alpha}$ is a regular $*$ -homomorphism, that is, $\tilde{\alpha}({\mathcal{N}}(\tilde{\mathcal{C}},\tilde{\mathcal{D}}))\subseteq{\mathcal{N}}(\tilde{\mathcal{A}},\tilde{\mathcal{B}})$ . For this, we use the relative strict topology on $\tilde{\mathcal{A}}$ , which we now describe.

For each $x\in{\mathcal{A}}$ , the map $\tilde{\mathcal{A}}\ni a\mapsto\left\lVert xa\right\rVert+\left\lVert ax\right\rVert$ is a seminorm on $\tilde{\mathcal{A}}$ , and the relative strict topology on $\tilde{\mathcal{A}}$ is the smallest topology on $\tilde{\mathcal{A}}$ making each of these seminorms continuous. Thus, a net $(a_{\lambda})$ in $\tilde{\mathcal{A}}$ converges to $a\in\tilde{\mathcal{A}}$ in the relative strict topology if and only if for every $x\in{\mathcal{A}}$ ,

\lim_{\lambda}(\left\lVert(a_{\lambda}-a)x\right\rVert+\left\lVert(x(a_{\lambda}-a)\right\rVert)=0.

We write $a=^{rs}\lim_{\lambda}a_{\lambda}$ when the net $(a_{\lambda})$ in $\tilde{\mathcal{A}}$ converges in the relative strict topology to $a\in\tilde{\mathcal{A}}$ . The relative strict topology on $\tilde{\mathcal{A}}$ is locally convex, and since ${\mathcal{A}}$ is an essential ideal in $\tilde{\mathcal{A}}$ , it is Hausdorff. Routine arguments show that when $\tilde{\mathcal{A}}$ is equipped with the relative strict topology, the adjoint operation is continuous and multiplication is jointly continuous on norm-bounded subsets of $\tilde{\mathcal{A}}$ .

We claim that if $(w_{\lambda})$ is a bounded net in ${\mathcal{N}}(\tilde{\mathcal{A}},\tilde{\mathcal{B}})$ which converges relative strictly to $w\in\tilde{\mathcal{A}}$ , then $w\in{\mathcal{N}}(\tilde{\mathcal{A}},\tilde{\mathcal{B}})$ . Indeed, for $h,k\in\tilde{\mathcal{B}}$ , as $w_{\lambda}^{*}hw_{\lambda}\in\tilde{\mathcal{B}}$ ,

w^{*}hwk={}^{rs}\lim_{\lambda}w_{\lambda}^{*}hw_{\lambda}k={}^{rs}\lim_{\lambda}kw_{\lambda}^{*}hw_{\lambda}=kw^{*}hw.

As $\tilde{\mathcal{B}}$ is a MASA in $\tilde{\mathcal{A}}$ , $w^{*}hw\in\tilde{\mathcal{B}}$ . Likewise $whw^{*}\in\tilde{\mathcal{B}}$ , so our claim holds.

Now let $(u_{\lambda})$ be a net in ${\mathcal{D}}$ which is an approximate unit for ${\mathcal{C}}$ . Choose $v\in{\mathcal{N}}(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ and write $v=(x,\xi)$ for some $x\in{\mathcal{C}}$ and $\xi\in{\mathbb{C}}$ . For each $\lambda$ , $u_{\lambda}\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ , so (since ${\mathcal{N}}(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is closed under multiplication) $vu_{\lambda}=(xu_{\lambda}+\xi u_{\lambda},0)\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ . The regularity of the map $\alpha$ and the fact that ${\mathcal{N}}({\mathcal{A}},{\mathcal{B}})\subseteq{\mathcal{N}}(\tilde{\mathcal{A}},\tilde{\mathcal{B}})$ (Lemma 3(c)) gives

{\mathcal{N}}({\mathcal{A}},{\mathcal{B}})\ni\alpha(vu_{\lambda})=\tilde{\alpha}(vu_{\lambda})\in{\mathcal{N}}(\tilde{\mathcal{A}},\tilde{\mathcal{B}}).

By Lemma 4.2, $\alpha(u_{\lambda})$ is an approximate unit for ${\mathcal{A}}$ . Therefore, $\alpha(u_{\lambda})$ converges in the relative strict topology to $I_{\tilde{\mathcal{A}}}$ . As $\tilde{\alpha}(vu_{\lambda})$ is a bounded net in ${\mathcal{N}}(\tilde{\mathcal{A}},\tilde{\mathcal{B}})$ , the claim gives

\tilde{\alpha}(v)=\tilde{\alpha}(v)\,{}^{rs}\lim_{\lambda}\alpha(u_{\lambda})={}^{rs}\lim_{\lambda}\tilde{\alpha}(vu_{\lambda})\in{\mathcal{N}}(\tilde{\mathcal{A}},\tilde{\mathcal{B}}).

Thus $\tilde{\alpha}$ is a regular $*$ -monomorphism. This completes the proof that $(\tilde{\mathcal{A}},\tilde{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\tilde{\alpha})$ is a Cartan package for $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ .

Now suppose that $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ . It remains to establish part (b) of Definition 3. By hypothesis, $({\mathcal{B}},\alpha({\mathcal{D}}))$ has the ideal intersection property, so $(\tilde{\mathcal{B}},\tilde{\alpha}(\tilde{\mathcal{D}}))=(\tilde{\mathcal{B}},(\alpha({\mathcal{D}}))^{\sim})$ has the ideal intersection property by Lemma 2.4. ∎

The following result shows that in some settings, a Cartan envelope for a non-unital inclusion may be constructed from a Cartan envelope for its unitization.

Lemma \the\numberby.

Let $({\mathcal{C}},{\mathcal{D}})$ be a regular inclusion having the AUP with ${\mathcal{C}}$ not unital. Suppose $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\pi)$ is a Cartan envelope for $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ and write $\Delta$ for the conditional expectation of ${\mathcal{A}}$ onto ${\mathcal{B}}$ . Put

\alpha:=\pi|_{\mathcal{C}},\quad{\mathcal{D}}_{1}:=C^{*}(\Delta(\alpha({\mathcal{C}}))),\quad\text{and}\quad{\mathcal{C}}_{1}:=C^{*}(\alpha({\mathcal{C}})\cup{\mathcal{D}}_{1}).

Then $({\mathcal{C}}_{1},{\mathcal{D}}_{1}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ ; further, ${\mathcal{C}}_{1}\unlhd{\mathcal{A}}$ and ${\mathcal{D}}_{1}\unlhd{\mathcal{B}}$ are ideals having codimension one.

Proof.

Note that since $({\mathcal{B}},\tilde{\mathcal{D}},\pi)$ and $(\tilde{\mathcal{D}},{\mathcal{D}})$ have the ideal intersection property, $({\mathcal{B}},{\mathcal{D}},\pi)$ also has the ideal intersection property. Proposition 3 shows $({\mathcal{C}}_{1},{\mathcal{D}}_{1}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ .

Let $I_{\mathcal{A}}$ denote the unit of ${\mathcal{A}}$ and consider the $C^{*}$ -algebras

{\mathcal{D}}_{1}^{+}={\mathcal{D}}_{1}+{\mathbb{C}}I_{\mathcal{A}}\quad\text{and}\quad{\mathcal{C}}_{1}^{+}={\mathcal{C}}_{1}+{\mathbb{C}}I_{\mathcal{A}}.

Obviously, ${\mathcal{D}}_{1}^{+}\subseteq{\mathcal{B}}$ , and ${\mathcal{C}}_{1}^{+}\subseteq{\mathcal{A}}$ ; Lemma 3 gives $\pi(0,1)=I_{\mathcal{A}}$ . Since $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan envelope for $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ , ${\mathcal{B}}$ is generated by $\pi(\Delta(\tilde{\mathcal{C}}))=\pi(\Delta({\mathcal{C}}))+{\mathbb{C}}I_{\mathcal{A}}$ and ${\mathcal{A}}$ is generated by ${\mathcal{B}}\cup\pi(\tilde{\mathcal{C}})={\mathcal{B}}\cup(\pi({\mathcal{C}})+{\mathbb{C}}I_{\mathcal{A}})$ . ${\mathcal{C}}_{1}^{+}={\mathcal{A}}$ . Thus,

(4.2.1)

(\tilde{\mathcal{C}}_{1},\tilde{\mathcal{D}}_{1})\simeq({\mathcal{C}}_{1}^{+},{\mathcal{D}}_{1}^{+})=({\mathcal{A}},{\mathcal{B}}).

That ${\mathcal{C}}_{1}\unlhd{\mathcal{A}}$ and ${\mathcal{D}}_{1}\unlhd{\mathcal{B}}$ are ideals of codimension one follows from (4.2.1). ∎

Theorem \the\numberby.

Let $({\mathcal{C}},{\mathcal{D}})$ be a regular inclusion with the AUP such that ${\mathcal{C}}$ is not unital. Then $({\mathcal{C}},{\mathcal{D}})$ is a pseudo-Cartan inclusion if and only if $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is a pseudo-Cartan inclusion. Moreover, the following statements hold.

(a)

If $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ , then $(\tilde{\mathcal{A}},\tilde{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\tilde{\alpha})$ is a Cartan envelope for $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ .

(b)

Suppose $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\pi)$ is a Cartan envelope for $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ and $\Delta:{\mathcal{A}}\rightarrow{\mathcal{B}}$ is the conditional expectation. Let

\alpha:=\pi|_{{\mathcal{C}}},\quad{\mathcal{D}}_{1}=C^{*}(\Delta(\pi({\mathcal{C}}))),\quad\text{and}\quad{\mathcal{C}}_{1}=C^{*}(\pi({\mathcal{C}})\cup{\mathcal{D}}_{1}).

Then ${\mathcal{D}}_{1}\unlhd{\mathcal{B}}$ and ${\mathcal{C}}_{1}\unlhd{\mathcal{A}}$ are ideals with codimension 1 and $({\mathcal{C}}_{1},{\mathcal{D}}_{1}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ .

Proof.

That $({\mathcal{C}},{\mathcal{D}})$ is a pseudo-Cartan inclusion if and only if $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is a pseudo-Cartan inclusion follows by combining Lemma 3(e) and Observation 4.2. Lemma 4.2 gives part (a), and Lemma 4.2 gives part (b). ∎

4.3. Some Properties Shared by a Pseudo-Cartan Inclusion and its Cartan Envelope

This subsection is devoted to establishing Theorem 4.3, which shows that certain desirable properties are common to both a pseudo-Cartan inclusion and its Cartan envelope.

Recall that Definition 3(a) states that $({\mathcal{C}}_{1},{\mathcal{D}}_{1}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is an essential expansion of $({\mathcal{C}},{\mathcal{D}})$ if the inclusion $({\mathcal{D}}_{1},{\mathcal{D}},\alpha|_{{\mathcal{D}}})$ has the ideal intersection property. We next observe that more can be said for essential expansions in the context of pseudo-Cartan inclusions.

Observation \the\numberby.

For $i=1,2$ , suppose $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ are pseudo-Cartan inclusions, and $({\mathcal{C}}_{2},{\mathcal{D}}_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is an essential expansion of $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ . Then the inclusion $({\mathcal{C}}_{2},{\mathcal{C}}_{1},\alpha)$ has the ideal intersection property.

Proof.

Suppose $J\unlhd{\mathcal{C}}_{2}$ and $J\cap\alpha({\mathcal{C}}_{1})=\{0\}$ . Then $J\cap\alpha({\mathcal{D}}_{1})=\{0\}$ . By hypothesis, $({\mathcal{D}}_{2},\alpha({\mathcal{D}}_{1}))$ has the ideal intersection property, so $J\cap{\mathcal{D}}_{2}=\{0\}$ . Since $({\mathcal{C}}_{2},{\mathcal{D}}_{2})$ is a pseudo-Cartan inclusion, both $({\mathcal{D}}_{2}^{c},{\mathcal{D}}_{2})$ and $({\mathcal{C}}_{2},{\mathcal{D}}_{2}^{c})$ have the ideal intersection property. As $(J\cap{\mathcal{D}}_{2}^{c})\cap{\mathcal{D}}_{2}\subseteq J\cap{\mathcal{D}}_{2}=\{0\}$ , we find $J\cap{\mathcal{D}}_{2}^{c}=\{0\}$ , whence $J=\{0\}$ . ∎

Lemma \the\numberby.

Let $({\mathcal{C}},{\mathcal{D}})$ be a pseudo-Cartan inclusion with Cartan envelope $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ . Let $J\unlhd{\mathcal{C}}$ , and define $J_{1}\subseteq{\mathcal{A}}$ to be the norm-closed ${\mathcal{B}}$ -bimodule generated by $\alpha(J)$ . Then $J_{1}\unlhd{\mathcal{A}}$ .

Proof.

Suppose $x\in J$ , $h\in{\mathcal{B}}$ and $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ . We first show that

(4.3.1)

\alpha(v)h\alpha(x)\in J_{1}.

Since $\alpha$ is a regular map, $\alpha(v)\in{\mathcal{N}}({\mathcal{A}},{\mathcal{B}})$ . By Lemma 3, given $u\in\overline{\alpha(v^{*}v){\mathcal{B}}}$ ,

\alpha(v)uh\alpha(x)=\theta_{\alpha(v)}^{-1}(uh)\alpha(v)\alpha(x).

As $\theta_{\alpha(v)}^{-1}(uh)\in\overline{\alpha(vv^{*}){\mathcal{B}}}$ and $\alpha(vx)\in\alpha(J)$ , we see that

(4.3.2)

\alpha(v)uh\alpha(x)\in J_{1}.

Since $({\mathcal{A}},{\mathcal{B}})$ is Cartan and $\alpha$ is a regular map, Lemma 3(b) gives $\alpha(v^{*}v)\in{\mathcal{B}}^{c}={\mathcal{B}}$ . By considering an approximate unit for ${\mathcal{B}}$ , we find $\alpha(v^{*}v)\in\overline{\alpha(v^{*}v){\mathcal{B}}}$ . Since $\alpha(v)\alpha(v^{*}v)^{1/n}\rightarrow\alpha(v)$ , taking $u=\alpha(v^{*}v)^{1/n}$ in (4.3.2), we obtain (4.3.1).

Next using (4.3.1), we see that if $n\in{\mathbb{N}}$ and $x_{i}\in J$ , $h_{i},k_{i}\in{\mathcal{B}}$ for $1\leq i\leq n$ and $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ ,

\alpha(v)\left(\sum_{i=1}^{n}h_{i}\alpha(x_{i})k_{i}\right)\in J_{1},

from which it follows that

(4.3.3)

\alpha(v)J_{1}\subseteq J_{1}.

Let $M$ be the set of all finite products of elements of ${\mathcal{B}}\cup\alpha({\mathcal{N}}({\mathcal{C}},{\mathcal{D}}))$ . An induction argument using (4.3.3) shows that for $w\in M$ and $x_{1}\in J_{1}$ , $wx_{1}\in J_{1}$ . As $\operatorname{span}M$ is dense in ${\mathcal{A}}$ , we conclude that $J_{1}$ is a left ideal of ${\mathcal{A}}$ . Similar arguments show $J_{1}$ is a right ideal, so $J_{1}\unlhd{\mathcal{B}}$ . ∎

Theorem \the\numberby.

Let $({\mathcal{C}},{\mathcal{D}})$ be a pseudo-Cartan inclusion with Cartan envelope $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ . Then

(a)

${\mathcal{C}}$ is simple if and only if ${\mathcal{A}}$ is simple; and
(b)

${\mathcal{C}}$ is unital if and only if ${\mathcal{A}}$ is unital.
(c)

${\mathcal{C}}$ is separable if and only if ${\mathcal{A}}$ is separable.

Proof.

Proposition 4.2 shows that $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is also a Cartan envelope for $({\mathcal{C}},{\mathcal{D}}^{c})$ . Thus, by replacing ${\mathcal{D}}$ with ${\mathcal{D}}^{c}$ if necessary, without loss of generality we may assume $({\mathcal{C}},{\mathcal{D}})$ is a pseudo-Cartan inclusion with the AUP.

Let $\Delta:{\mathcal{A}}\rightarrow{\mathcal{B}}$ be the conditional expectation and let $E:{\mathcal{C}}\rightarrow{\mathcal{B}}$ be the map

E:=\Delta\circ\alpha.

(a) Suppose ${\mathcal{C}}$ is simple and let $J\unlhd{\mathcal{A}}$ . Then $J\cap\alpha({\mathcal{C}})\in\{\{0\},\alpha({\mathcal{C}})\}$ . If $J\cap\alpha({\mathcal{C}})=\{0\}$ , Observation 4.3 shows $J=\{0\}$ . On the other hand, suppose $J\cap\alpha({\mathcal{C}})=\alpha({\mathcal{C}})$ . Since we are assuming $({\mathcal{C}},{\mathcal{D}})$ has the AUP, we may choose a net $(e_{\lambda})$ in ${\mathcal{D}}$ which is an approximate unit for ${\mathcal{C}}$ . As ${\mathcal{B}}=C^{*}(E({\mathcal{C}}))$ , we find $(\alpha(e_{\lambda}))$ is an approximate unit for ${\mathcal{B}}$ . But $({\mathcal{A}},{\mathcal{B}})$ is a Cartan inclusion, so by [PittsNoApUnInC*Al, Theorem 2.6], every approximate unit for ${\mathcal{B}}$ is an approximate unit for ${\mathcal{A}}$ . As $\alpha({\mathcal{D}})\subseteq\alpha({\mathcal{C}})\subseteq J$ , we conclude that $J$ contains an approximate unit for ${\mathcal{A}}$ , whence $J={\mathcal{A}}$ . Thus ${\mathcal{A}}$ is simple.

For the converse, suppose ${\mathcal{A}}$ is simple. Let $J\unlhd{\mathcal{C}}$ be a non-zero ideal, let $K$ be the ideal in ${\mathcal{B}}$ generated by $\alpha(J\cap{\mathcal{D}})$ and put

J^{\prime}:=\{x\in{\mathcal{C}}:E(x^{*}x)\in K\}.

We aim to show $J^{\prime}$ is a non-zero ideal in ${\mathcal{C}}$ . By construction, $J^{\prime}$ is a closed subset of ${\mathcal{C}}$ .

Next we show $J^{\prime}$ is a non-zero subspace of ${\mathcal{C}}$ . For $\tau\in\hat{\mathcal{B}}$ , $\tau\circ E$ is a state on ${\mathcal{C}}$ , so the map,

{\mathcal{C}}\ni x\mapsto(\tau(E(x^{*}x)))^{1/2}

is a semi-norm on ${\mathcal{C}}$ . Thus for $x,y\in{\mathcal{C}}$ , $\tau(E((x+y)^{*}(x+y)))^{1/2}\leq\tau(E(x^{*}x))^{1/2}+\tau(E(y^{*}y))^{1/2}$ . Allowing $\tau$ to vary throughout $\hat{\mathcal{B}}$ , we conclude

E((x+y)^{*}(x+y))\leq(E(x^{*}x)^{1/2}+E(y^{*}y)^{1/2})^{2}.

Thus when $x,y\in J^{\prime}$ we obtain $x+y\in J^{\prime}$ . That $J^{\prime}$ is invariant under scalar multiplication is obvious, so $J^{\prime}$ is a closed linear subspace of ${\mathcal{C}}$ . Clearly $J\cap{\mathcal{D}}\subseteq J^{\prime}$ . Since $({\mathcal{C}},{\mathcal{D}}^{c})$ and $({\mathcal{D}}^{c},{\mathcal{D}})$ both have the ideal intersection property, $J\cap{\mathcal{D}}\neq\{0\}$ , and hence $J^{\prime}\neq\{0\}$ .

We are now ready to show $J^{\prime}$ is an ideal. For $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ , $v^{*}(J\cap{\mathcal{D}})v\subseteq J\cap{\mathcal{D}}$ , and regularity of $\alpha$ gives $\alpha(v)\in{\mathcal{N}}({\mathcal{A}},{\mathcal{B}})$ . So for $b\in{\mathcal{B}}$ and $d\in J\cap D$ , we have

	$\displaystyle\alpha(v^{*})\alpha(d)b\alpha(v)$	$\displaystyle=\lim\alpha(v^{})\alpha(d)b\alpha(vv^{})^{1/n}\alpha(v)$
		$\displaystyle=\lim(\alpha(v)^{}\alpha(d)\alpha(v))\theta_{\alpha(v)}(\alpha(vv^{})^{1/n}b)\in K.$

It follows that

\alpha(v)^{*}K\alpha(v)\subseteq K,\quad v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}}).

Therefore, when $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ and $x\in J^{\prime}$ , $xv\in J^{\prime}$ because

	$\displaystyle E(v^{}x^{}xv)$	$\displaystyle=\Delta(\alpha(v)^{}\alpha(x^{}x)\alpha(v))=\alpha(v)^{}\Delta(\alpha(x^{}x))\alpha(v)$
		$\displaystyle=\alpha(v)^{}E(x^{}x)\alpha(v)\in K.$

Since $\operatorname{span}{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ is dense in ${\mathcal{C}}$ , we conclude that $J^{\prime}$ is a right ideal in ${\mathcal{C}}$ . For $x\in J^{\prime}$ and $y\in{\mathcal{C}}$ , $E(x^{*}y^{*}yx)\leq\left\lVert y\right\rVert^{2}E(x^{*}x)$ , whence $J^{\prime}$ is a left ideal in ${\mathcal{C}}$ . Thus $\{0\}\neq J^{\prime}\unlhd{\mathcal{C}}$ .

Next we show that $J\cap{\mathcal{D}}={\mathcal{D}}$ . For this, let $\sigma\in\hat{\mathcal{D}}$ and let $\tau\in\hat{\mathcal{B}}$ satisfy $\tau\circ\alpha=\sigma$ . Let $M$ be the norm-closed ${\mathcal{B}}$ -bimodule generated by $\alpha(J^{\prime})$ . By Lemma 4.3 and the hypothesis that ${\mathcal{A}}$ is simple,

M={\mathcal{A}}.

Let $0\leq b\in{\mathcal{B}}$ be chosen so that $\left\lVert b\right\rVert=1$ and $\tau(b)=1$ . Since $b\in M$ there is $n\in{\mathbb{N}}$ , $x_{i}\in J^{\prime}$ , and $h_{i},k_{i}\in{\mathcal{B}}$ ( $1\leq i\leq n$ ) so that

\left\lVert b-\sum_{i=1}^{n}h_{i}\alpha(x_{i})k_{i}\right\rVert<1/2.

Then

\left|\tau\left(\Delta\left(b-\sum_{i=1}^{n}h_{i}\alpha(x_{i})k_{i}\right)\right)\right|=\left|1-\sum_{i=1}^{n}\tau(h_{i})\tau(\Delta(\alpha(x_{i})))\tau(k_{i})\right|<1/2.

It follows that there exists $x\in J^{\prime}$ such that $\tau(\Delta(\alpha(x)))\neq 0$ . Hence

0\neq|\tau(\Delta(\alpha(x)))|^{2}=\tau(E(x^{*})E(x))\leq\tau(E(x^{*}x)).

Since $x\in J^{\prime}$ , $E(x^{*}x)\in K$ , and thus $\tau$ does not annihilate $K$ . But $K$ is the ideal of ${\mathcal{B}}$ generated by $\alpha(J\cap{\mathcal{D}})$ , so $\tau$ does not annihilate $\alpha(J\cap{\mathcal{D}})$ . Therefore, $\sigma|_{J\cap{\mathcal{D}}}\neq 0$ . Since this holds for all $\sigma\in\hat{\mathcal{D}}$ , we conclude that $J\cap{\mathcal{D}}={\mathcal{D}}$ .

Finally, the fact that ${\mathcal{D}}$ contains an approximate unit for ${\mathcal{C}}$ implies that $J={\mathcal{C}}$ . Thus ${\mathcal{C}}$ is simple.

(b) Suppose ${\mathcal{C}}$ is unital. Lemma 3 shows ${\mathcal{A}}$ is unital and $\alpha(I_{\mathcal{C}})=I_{\mathcal{A}}$ .

Now suppose ${\mathcal{A}}$ is unital. Let $(u_{\lambda})\subseteq{\mathcal{D}}$ be an approximate unit for ${\mathcal{C}}$ . Choose $x_{1},\dots,x_{n}\in{\mathcal{C}}$ and let

z=\prod_{k=1}^{n}\Delta(\alpha(x_{k})).

Since $\Delta(\alpha(x_{n}))\,\alpha(u_{\lambda})=\Delta(\alpha(x_{n}u_{\lambda}))$ , $z\alpha(u_{\lambda})\rightarrow z$ ; likewise, $\alpha(u_{\lambda})z\rightarrow z$ . As the collection of all finite products of $\Delta(\alpha({\mathcal{C}}))$ has dense span in ${\mathcal{B}}$ , it follows that $\alpha(u_{\lambda})$ is an approximate unit for ${\mathcal{B}}$ , and hence also for ${\mathcal{A}}$ (because $({\mathcal{A}},{\mathcal{B}})$ has the AUP). Therefore,

\left\lVert\alpha(u_{\lambda})I_{\mathcal{A}}-I_{\mathcal{A}}\right\rVert=\left\lVert\alpha(u_{\lambda})-I_{\mathcal{A}}\right\rVert\rightarrow 0.

Hence $(u_{\lambda})$ converges to an element $e\in{\mathcal{C}}$ . Since $\alpha(e)=I_{\mathcal{A}}$ , we find $e=I_{\mathcal{C}}$ .

(c) Since every non-empty subset of a separable metric space is a separable metric space, separability of ${\mathcal{A}}$ implies $\alpha({\mathcal{C}})$ is separable. So ${\mathcal{C}}$ is separable.

Now suppose ${\mathcal{C}}$ is separable and let $Q\subseteq{\mathcal{C}}$ be countable and dense. Then ${\mathcal{A}}$ is generated by $\alpha(Q)\cup\Delta(\alpha(Q))$ , so ${\mathcal{A}}$ is separable. ∎

It would be desirable to include nuclearity in the list of properties shared by ${\mathcal{C}}$ and ${\mathcal{A}}$ .

Conjecture \the\numberby.

Let $({\mathcal{C}},{\mathcal{D}})$ be a pseudo-Cartan inclusion and let $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ be a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ . Then ${\mathcal{C}}$ is nuclear if and only if ${\mathcal{A}}$ is nuclear.

4.4. Constructing Pseudo-Cartan Inclusions from Cartan Inclusions

We have seen that every pseudo-Cartan inclusion has a Cartan envelope, and we now consider the reverse process, that of constructing pseudo-Cartan inclusions from a given Cartan inclusion. We will work more generally, starting instead with a given pseudo-Cartan inclusion and constructing new pseudo-Cartan inclusions from it. Our next result, Proposition 4.4, extends parts of [PittsStReInII, Lemma 5.26 and Proposition 5.31] to the setting of pseudo-Cartan inclusions.

Proposition \the\numberby.

Suppose $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ is a pseudo-Cartan inclusion and ${\mathcal{D}}$ is a $C^{*}$ -subalgebra of ${\mathcal{D}}_{1}$ such that ${\mathcal{D}}\subseteq{\mathcal{D}}_{1}$ has the ideal intersection property. Let ${\mathcal{M}}\subseteq{\mathcal{N}}({\mathcal{C}}_{1},{\mathcal{D}})$ be a $*$ -semigroup such that ${\mathcal{D}}\subseteq\overline{\operatorname{span}}\,{\mathcal{M}}$ and set ${\mathcal{C}}=\overline{\operatorname{span}}\,{\mathcal{M}}$ . The following statements hold.

(a)

$({\mathcal{C}},{\mathcal{D}})$ is a pseudo-Cartan inclusion.

(b)

Let $({\mathcal{A}}_{1},{\mathcal{B}}_{1}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ be a Cartan envelope for $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ and let $\Delta_{1}:{\mathcal{A}}_{1}\rightarrow{\mathcal{B}}_{1}$ be the conditional expectation. Set

{\mathcal{B}}:=C^{*}(\Delta_{1}(\alpha({\mathcal{C}})))\quad\text{and}\quad{\mathcal{A}}:=C^{*}(\alpha({\mathcal{C}})\cup{\mathcal{B}}).

Then $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha|_{\mathcal{C}})$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ .

Proof.

(a) Since ${\mathcal{M}}$ is a $*$ -semigroup, ${\mathcal{C}}$ is a $C^{*}$ -algebra. Therefore, $({\mathcal{C}},{\mathcal{D}})$ is a regular inclusion. Since $({\mathcal{C}}_{1},{\mathcal{D}}_{1}\hbox{\,:\hskip-1.0pt:\,}\subseteq)$ is an essential expansion of $({\mathcal{C}},{\mathcal{D}})$ , Lemma 3(a) shows $({\mathcal{C}},{\mathcal{D}})$ has the faithful unique pseudo-expectation property. Hence $({\mathcal{C}},{\mathcal{D}})$ is a pseudo-Cartan inclusion.

(b) As $({\mathcal{D}}_{1},{\mathcal{D}})$ and $({\mathcal{B}}_{1},\alpha({\mathcal{D}}_{1}))$ have the ideal intersection property, so does $({\mathcal{B}}_{1},\alpha({\mathcal{D}}))$ . Therefore $({\mathcal{A}}_{1},{\mathcal{B}}_{1}\hbox{\,:\hskip-1.0pt:\,}\alpha|_{{\mathcal{D}}})$ is an essential and Cartan expansion of $({\mathcal{C}},{\mathcal{D}})$ . Applying Proposition 3, we obtain (b). ∎

While Proposition 4.4 applies when $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ is a Cartan inclusion, in general, $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ need not be the Cartan envelope of $({\mathcal{C}},{\mathcal{D}})$ . In fact, $({\mathcal{C}},{\mathcal{D}})$ can itself be Cartan. A trivial example of this occurs when ${\mathcal{M}}$ is taken to be ${\mathcal{D}}$ , in which case, $({\mathcal{C}},{\mathcal{D}})=({\mathcal{D}},{\mathcal{D}})$ is a Cartan inclusion. Here is a more interesting example where $({\mathcal{C}},{\mathcal{D}})$ is a Cartan inclusion.

Example \the\numberby. Let the amenable discrete group $\Gamma$ act topologically freely on the compact Hausdorff space $X$ . Suppose $Z\subseteq X$ is a non-empty closed invariant set which is nowhere dense in $X$ . Then

{\mathcal{D}}:=\{f\in C(X):f|_{Z}\text{ is constant}\}

has the ideal intersection property in $C(X)$ . Let $Y=\hat{\mathcal{D}}$ and fix $x_{0}\in Z$ . Since every $f\in{\mathcal{D}}$ is constant on $Z$ , for every $f\in{\mathcal{D}}$ , $f-f(x_{0})\in C_{0}(X\setminus Z)$ , and it follows that $Y$ is homeomorphic to the one point compactification of $X\setminus Z$ . Let $\pi:C(Y)\rightarrow{\mathcal{D}}$ be the inverse of the Gelfand transformation and let $\{U_{s}:s\in\Gamma\}$ be the canonical copy of $\Gamma$ in $C(X)\rtimes_{r}\Gamma$ . The restriction of the action of $\Gamma$ to $X\setminus Z$ is topologically free, and hence determines a topologically free action of $\Gamma$ on $Y$ .

Letting $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ be the Cartan inclusion $(C(X)\rtimes_{r}\Gamma,C(X))$ , apply Proposition 4.4 to $({\mathcal{D}}_{1},{\mathcal{D}})$ to produce a pseudo-Cartan inclusion $({\mathcal{C}},{\mathcal{D}})$ . Since $Z$ is a $\Gamma$ invariant set, ${\mathcal{D}}$ is invariant under $\{U_{s}\}_{s\in\Gamma}$ . Therefore, the pair $(\pi,U)$ is a covariant representation for the action of $\Gamma$ on $Y$ . Since ${\mathcal{C}}$ is generated by ${\mathcal{D}}$ and $\{U_{s}\}_{s\in\Gamma}$ , we obtain a $*$ -epimorphism $(\pi\times U):C(Y)\rtimes_{f}\Gamma=C(Y)\rtimes_{r}\Gamma\rightarrow{\mathcal{C}}$ . The topological freeness of $\Gamma$ acting on $Y$ and Proposition 4.4 implies that $(C(Y)\rtimes_{r}\Gamma,C(Y))$ is a Cartan pair, and in particular, has the ideal intersection property. As $\pi$ is faithful, we conclude that $(C(Y)\rtimes\Gamma,C(Y))$ is isomorphic to $({\mathcal{C}},{\mathcal{D}})$ . Thus $({\mathcal{C}},{\mathcal{D}})$ is already Cartan, so is its own Cartan envelope. This shows $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ is not the Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ .

It is natural to wonder when a reduced crossed product is a pseudo-Cartan inclusion. This question is answered by the following result.

Proposition \the\numberby.

Let $X$ be a locally compact Hausdorff space and suppose $\Gamma$ is a discrete group acting as homeomorphisms on $X$ . With the corresponding action of $\Gamma$ on $C_{0}(X)$ , the following statements are equivalent:

(a)

$(C_{0}(X)\rtimes_{r}\Gamma,C_{0}(X))$ is a pseudo-Cartan inclusion;
(b)

$(C_{0}(X)\rtimes_{r}\Gamma,C_{0}(X))$ is a Cartan inclusion; and
(c)

the action of $\Gamma$ on $X$ is topologically free.

Proof.

For each of $(a)\Leftrightarrow(b)$ and $(b)\Leftrightarrow(c)$ , we use the fact that $(C_{0}(X)\rtimes_{r}\Gamma,C_{0}(X))$ is a regular inclusion and there is a faithful conditional expectation $\Delta:C_{0}(X)\rtimes\Gamma\rightarrow C_{0}(X)$ .

$(a)\Leftrightarrow(b)$ . Apply Proposition 2.3.

$(b)\Leftrightarrow(c)$ . By [Zeller-MeierPrCrCstAlGrAu, Proposition 4.14], $(C_{0}(X)\rtimes_{r}\Gamma,C_{0}(X))$ is a MASA inclusion if and only if the action of $\Gamma$ on $X$ is topologically free. ∎

Remark \the\numberby. Despite the equivalence of parts (a) and (b) in Proposition 4.4, reduced crossed products can be used to construct examples of pseudo-Cartan inclusions which are not Cartan inclusions. Indeed, [PittsStReInI, Theorem 6.15] shows that reduced crossed products can be used to construct unital virtual Cartan inclusions. Thus by combining Proposition 4.4 with [PittsStReInI, Theorem 6.15], we obtain a wide variety of pseudo-Cartan inclusions.

5. The Twisted Groupoid of the Cartan Envelope

In [PittsStReInII, Section 7], we described the twist for the Cartan envelope of a unital regular inclusion. Combining Theorem 4.2 with results of [PittsStReInII] allows us to describe the twist associated to the Cartan envelope for any pseudo-Cartan inclusion regardless of whether it is unital. This description is found in Theorem 5.2 below.

5.1. Reprising the Unital Case

We begin with reprising some definitions and results from [PittsStReInII, Section 7]. Because we shall also require these notions in Section 6.3, we shall first consider unital regular inclusions with the unique (but not necessarily faithful) pseudo-expectation property. We then turn our attention to unital pseudo-Cartan inclusions.

Let $({\mathcal{C}},{\mathcal{D}})$ be a regular and unital inclusion with the unique pseudo-expectation property and let $(I({\mathcal{D}}),\iota)$ be an injective envelope for $({\mathcal{C}},{\mathcal{D}})$ . Let $E:{\mathcal{C}}\rightarrow I({\mathcal{D}})$ be the pseudo-expectation.

(a)

An eigenfunctional (see [DonsigPittsCoSyBoIs, Definition 2.1]) on ${\mathcal{C}}$ is a non-zero bounded linear functional $\phi$ on ${\mathcal{C}}$ which is an eigenvector for the left and right actions of ${\mathcal{D}}$ on the dual space of ${\mathcal{C}}$ . When $\phi$ is an eigenfunctional, there exist unique $r(\phi)\in\hat{\mathcal{D}}$ and $s(\phi)\in\hat{\mathcal{D}}$ such that for every $d_{1},d_{2}\in{\mathcal{D}}$ and $x\in{\mathcal{C}}$ ,

$\phi(d_{1}xd_{2})=r(\phi)(d_{1})\,\phi(x)\,s(\phi)(d_{2}).$

If for every $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ ,

$|\phi(v)|^{2}\in\{0,s(\phi)(v^{*}v)\},$

$\phi$ is a compatible eigenfunctional ([PittsStReInII, Definition 7.6]). A compatible state is a state on ${\mathcal{C}}$ which is also a compatible eigenfunctional. The collection of all compatible eigenfunctionals of unit norm is denoted ${\mathcal{E}}^{1}_{c}({\mathcal{C}},{\mathcal{D}})$ and ${\mathfrak{S}}({\mathcal{C}},{\mathcal{D}})$ denotes the set of compatible states.

(b)

For $\phi\in{\mathcal{E}}^{1}_{c}({\mathcal{C}},{\mathcal{D}})$ , [PittsStReInII, Theorem 7.9] shows that there are unique states ${\mathfrak{s}}(\phi),{\mathfrak{r}}(\phi)$ on ${\mathcal{C}}$ with the following properties:

•

$s(\phi)={\mathfrak{s}}(\phi)|_{\mathcal{D}}$ , $r(\phi)={\mathfrak{r}}(\phi)|_{\mathcal{D}}$ ;
•

when $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ satisfies $\phi(v)\neq 0$ and $x\in{\mathcal{C}}$ ,

$\phi(vx)=\phi(v)\,{\mathfrak{s}}(\phi)(x)\quad\text{and}\quad\phi(xv)={\mathfrak{r}}(\phi)(x)\,\phi(v).$

For later use, here are formulas for ${\mathfrak{s}}(\phi)$ and ${\mathfrak{r}}(\phi)$ : for $x\in{\mathcal{C}}$ and $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ with $\phi(v)>0$ ,

(5.1.1)

{\mathfrak{s}}(\phi)(x)=\frac{\phi(vx)}{\phi(v)}\quad\text{and}\quad{\mathfrak{r}}(\phi)(x)=\frac{\phi(xv)}{\phi(v)}.

In addition, note that (still assuming $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ satisfies $\phi(v)>0$ ) ${\mathfrak{s}}(\phi)(v^{*}v)\neq 0$ and for every $x\in{\mathcal{C}}$ ,

\phi(x)=\frac{{\mathfrak{s}}(\phi)(v^{*}x)}{{\mathfrak{s}}(\phi)(v^{*}v)^{1/2}}.

(c)

For $\rho\in{\mathfrak{S}}({\mathcal{C}},{\mathcal{D}})$ and $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ with $\rho(v^{*}v)\neq 0$ , we use the notation $[v,\rho]$ for the linear functional

[v,\rho](x):=\frac{\rho(v^{*}x)}{\rho(v^{*}v)^{1/2}}.

By [PittsStReInII, Corollary 7.11],

{\mathcal{E}}^{1}_{c}({\mathcal{C}},{\mathcal{D}})=\{[v,\rho]:\rho\in{\mathfrak{S}}({\mathcal{C}},{\mathcal{D}})\text{ and }\rho(v^{*}v)\neq 0\}.

(d)

A strongly compatible state on ${\mathcal{C}}$ is a state $\rho$ on ${\mathcal{C}}$ for which there exists $\sigma\in\widehat{I({\mathcal{D}})}$ such that

$\rho=\sigma\circ E.$

We denote the family of all strongly compatible states on ${\mathcal{C}}$ by ${\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})$ . Since

$\hat{\mathcal{D}}=\{\rho|_{\mathcal{D}}:\rho\in{\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})\},$

there is a rich supply of strongly compatible states.

By [PittsStReInII, Theorem 6.9] (whose statement is reproduced in Theorem A) every strongly compatible state is a compatible state, that is,

${\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})\subseteq{\mathfrak{S}}({\mathcal{C}},{\mathcal{D}}).$
(e)

A compatible eigenfunctional $\phi\in{\mathcal{E}}^{1}_{c}({\mathcal{C}},{\mathcal{D}})$ is called a strongly compatible eigenfunctional if ${\mathfrak{s}}(\phi)\in{\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})$ . When this occurs, ${\mathfrak{r}}(\phi)$ also belongs to ${\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})$ .
(f)

Let $\Sigma({\mathcal{C}},{\mathcal{D}})$ be the collection of all norm-one strongly compatible eigenfunctionals and put

$G({\mathcal{C}},{\mathcal{D}}):=\{|\phi|:\phi\in\Sigma({\mathcal{C}},{\mathcal{D}})\},$

where $|\phi|$ denotes the composition of the absolute value function on ${\mathbb{C}}$ with $\phi$ .

(g)

The representation of $\phi\in\Sigma({\mathcal{C}},{\mathcal{D}})$ as $\phi=[v,{\mathfrak{s}}(\phi)]$ , allows the sets $\Sigma({\mathcal{C}},{\mathcal{D}})$ and $G({\mathcal{C}},{\mathcal{D}})$ to be equipped with groupoid operations [PittsStReInII, Definition 7.17]. Upon doing so, the unit space, $G({\mathcal{C}},{\mathcal{D}})^{(0)}$ , may be identified with ${\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})$ . With the topology of pointwise convergence, $\Sigma({\mathcal{C}},{\mathcal{D}})$ and $G({\mathcal{C}},{\mathcal{D}})$ become Hausdorff topological groupoids (take $F={\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})$ in [PittsStReInII, Theorem 7.18]) and we obtain a twist

G({\mathcal{C}},{\mathcal{D}})^{(0)}\times{\mathbb{T}}\hookrightarrow\Sigma({\mathcal{C}},{\mathcal{D}})\twoheadrightarrow G({\mathcal{C}},{\mathcal{D}})

associated with $({\mathcal{C}},{\mathcal{D}})$ .

(h)

For each $a\in{\mathcal{C}}$ , let $\mathfrak{g}(a):\Sigma({\mathcal{C}},{\mathcal{D}})\rightarrow{\mathbb{C}}$ be given by

$\mathfrak{g}(a)(\phi)=\phi(a).$

(i)

Set

{\mathcal{L}}({\mathcal{C}},{\mathcal{D}}):=\{a\in{\mathcal{C}}:\rho(a^{*}a)=0\text{ for all }\rho\in{\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})\}=\{a\in{\mathcal{C}}:E(a^{*}a)=0\}.

By [PittsStReInII, Theorem 6.5], ${\mathcal{L}}({\mathcal{C}},{\mathcal{D}})$ is an ideal of ${\mathcal{C}}$ such that ${\mathcal{L}}({\mathcal{C}},{\mathcal{D}})\cap{\mathcal{D}}=\{0\}$ . Also, if $J\unlhd{\mathcal{C}}$ satisfies $J\cap{\mathcal{D}}=\{0\}$ , then $J\subseteq{\mathcal{L}}({\mathcal{C}},{\mathcal{D}})$ .

Now suppose $({\mathcal{C}},{\mathcal{D}})$ is a unital pseudo-Cartan inclusion. Recall from Proposition 2.4 that $({\mathcal{D}}^{c},{\mathcal{D}})$ has the faithful unique pseudo-expectation property and $E|_{{\mathcal{D}}^{c}}$ is a $*$ -monomorphism; also $E|_{{\mathcal{D}}^{c}}$ is the pseudo-expectation for $({\mathcal{D}}^{c},{\mathcal{D}})$ . Simplify the notation of 5.1(f) somewhat by writing $\Sigma$ and $G$ instead of $\Sigma({\mathcal{C}},{\mathcal{D}})$ and $G({\mathcal{C}},{\mathcal{D}})$ . As in 5.1(g), let

G^{(0)}\times{\mathbb{T}}\hookrightarrow\Sigma\twoheadrightarrow G,

be the twist for $({\mathcal{C}},{\mathcal{D}})$ . We remind the reader that $G^{(0)}$ is identified with ${\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})$ .

By [PittsStReInII, Theorem 7.24], $\mathfrak{g}$ induces a regular $*$ -monomorphism $\theta:{\mathcal{C}}\rightarrow C^{*}_{r}(\Sigma,G)$ and by [PittsStReInII, Corollary 7.30],

(5.1.2)

(C^{*}_{r}(\Sigma,G),C(G^{(0)})\hbox{\,:\hskip-1.0pt:\,}\theta)

is a package for $({\mathcal{C}},{\mathcal{D}})$ . While [PittsStReInII, Corollary 7.31] states $(C^{*}_{r}(\Sigma,G),C(G^{(0)})\hbox{\,:\hskip-1.0pt:\,}\theta)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ , full details were omitted in [PittsStReInII]; [PittsCoStReInII, Remark 1.2] provides the missing details.

5.2. The Non-Unital Case

Let $({\mathcal{C}},{\mathcal{D}})$ be a non-unital pseudo-Cartan inclusion, and fix an injective envelope $(I({\mathcal{D}}),\iota)$ for ${\mathcal{D}}$ . As $({\mathcal{D}}^{c},{\mathcal{D}})$ has the ideal intersection property, there is a unique $*$ -monomorphism $u:{\mathcal{D}}^{c}\rightarrow I({\mathcal{D}})$ such that $u|_{\mathcal{D}}=\iota|_{\mathcal{D}}$ . Thus $(I({\mathcal{D}}),\tilde{u})$ is an injective envelope for ${\mathcal{D}}^{c}$ . It follows that $E:{\mathcal{C}}\rightarrow I({\mathcal{D}})$ is a pseudo-expectation for $({\mathcal{C}},{\mathcal{D}})$ relative to $(I({\mathcal{D}}),\iota)$ if and only if $E$ is a pseudo-expectation for $({\mathcal{C}},{\mathcal{D}}^{c})$ relative to $(I({\mathcal{D}}),u)$ . Since $({\mathcal{C}},{\mathcal{D}}^{c})$ and $({\mathcal{C}},{\mathcal{D}})$ have the faithful unique pseudo-expectation property, we may define ${\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})$ and ${\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}}^{c})$ as in 5.1(d). Then

(5.2.1)

{\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})={\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}}^{c}).

Recalling from Proposition 4.2 that $({\mathcal{C}},{\mathcal{D}})$ and $({\mathcal{C}},{\mathcal{D}}^{c})$ have the same Cartan envelope, we will give a groupoid description for a Cartan envelope of $({\mathcal{C}},{\mathcal{D}}^{c})$ .

Let $\tilde{\Sigma}$ be the set of norm-one strongly compatible eigenfunctionals for $(\tilde{\mathcal{C}},({\mathcal{D}}^{c})^{\sim})$ and let $\tilde{G}=\{|\phi|:\phi\in\tilde{\Sigma}\}$ . Applying the discussion for the unital case to $(\tilde{\mathcal{C}},({\mathcal{D}}^{c})^{\sim})$ we obtain the twist,

\tilde{G}^{(0)}\times{\mathbb{T}}\rightarrow\tilde{\Sigma}\twoheadrightarrow\tilde{G}

with unit space $\tilde{G}^{(0)}={\mathfrak{S}}_{s}(\tilde{\mathcal{C}},({\mathcal{D}}^{c})^{\sim})$ . Taking $\theta$ as in (5.1.2), $(C^{*}_{r}(\tilde{\Sigma},\tilde{G}),C(\tilde{G}^{(0)})\hbox{\,:\hskip-1.0pt:\,}\theta)$ is a Cartan envelope for $(\tilde{\mathcal{C}},({\mathcal{D}}^{c})^{\sim})$ . We use $\Delta$ to denote the conditional expectation of $C^{*}_{r}(\tilde{\Sigma},\tilde{G})$ onto $C(\tilde{G}^{(0)})$ ; note that $\Delta$ arises from the restriction of an element in $C_{c}(\tilde{\Sigma},\tilde{G})$ to $\tilde{G}^{(0)}$ .

Let $q$ denote the linear functional, $\tilde{\mathcal{C}}\ni(c,\lambda)\mapsto\lambda$ . Since $q$ is a multiplicative linear functional, it is a compatible state for $(\tilde{\mathcal{C}},({\mathcal{D}}^{c})^{\sim})$ , and we claim it is actually a strongly compatible state. Let $\psi$ be a multiplicative linear functional on $I({\mathcal{D}}^{c})$ satisfying $\psi\circ\tilde{u}=q|_{({\mathcal{D}}^{c})^{\sim}}$ . Let $E:{\mathcal{C}}\rightarrow I({\mathcal{D}})$ be the pseudo-expectation and $\tilde{E}$ its unitization. Given $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}}^{c})$ , we have

\tilde{E}(v)^{*}\tilde{E}(v)\leq\tilde{E}(v^{*}v)=u(v^{*}v).

Applying $\psi$ yields,

|\psi(\tilde{E}(v))|^{2}\leq\psi(u(v^{*}v))=q(v^{*}v)=0.

It follows that $\psi\circ\tilde{E}$ annihilates ${\mathcal{N}}({\mathcal{C}},{\mathcal{D}}^{c})$ , and hence annihilates ${\mathcal{C}}$ . As $q$ and $\psi\circ\tilde{E}$ are states on $\tilde{\mathcal{C}}$ which have the same kernel, we obtain

q=\psi\circ\tilde{E},

showing that $q$ is a strongly compatible state.

Let

\Sigma:=\{\phi\in\tilde{\Sigma}:\phi|_{\mathcal{C}}\neq 0\}\quad\text{and}\quad G:=\{|\phi|:\phi\in\tilde{\Sigma}\text{ and }\phi|_{\mathcal{C}}\neq 0\}

be equipped with their relative topologies. Note that if $\phi\in\tilde{\Sigma}$ satisfies $\phi|_{\mathcal{C}}=0$ , then $\ker\phi=\ker q={\mathcal{C}}$ , so $\phi$ is a scalar multiple of $q$ . It follows that

\Sigma=\tilde{\Sigma}\setminus{\mathbb{T}}q\quad\text{and}\quad G=\tilde{G}\setminus\{|q|\}.

The definition of the topology and groupoid operations on $G$ show that $G$ is an open subgroupoid of $\tilde{G}$ . Since $\tilde{G}^{(0)}={\mathfrak{S}}_{s}(\tilde{\mathcal{C}},({\mathcal{D}}^{c})^{\sim})$ ,

G^{(0)}=\{\rho\in{\mathfrak{S}}_{s}(\tilde{\mathcal{C}},({\mathcal{D}}^{c})^{\sim}):\rho|_{\mathcal{C}}\neq 0\}.

By [BrownFullerPittsReznikoffGrC*AlTwGpC*Al, Lemma 2.7]. this produces the twist,

(5.2.2)

G^{(0)}\times{\mathbb{T}}\hookrightarrow\Sigma\twoheadrightarrow G.

Routine modifications to the proof of [PittsStReInII, Theorem 7.24] show that with $\theta:\tilde{\mathcal{C}}\rightarrow C^{*}_{r}(\tilde{\Sigma},\tilde{G})$ arising from $\mathfrak{g}$ as described in the unital case above,

C^{*}(\theta({\mathcal{C}})\cup C_{0}(G^{(0)}))=C^{*}_{r}(\Sigma,G).

We now show that $C^{*}(\Delta(\theta({\mathcal{C}})))=C_{0}(G^{(0)})$ . Let

{\mathcal{N}}_{0}({\mathcal{C}},{\mathcal{D}}^{c}):=\{v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}}^{c}):\mathfrak{g}(v)\text{ is compactly supported}\}

and let ${\mathcal{C}}_{0}=\operatorname{span}{\mathcal{N}}_{0}({\mathcal{C}},{\mathcal{D}}^{c})$ . As in the proof of [PittsStReInII, Theorem 7.24], ${\mathcal{N}}_{0}({\mathcal{C}},{\mathcal{D}}^{c})$ is a $*$ -semigroup and ${\mathcal{C}}_{0}$ is dense in ${\mathcal{C}}$ . Now suppose $\rho_{1}$ and $\rho_{2}$ are distinct elements of $G^{(0)}$ , in other words, $\rho_{1}$ and $\rho_{2}$ are distinct strongly compatible states on $\tilde{\mathcal{C}}$ , neither of which vanish on ${\mathcal{C}}$ . Then $\rho_{1}|_{\mathcal{C}}\neq\rho_{2}|_{\mathcal{C}}$ , so we may find an element $v\in{\mathcal{N}}_{0}({\mathcal{C}},{\mathcal{D}}^{c})$ such that $\rho_{1}(v)\neq\rho_{2}(v)$ . Thus, $\mathfrak{g}(v)(\rho_{1})\neq\mathfrak{g}(v)(\rho_{2})$ . Since $\Delta$ is determined by restriction to $G^{(0)}$ , we obtain $\Delta(\mathfrak{g}(v)(\rho_{1})\neq\Delta(\mathfrak{g}(v))(\rho_{2})$ . As $\theta(v)=\mathfrak{g}(v)$ , we conclude that $\Delta(\theta({\mathcal{C}}))$ separates points of $G^{(0)}$ . Also, recalling that ${\mathcal{D}}^{c}$ contains an approximate unit for ${\mathcal{C}}$ , we see that given $\rho\in G^{(0)}$ , there exists $h\in{\mathcal{D}}^{c}$ such that $\rho(h)=\theta(h)(\rho)\neq 0$ . By the Stone-Wierstrauß theorem, $C^{*}(\Delta(\theta({\mathcal{C}})))=C_{0}(G^{(0)})$ .

An application of Theorem 4.2 shows that $(C^{*}_{r}(\Sigma,G),C_{0}(G^{(0)})\hbox{\,:\hskip-1.0pt:\,}\theta|_{\mathcal{C}})$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}}^{c})$ , giving a groupoid description for $({\mathcal{C}},{\mathcal{D}}^{c})$ (and $({\mathcal{C}},{\mathcal{D}})$ ).

The definition of compatible eigenfunctional makes sense regardless of whether the pseudo-Cartan inclusion $({\mathcal{C}},{\mathcal{D}})$ is unital. Since ${\mathcal{N}}({\mathcal{C}},{\mathcal{D}})\subseteq{\mathcal{N}}({\mathcal{C}},{\mathcal{D}}^{c})$ ,

{\mathcal{E}}^{1}_{c}({\mathcal{C}},{\mathcal{D}}^{c})\subseteq{\mathcal{E}}^{1}_{c}({\mathcal{C}},{\mathcal{D}}).

Actually, equality holds. To see this, let $\phi\in{\mathcal{E}}^{1}_{c}({\mathcal{C}},{\mathcal{D}})$ . Regularity of $({\mathcal{C}},{\mathcal{D}})$ implies there exists $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ such that $\phi(v)>0$ . With ${\mathfrak{s}}(\phi)$ as defined in (5.1.1), we obtain

\phi=[v,{\mathfrak{s}}(\phi)]\in{\mathcal{E}}^{1}_{c}({\mathcal{C}},{\mathcal{D}}^{c}).

Using (5.2.1), we conclude that the collection of norm-one strongly compatible eigenfunctionals for $({\mathcal{C}},{\mathcal{D}})$ is the same as the set of norm-one strongly compatible eigenfunctionals for $({\mathcal{C}},{\mathcal{D}}^{c})$ .

We summarize our discussion with the following result, which extends the groupoid description of the Cartan envelope given in [PittsStReInII, Corollary 7.31] (once again, with $F={\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}}))$ ) from the unital setting to include both the unital and non-unital cases.

Theorem \the\numberby.

Let $({\mathcal{C}},{\mathcal{D}})$ be a pseudo-Cartan inclusion, let $\Sigma$ be the collection of all norm-one strongly compatible eigenfunctionals for $({\mathcal{C}},{\mathcal{D}})$ , and let $G=\{|\phi|:\phi\in\Sigma\}$ . Then the following statements hold.

(a)

$\Sigma$ and $G$ are Hausdorff topological groupoids with $G$ effective and étale, $G^{(0)}={\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})$ and

$G^{(0)}\times{\mathbb{T}}\hookrightarrow\Sigma\twoheadrightarrow G$

is a twist.
(b)

The map $\mathfrak{g}$ extends to a regular $*$ -monomorphism $\theta:{\mathcal{C}}\rightarrow C^{*}_{r}(\Sigma,G)$ and

$(C^{*}_{r}(\Sigma,G),C_{0}(G^{(0)})\hbox{\,:\hskip-1.0pt:\,}\theta)$

is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ .

6. Constructions and Properties of Pseudo-Cartan Inclusions

In this section, we explore the behavior of pseudo-Cartan inclusions and their Cartan envelopes under mappings and also some familiar constructions. Theorem 6.1 shows that under a suitable hypothesis, a regular map between pseudo-Cartan inclusions extends to the Cartan envelopes and Proposition 6.1 shows that a regular automorphism of a pseudo-Cartan inclusion uniquely extends to its Cartan envelope. We examine the behavior of pseudo-Cartan inclusions and their inductive limits for suitable connecting maps in Theorems 6.2 and 6.2, and Theorem 6.3 describes behavior of pseudo-Cartan inclusions and their Cartan envelops under the minimal tensor product. In some of the results, obtaining regularity of maps under these constructions is a delicate and technical issue.

6.1. Mapping Results

The purposes of this subsection are to establish a mapping property, Theorem 6.1, for Cartan envelopes and to show that a regular automorphism of a pseudo-Cartan inclusion extends uniquely to its Cartan envelope, Proposition 6.1. While interesting in its own right, Theorem 6.1 is a key tool for studying inductive limits of pseudo-Cartan inclusions.

We require some preparation, beginning with a simple fact about sums of normalizers. The sum of normalizers is not usually a normalizer. However, when the normalizers are “orthogonal,” their sum is again a normalizer. Here is the precise statement.

Fact \the\numberby.

Suppose $({\mathcal{C}},{\mathcal{D}})$ is an inclusion and $w,v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ . If $w$ and $v$ are orthogonal in the sense that $v^{*}w=0=wv^{*}$ , then $v+w\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ .

Proof.

Let $d\in{\mathcal{D}}$ . We have

\displaystyle\begin{split}(v+w)^{*}d(v+w)&=v^{*}dv+w^{*}dw+v^{*}dw+w^{*}dv\\ &=v^{*}dv+w^{*}dw+\lim v^{*}d(ww^{*})^{1/n}w+\lim w^{*}d(vv^{*})^{1/n}v\\ &=v^{*}dv+w^{*}dw+\lim v^{*}w\theta_{w}(d(ww^{*})^{1/n})\\ &\qquad+\lim w^{*}v\theta_{v}(d(vv^{*})^{1/n})v\\ &=v^{*}dv+w^{*}dw\in{\mathcal{D}}.\end{split}

Similarly $(v+w)d(v+w)^{*}\in{\mathcal{D}}$ , so $v+w\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ . ∎

Suppose $({\mathcal{C}},{\mathcal{D}})$ is an unital inclusion and $(I({\mathcal{D}}),\iota)$ is an injective envelope for ${\mathcal{D}}$ . Given $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ , let $P,Q$ be the support projections in $I({\mathcal{D}})$ for the ideals $\overline{vv^{*}{\mathcal{D}}}$ and $\overline{v^{*}v{\mathcal{D}}}$ respectively. (Given any ideal $J\unlhd{\mathcal{D}}$ , the supremum in $I({\mathcal{D}})_{s.a.}$ of $\{\iota(x):0\leq x\in J\text{ and }\left\lVert x\right\rVert\leq 1\}$ is called the support projection for $J$ .) Recall from [PittsStReInI, Proposition 1.11] that the isomorphism $\theta_{v}:\overline{vv^{*}{\mathcal{D}}}\rightarrow\overline{v^{*}v{\mathcal{D}}}$ uniquely extends to a $*$ -isomorphism $\tilde{\theta}_{v}:I({\mathcal{D}})P\rightarrow I({\mathcal{D}})Q$ . We then obtain a Frolík decomposition $\{R_{j}\}_{j=0}^{4}$ for $\tilde{\theta}_{v}$ , see [PittsStReInI, Definition 2.12], and also the Frolík ideals $\{K_{i}(v)\}_{i=0}^{4}$ , [PittsStReInI, Definition 2.13]. Our immediate interest will be with $K_{0}(v)$ . This is the fixed point ideal for $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ ; its definition is

(6.1.1)

K_{0}(v):=\iota^{-1}(R_{0}I({\mathcal{D}})).

This is a regular ideal in ${\mathcal{D}}$ , and it has the following alternate description:

(6.1.2)

K_{0}(v)=\{d\in(vv^{*}{\mathcal{D}})^{\perp\perp}:vd=dv\in{\mathcal{D}}^{c}\}=\{d\in(v^{*}v{\mathcal{D}})^{\perp\perp}:vd=dv\in{\mathcal{D}}^{c}\}.

See [PittsStReInI, Definition 2.13 and Lemma 2.15] for further details.

To extend the previous considerations to weakly non-degenerate inclusions, recall that if $({\mathcal{C}},{\mathcal{D}})$ is a weakly non-degenerate inclusion, then $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is a unital inclusion and the standard embeddings (described in (2.1.2)) satisfy $u_{\mathcal{C}}|_{\mathcal{D}}=u_{\mathcal{D}}$ . For $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ , define

(6.1.3)

K_{0}(v):=u_{\mathcal{D}}^{-1}(K_{0}(u_{\mathcal{C}}(v)))=\{d\in{\mathcal{D}}:u_{\mathcal{D}}(d)\in K_{0}(u_{\mathcal{C}}(v))\}.

When a normalizer $v$ for a regular MASA inclusion is not annihilated by the pseudo-expectation, $K_{0}(v)$ is non-trivial. In fact, a bit more is true.

Lemma \the\numberby.

Let $({\mathcal{C}},{\mathcal{D}})$ be a regular MASA inclusion, let $(I({\mathcal{D}}),\iota)$ be an injective envelope for ${\mathcal{D}}$ and let $E:{\mathcal{C}}\rightarrow I({\mathcal{D}})$ be the pseudo-expectation. Suppose $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ satisfies $E(v)\neq 0$ . Then $K_{0}(v)\neq\{0\}$ and there exists $k\in K_{0}(v)\cap\overline{v^{*}v{\mathcal{D}}}$ such that $vk\neq 0$ .

Proof.

Assume first that in addition to the hypotheses stated, $({\mathcal{C}},{\mathcal{D}})$ is a unital inclusion. By [PittsStReInI, Theorem 3.5(iii)], $|E(v)|^{2}=R_{0}(\iota(v^{*}v))$ , so $R_{0}\neq 0$ . Therefore, $K_{0}(v)\neq\{0\}$ by (6.1.1) and the fact that $(I({\mathcal{D}}),{\mathcal{D}},\iota)$ has the ideal intersection property. Next, [PittsStReInII, Lemma 2.15] shows $K_{0}(v)=(K_{0}(v)\cap\overline{v^{*}v{\mathcal{D}}})^{\perp\perp}$ , thus $K_{0}(v)\cap\overline{v^{*}v{\mathcal{D}}}\neq\{0\}$ . Choose $0\neq k\in K_{0}(v)\cap\overline{v^{*}v{\mathcal{D}}}$ . Then $vk\neq 0$ . (Otherwise, $v^{*}vk=0$ , whence $k\in\{v^{*}v{\mathcal{D}}\}^{\perp}\cap\{v^{*}v{\mathcal{D}}\}^{\perp\perp}$ , contrary to $k\neq 0$ .) This completes the unital case.

Now assume $({\mathcal{C}},{\mathcal{D}})$ is not unital. First observe that $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is a unital regular MASA inclusion. Indeed, [PittsNoApUnInC*Al, Theorem 2.5] shows $({\mathcal{C}},{\mathcal{D}})$ has the AUP, so Lemma 3(e) yields regularity of $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ and a routine argument shows $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is a MASA inclusion. Since $({\mathcal{C}},{\mathcal{D}})$ has the AUP, the standard embedding $u_{\mathcal{C}}:{\mathcal{C}}\rightarrow\tilde{\mathcal{C}}$ is a regular map by Lemma 3(c). Thus, $(v,0)=u_{\mathcal{C}}(v)\in{\mathcal{N}}(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ . By the unital case, there exists $(k,\lambda)\in K_{0}((v,0))\cap\overline{(v^{*}v,0)\tilde{\mathcal{D}}}$ , with $(v,0)(k,\lambda)\neq(0,0)$ . Since $(k,\lambda)\in\overline{(v^{*}v,0)\tilde{\mathcal{D}}}$ , we must have $\lambda=0$ . In other words, there exists $k\in K_{0}(v)\cap\overline{v^{*}v{\mathcal{D}}}$ such that $vk\neq 0$ . ∎

Sometimes it is possible to establish regularity of a $*$ -monomorphism between inclusions with only partial knowledge of the normalizers in the domain of the map. The following useful technical result gives a setting where this can be done.

Proposition \the\numberby.

Let $({\mathcal{A}}_{2},{\mathcal{B}}_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ be an essential and Cartan expansion for the pseudo-Cartan inclusion $({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ and suppose $N\subseteq{\mathcal{N}}({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ is a $*$ -semigroup such that ${\mathcal{B}}_{1}\subseteq N$ and $\operatorname{span}N$ is dense in ${\mathcal{A}}_{1}$ . If $\alpha(w)\in{\mathcal{N}}({\mathcal{A}}_{2},{\mathcal{B}}_{2})$ for every $w\in N$ , then $\alpha:({\mathcal{A}}_{1},{\mathcal{B}}_{1})\rightarrow({\mathcal{A}}_{2},{\mathcal{B}}_{2})$ is a regular $*$ -monomorphism.

Proof.

Throughout the proof, let $E_{2}:{\mathcal{A}}_{2}\rightarrow{\mathcal{B}}_{2}$ be the conditional expectation. The first step in the proof is to establish the following lemma.

Lemma \the\numberby.

For $v\in{\mathcal{N}}({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ , let

{\mathcal{Z}}_{v}:=\{b_{1}\in{\mathcal{B}}_{1}:\alpha(vb_{1})\in{\mathcal{N}}({\mathcal{A}}_{2},{\mathcal{B}}_{2})\}.

If $J\unlhd{\mathcal{B}}_{1}$ is an essential ideal such that $J\subseteq{\mathcal{Z}}_{v}$ , then for every $b_{2}\in{\mathcal{B}}_{2}$ , $\alpha(v)^{*}b_{2}\alpha(v)\in{\mathcal{B}}_{2}$ .

Proof.

We first show that for any $h\in{\mathcal{B}}_{2}$ and $x\in J$ ,

(6.1.4)

\alpha(v)^{*}h\alpha(v)\alpha(x)\in{\mathcal{B}}_{2}.

Indeed for $y_{n}:=\alpha((vx)(vx)^{*}))^{1/n}\in\overline{\alpha((vx)(vx)^{*}){\mathcal{B}}_{2}}$ ,

\alpha(v)^{*}h\alpha(vx)=\lim\alpha(v)^{*}hy_{n}\alpha(vx)=\lim\alpha(v^{*}vx)\theta_{\alpha(vx)}(hy_{n})\in{\mathcal{B}}_{2};

thus (6.1.4) holds.

Fixing $b_{2}\in{\mathcal{B}}_{2}$ , put

m:=\alpha(v^{*})b_{2}\alpha(v)-E_{2}(\alpha(v)^{*}b_{2}\alpha(v)).

We shall show $m=0$ . To do this, let

M:=\{h\in{\mathcal{B}}_{2}:E_{2}(m^{*}m)h=0\}.

Then $M\unlhd{\mathcal{B}}_{2}$ , and because $M=\{E_{2}(m^{*}m)\}^{\perp}$ , $M=M^{\perp\perp}$ . By (6.1.4),

\overline{\alpha(J){\mathcal{B}}_{2}}\subseteq M.

We claim $M$ is an essential ideal of ${\mathcal{B}}_{2}$ . Indeed, suppose $K\unlhd{\mathcal{B}}_{2}$ satisfies $K\cap M=\{0\}$ . Put

L:=\alpha^{-1}(K)\unlhd{\mathcal{B}}_{1}

and note that $L\cap J=\{0\}$ because

\alpha(L\cap J)\subseteq K\cap\alpha(J)\subseteq K\cap\overline{\alpha(J){\mathcal{B}}_{2}}\subseteq K\cap M=\{0\}.

But $J$ is an essential ideal in ${\mathcal{B}}_{1}$ , so $L=\{0\}$ . This gives $K\cap\alpha({\mathcal{B}}_{1})=\{0\}$ . Then $K=\{0\}$ because $({\mathcal{B}}_{2},{\mathcal{B}}_{1},\alpha|_{{\mathcal{B}}_{1}})$ has the ideal intersection property. Thus, $M$ is an essential ideal.

Since $M$ is an essential ideal, $M^{\perp}=\{0\}$ . As $M=M^{\perp\perp}$ , $M={\mathcal{B}}_{2}$ . Hence $E_{2}(m^{*}m)=0$ , so $m=0$ by faithfulness of $E_{2}$ . This gives $\alpha(v^{*})b_{2}\alpha(v)\in{\mathcal{B}}_{2}$ . ∎

We now return to the proof of Proposition 6.1. Suppose first that $({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ is a virtual Cartan inclusion. Given $v\in{\mathcal{N}}({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ , we shall show that ${\mathcal{Z}}_{v}$ contains an essential ideal. We require two claims. Claim 1. Recall that $\{v^{*}v\}^{\perp}=\{b_{1}\in{\mathcal{B}}_{1}:v^{*}vb_{1}=0\}$ . Then

\{v^{*}v\}^{\perp}=\{b_{1}\in{\mathcal{B}}_{1}:vb_{1}=0\}\subseteq{\mathcal{Z}}_{v}.

Proof. To see $\{v^{*}v\}^{\perp}\subseteq\{b_{1}\in{\mathcal{B}}_{1}:vb_{1}=0\}$ , note that for any polynomial $p$ with $p(0)=0$ , $0=vp(v^{*}v)b_{1}=p(vv^{*})vb_{1}$ ; then use

0=\inf\{\left\lVert p(vv^{*})v-v\right\rVert:p\text{ is a polynomial with }p(0)=0\}.

The opposite inclusion is trivial, so $\{v^{*}v\}^{\perp}=\{b_{1}\in{\mathcal{B}}_{1}:vb_{1}=0\}$ ; this description gives $\{v^{*}v\}^{\perp}\subseteq{\mathcal{Z}}_{v}$ . $\diamondsuit$

Claim 2. Suppose $\Lambda$ is a non-empty set and $\{J_{\lambda}:\lambda\in\Lambda\}$ is a collection of ideals in ${\mathcal{B}}_{1}$ such that: for every $\lambda\in\Lambda$ , $J_{\lambda}\subseteq{\mathcal{Z}}_{v}$ ; and for distinct $\lambda,\mu\in\Lambda$ , $J_{\lambda}\cap J_{\mu}=\{0\}$ . Write $\bigvee_{\lambda\in\Lambda}J_{\lambda}:=\overline{\operatorname{span}}\bigcup_{\lambda\in\Lambda}J_{\lambda}$ . Then $\bigvee_{\lambda\in\Lambda}J_{\lambda}\unlhd{\mathcal{B}}_{1}$ and $\bigvee_{\lambda\in\Lambda}J_{\lambda}\subseteq{\mathcal{Z}}_{v}$ .Proof. That $\bigvee_{\lambda\in\Lambda}J_{\lambda}$ is an ideal in ${\mathcal{B}}_{1}$ is clear. Let $h\in\operatorname{span}\bigcup_{\lambda\in\Lambda}J_{\lambda}$ . Then we may find a finite set $F\subseteq\Lambda$ and for each $\lambda\in F$ , an element $h_{\lambda}\in J_{\lambda}$ so that

h=\sum_{\lambda\in F}h_{\lambda}.

For distinct $\lambda,\mu\in F$ , $h_{\lambda}h_{\mu}\in J_{\lambda}\cap J_{\mu}=\{0\}$ , so $h_{\lambda}h_{\mu}=0$ . A calculation then shows that $\{vh_{\lambda}:\lambda\in F\}$ is a collection of pairwise orthogonal normalizers in ${\mathcal{N}}({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ . But $h_{\lambda}\in{\mathcal{Z}}_{v}$ for every $\lambda$ , whence $\{\alpha(vh_{\lambda})\}_{\lambda\in F}$ is a family of pairwise orthogonal normalizers in ${\mathcal{N}}({\mathcal{A}}_{2},{\mathcal{B}}_{2})$ . Fact 6.1 shows that $\sum_{\lambda\in F}\alpha(h_{\lambda})\in{\mathcal{N}}({\mathcal{A}}_{2},{\mathcal{B}}_{2})$ . Since

\alpha(v)\left(\sum_{\lambda\in F}\alpha(h_{\lambda})\right)\in{\mathcal{N}}({\mathcal{A}}_{2},{\mathcal{B}}_{2}),

$h\in{\mathcal{Z}}_{v}$ . The proof of Claim 2 is completed by observing that ${\mathcal{Z}}_{v}$ is a closed set. (Indeed, if $z_{i}\in{\mathcal{Z}}_{v}$ converges to $z\in{\mathcal{B}}_{1}$ , then

\alpha(vz)=\lim\alpha(vz_{i})\in\overline{{\mathcal{N}}({\mathcal{A}}_{2},{\mathcal{B}}_{2})}={\mathcal{N}}({\mathcal{A}}_{2},{\mathcal{B}}_{2}),

so $z\in{\mathcal{Z}}_{v}$ .) $\diamondsuit$ We are now prepared to show ${\mathcal{Z}}_{v}$ contains an essential ideal. This fact is trivial when $v=0$ , so assume $v\neq 0$ .

Let ${\mathcal{Q}}$ be a collection of ideals in ${\mathcal{B}}_{1}$ . We shall say ${\mathcal{Q}}$ is a nice collection of ideals if:

•

for each $J\in{\mathcal{Q}}$ , $J\subseteq{\mathcal{Z}}_{v}$ ;
•

If $J,K\in{\mathcal{Q}}$ and $J\neq K$ , then $J\cap K=\{0\}$ ; and
•

$\{v^{*}v\}^{\perp}\in{\mathcal{Q}}$ .

By Claim 1, the singleton set, $\{\{v^{*}v\}^{\perp}\}$ , is a nice collection, so the family ${\mathfrak{Q}}$ consisting of all nice collections of ideals is non-empty. Partially order ${\mathfrak{Q}}$ by inclusion. Zorn’s lemma provides a maximal nice collection of ideals; call it ${\mathfrak{M}}$ . Consider the ideal,

{\mathfrak{J}}:=\bigvee_{J\in{\mathfrak{M}}}J.

Claim 2 shows ${\mathfrak{J}}\subseteq{\mathcal{Z}}_{v}$ . We will show ${\mathfrak{J}}$ is an essential ideal in ${\mathcal{B}}_{1}$ by showing ${\mathfrak{J}}^{\perp}=\{0\}$ .

Suppose to the contrary that ${\mathfrak{J}}^{\perp}\neq\{0\}$ . Since $\{v^{*}v\}^{\perp}\subseteq{\mathfrak{J}}$ ,

{\mathfrak{J}}^{\perp}\subseteq\{v^{*}v\}^{\perp\perp}.

Note that $\{v^{*}v\}^{\perp\perp}=\overline{v^{*}v{\mathcal{B}}_{1}}^{\perp\perp}$ , and, since $\overline{v^{*}v{\mathcal{B}}_{1}}\subseteq\{v^{*}v\}^{\perp\perp}$ has the ideal intersection property, we find

{\mathfrak{J}}^{\perp}\cap\overline{v^{*}v{\mathcal{B}}_{1}}\neq\{0\}.

Let $0\neq b\in{\mathfrak{J}}^{\perp}\cap\overline{v^{*}v{\mathcal{B}}_{1}}$ . If $v^{*}vb=0$ , then $b\in\{v^{*}v\}^{\perp}\cap\{v^{*}v\}^{\perp\perp}=\{0\}$ , which is impossible as $b\neq 0$ . Therefore, $vb\neq 0$ . Since $({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ is a pseudo-Cartan inclusion, it has the faithful unique pseudo-expectation property. Thus, letting $E:{\mathcal{A}}_{1}\rightarrow I({\mathcal{B}}_{1})$ be the pseudo-expectation, $E(b^{*}v^{*}vb)\neq 0$ . Since $\operatorname{span}N$ is dense in ${\mathcal{A}}_{1}$ , there exists a sequence $t_{\ell}\in\operatorname{span}N$ such that $t_{\ell}\rightarrow vb$ . Then $E(t_{\ell}^{*}vb)\neq 0$ for sufficiently large $\ell$ , so there exists $w\in N$ such that

E(w^{*}vb)\neq 0.

Because we have assumed $({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ is a virtual Cartan inclusion, we may apply Lemma 6.1 to obtain $k\in K_{0}(w^{*}vb)\cap\overline{(w^{*}vb)^{*}(w^{*}vb){\mathcal{B}}_{1}}$ such that

0\neq w^{*}vbk\in{\mathcal{B}}_{1}.

If $(e_{\lambda})$ is an approximate unit for ${\mathfrak{J}}^{\perp}\cap\overline{v^{*}v{\mathcal{B}}_{1}}$ , then

w^{*}vbk=\lim w^{*}vk(be_{\lambda}),\quad\text{whence }\quad w^{*}vbk\in{\mathfrak{J}}^{\perp}\cap\overline{v^{*}v{\mathcal{B}}_{1}}.

Therefore $v(w^{*}vbk)^{*}\neq 0$ , for otherwise $(w^{*}vbk)^{*}\in\overline{v^{*}v{\mathcal{B}}_{1}}^{\perp}\cap\overline{v^{*}v{\mathcal{B}}_{1}}$ . Then

0\neq v(w^{*}vbk)^{*}=(vk^{*}b^{*}v^{*})w\in N.

Since $\alpha(N)\subseteq{\mathcal{N}}({\mathcal{A}}_{2},{\mathcal{B}}_{2})$ , we conclude that

h:=(w^{*}vbk)^{*}

is a non-zero element of $J_{v}$ .

Let $L:=\overline{h{\mathcal{B}}_{1}}$ be the ideal generated by $h$ . Then $L\subseteq{\mathcal{Z}}_{v}$ , and since we noted above that $h^{*}\in{\mathfrak{J}}^{\perp}\cap\overline{v^{*}v{\mathcal{B}}_{1}}$ , we see that for every $J\in{\mathfrak{M}}$ , $L\cap J=\{0\}$ . Thus ${\mathfrak{M}}\cup\{L\}$ is a nice collection of ideals properly containing ${\mathfrak{M}}$ . As this contradicts maximality of ${\mathfrak{M}}$ , we obtain ${\mathfrak{J}}^{\perp}=\{0\}$ . Therefore, ${\mathfrak{J}}$ is an essential ideal.

Apply Lemma 6.1 to ${\mathcal{Z}}_{v}$ and ${\mathcal{Z}}_{v^{*}}$ to conclude that $\alpha(v)\in{\mathcal{N}}({\mathcal{A}}_{2},{\mathcal{B}}_{2})$ . If follows that $\alpha$ is a regular $*$ -monomorphism. This completes the proof of the proposition when $({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ is a virtual Cartan inclusion.

Now suppose $({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ is a general pseudo-Cartan inclusion. By Observation 4.1, $({\mathcal{A}}_{1},{\mathcal{B}}_{1}^{c})$ is a virtual Cartan inclusion. Lemma 3(a) shows ${\mathcal{N}}({\mathcal{A}}_{1},{\mathcal{B}}_{1})\subseteq{\mathcal{N}}({\mathcal{A}}_{1},{\mathcal{B}}_{1}^{c})$ and Lemma 3(b) gives $\alpha({\mathcal{B}}_{1}^{c})\subseteq{\mathcal{B}}_{2}$ . Let $N^{\prime}$ be the $*$ -semigroup generated by $N\cup{\mathcal{B}}_{1}^{c}$ . Then ${\mathcal{B}}_{1}^{c}\subseteq N^{\prime}\subseteq{\mathcal{N}}({\mathcal{A}}_{1},{\mathcal{B}}_{1}^{c})$ . Writing $w^{\prime}\in N^{\prime}$ as a finite product with factors belonging to $N\cup{\mathcal{B}}_{1}^{c}$ , we find $\alpha(w^{\prime})\in{\mathcal{N}}({\mathcal{A}}_{2},{\mathcal{B}}_{2})$ . Therefore,

\alpha({\mathcal{N}}({\mathcal{A}}_{1},{\mathcal{B}}_{1}))\subseteq\alpha({\mathcal{N}}({\mathcal{A}}_{1},{\mathcal{B}}_{1}^{c}))\subseteq{\mathcal{N}}({\mathcal{A}}_{2},{\mathcal{B}}_{2}),

with the second inclusion obtained from the virtual Cartan case applied to $({\mathcal{A}}_{1},{\mathcal{B}}_{1}^{c})$ and $N^{\prime}$ . This completes the proof. ∎

We need an intertwining property for conditional expectations before stating and proving Theorem 6.1. We will discuss this property further in the introduction to Section 6.2.

Lemma \the\numberby.

Let $({\mathcal{A}}_{2},{\mathcal{B}}_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ be an essential and Cartan expansion for the Cartan inclusion $({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ , and for $i=1,2$ , let $E_{i}:{\mathcal{A}}_{i}\rightarrow{\mathcal{B}}_{i}$ be the conditional expectations. Then $\alpha\circ E_{1}=E_{2}\circ\alpha$ .

Proof.

Let $(I({\mathcal{B}}_{1}),\iota_{1})$ be an injective envelope for ${\mathcal{B}}_{1}$ . Since $({\mathcal{A}}_{2},{\mathcal{B}}_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is an essential expansion of $({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ , $({\mathcal{B}}_{2},{\mathcal{B}}_{1},\alpha|_{{\mathcal{B}}_{1}})$ is an inclusion with the ideal intersection property. Proposition 2.4 shows there is a unique $*$ -monomorphism $\iota_{2}:{\mathcal{B}}_{2}\rightarrow I({\mathcal{B}}_{1})$ such that

(6.1.5)

\iota_{1}=\iota_{2}\circ(\alpha|_{{\mathcal{B}}_{1}}).

For $b_{1}\in{\mathcal{B}}_{1}$ , we have

(\iota_{2}\circ E_{2}\circ\alpha)(b_{1})=\iota_{2}(\alpha(b_{1}))=\iota_{1}(b_{1}),

so $\iota_{2}\circ E_{2}\circ\alpha$ is a pseudo-expectation for $({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ . Since $\iota_{1}\circ E_{1}$ is the pseudo-expectation for $({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ , we have

\iota_{2}\circ\alpha\circ E_{1}\stackrel{{\scriptstyle\eqref{EWork1}}}{{=}}\iota_{1}\circ E_{1}=\iota_{2}\circ E_{2}\circ\alpha.

Since $\iota_{2}$ is one-to-one, the lemma follows. ∎

We are now ready to establish a key mapping property of Cartan envelopes.

Theorem \the\numberby.

Suppose $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ and $({\mathcal{C}}_{2},{\mathcal{D}}_{2})$ are pseudo-Cartan inclusions and $({\mathcal{C}}_{2},{\mathcal{D}}_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ is a regular and essential expansion of $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ . For $i=1,2$ , let $({\mathcal{A}}_{i},{\mathcal{B}}_{i}\hbox{\,:\hskip-1.0pt:\,}\tau_{i})$ be Cartan envelopes for $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ . The following statements hold.

(a)

There is a unique $*$ -monomorphism $\breve{\alpha}:{\mathcal{A}}_{1}\rightarrow{\mathcal{A}}_{2}$ such that

(6.1.6) $\breve{\alpha}\circ\tau_{1}=\tau_{2}\circ\alpha.$
(b)

$({\mathcal{A}}_{2},{\mathcal{B}}_{2}\hbox{\,:\hskip-1.0pt:\,}\breve{\alpha})$ is a regular, essential expansion of $({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ .
(c)

For $i=1,2$ , let $\Delta_{i}:{\mathcal{A}}_{i}\rightarrow{\mathcal{B}}_{i}$ denote the conditional expectation. Then

(6.1.7) $\breve{\alpha}\circ\Delta_{1}=\Delta_{2}\circ\breve{\alpha}$

and

(6.1.8) $\breve{\alpha}\circ\Delta_{1}\circ\tau_{1}=\Delta_{2}\circ\tau_{2}\circ\alpha.$

The following commutative diagram illustrates the maps involved; the unlabelled vertical arrows represent inclusion maps.

(6.1.9)

Proof.

(a) Note that $({\mathcal{A}}_{2},{\mathcal{B}}_{2}\hbox{\,:\hskip-1.0pt:\,}\tau_{2}\circ\alpha)$ is a Cartan expansion of $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ . Also, the inclusions $({\mathcal{B}}_{2},{\mathcal{D}}_{2},\tau_{2}|_{{\mathcal{D}}_{2}})$ and $({\mathcal{D}}_{2},{\mathcal{D}}_{1},\alpha|_{{\mathcal{D}}_{1}})$ have the ideal intersection property, and therefore $({\mathcal{B}}_{2},{\mathcal{D}}_{1},\tau_{2}\circ\alpha|_{{\mathcal{D}}_{1}})$ also has the ideal intersection property. Since $\tau_{2}\circ\alpha$ is a regular map, $({\mathcal{A}}_{2},{\mathcal{B}}_{2}\hbox{\,:\hskip-1.0pt:\,}\tau_{2}\circ\alpha)$ is an essential and regular Cartan expansion of $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ .

Let

(6.1.10)

{\mathcal{B}}_{1}^{\prime}:=C^{*}((\Delta_{2}\circ\tau_{2}\circ\alpha)({\mathcal{C}}_{1}))\quad\text{and}\quad{\mathcal{A}}_{1}^{\prime}:=C^{*}({\mathcal{B}}_{1}^{\prime}\cup\tau_{2}(\alpha({\mathcal{C}}_{1})).

Proposition 3 shows $({\mathcal{A}}_{1}^{\prime},{\mathcal{B}}_{1}^{\prime}\hbox{\,:\hskip-1.0pt:\,}\tau_{2}\circ\alpha)$ is a Cartan envelope for $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ .

By the uniqueness of the Cartan envelope, there exists a unique $*$ -isomorphism $\psi:{\mathcal{A}}_{1}\rightarrow{\mathcal{A}}_{1}^{\prime}$ such that $\psi$ is regular and $\psi\circ\tau_{1}=\tau_{2}\circ\alpha$ . Let $f:{\mathcal{A}}_{1}^{\prime}\hookrightarrow{\mathcal{A}}_{2}$ be the inclusion map. Taking

\breve{\alpha}:=f\circ\psi,

we obtain (6.1.6).

(b) By construction, $\breve{\alpha}({\mathcal{B}}_{1})=\psi({\mathcal{B}}_{1})\subseteq{\mathcal{B}}_{2}$ . Let us show $({\mathcal{B}}_{2},\breve{\alpha}({\mathcal{B}}_{1}))$ has the ideal intersection property. The proof of part (a) shows that $({\mathcal{B}}_{2},\tau_{2}(\alpha({\mathcal{D}}_{1})))$ has the ideal intersection property. Since

\tau_{2}(\alpha({\mathcal{D}}_{1}))\subseteq\psi({\mathcal{B}}_{1})=\breve{\alpha}({\mathcal{B}}_{1})\subseteq{\mathcal{B}}_{2},

$({\mathcal{B}}_{2},\breve{\alpha}({\mathcal{B}}_{1}))$ has the ideal intersection property. Thus $({\mathcal{A}}_{2},{\mathcal{B}}_{2}\hbox{\,:\hskip-1.0pt:\,}\breve{\alpha})$ is an essential expansion of $({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ .

Let $N$ be the $*$ -semigroup generated by ${\mathcal{B}}_{1}\cup\tau_{1}({\mathcal{N}}({\mathcal{C}}_{1},{\mathcal{D}}_{1}))$ . Since $({\mathcal{A}}_{1},{\mathcal{B}}_{1}\hbox{\,:\hskip-1.0pt:\,}\tau_{1})$ is a Cartan envelope for $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ , $N\subseteq{\mathcal{N}}({\mathcal{A}}_{1},{\mathcal{B}}_{1})$ , and $\operatorname{span}N$ is dense in ${\mathcal{A}}_{1}$ . Proposition 6.1 shows $\breve{\alpha}$ is a regular $*$ -monomorphism. Thus the proof of (b) is complete.

\breve{\alpha}\circ\Delta_{1}\circ\tau_{1}=\Delta_{2}\circ\breve{\alpha}\circ\tau_{1}=\Delta_{2}\circ\tau_{2}\circ\alpha,

with the last equality following from (6.1.6). Thus (6.1.8) holds, and the proof is complete. ∎

Example \the\numberby. The conclusions of Theorem 6.1 need not hold without the hypothesis that $\alpha$ is an essential map. Indeed, Section 2 of [PittsCoStReInII] gives an example of a virtual Cartan inclusion $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ and describes a Cartan package $({\mathcal{C}}_{2},{\mathcal{D}}_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ for $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ which is not the Cartan envelope for $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ . A Cartan envelope $({\mathcal{A}}_{1},{\mathcal{B}}_{1}\hbox{\,:\hskip-1.0pt:\,}\tau_{1})$ for $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ is also described in [PittsCoStReInII, Section 2]. For convenience, we summarize a particular case of the discussion in [PittsCoStReInII, Section 2]. Let $S_{-}=\begin{bmatrix}0&1\\ 1&0\end{bmatrix}$ , $S_{+}=\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}$ , and put $S_{0}:=iS_{-}S_{+}$ . Then $S_{-},S_{+}$ and $S_{0}$ are self-adjoint unitary matrices. Put

	$\displaystyle{\mathcal{C}}_{1}$	$\displaystyle:=C([-1,1],M_{2}({\mathbb{C}}))$
and
	$\displaystyle{\mathcal{D}}_{1}$	$\displaystyle:=\{f\in{\mathcal{C}}_{1}:f(t)\in C^{}(S_{-})\text{ if }t<0\text{ and }f(t)\in C^{}(S_{+})\text{ if }t>0\};$

continuity gives $f(0)\in{\mathbb{C}}I$ for $f\in{\mathcal{D}}_{1}$ . A Cartan package $({\mathcal{C}}_{2},{\mathcal{D}}_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ for $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ is

	$\displaystyle{\mathcal{C}}_{2}$	$\displaystyle:=C([-1,0],M_{2}({\mathbb{C}}))\oplus M_{2}({\mathbb{C}})\oplus C([0,1],M_{2}({\mathbb{C}})),$
	$\displaystyle{\mathcal{D}}_{2}$	$\displaystyle:=C([-1,0],C^{}(S_{-}))\oplus C^{}(S_{0})\oplus C([0,1],C^{*}(S_{+})),\quad\text{and}\quad$
	$\displaystyle\alpha(f)$	$\displaystyle:=f\|_{[-1,0]}\oplus f(0)\oplus f\|_{[0,1]},\quad f\in{\mathcal{C}}_{1}.$

A Cartan envelope for $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ is $({\mathcal{A}}_{1},{\mathcal{B}}_{1}\hbox{\,:\hskip-1.0pt:\,}\tau_{1})$ , where

	$\displaystyle{\mathcal{A}}_{1}$	$\displaystyle:=C([-1,0],M_{2}({\mathbb{C}}))\oplus C([0,1],M_{2}({\mathbb{C}})),$
	$\displaystyle{\mathcal{B}}_{1}$	$\displaystyle:=C([-1,0],C^{}(S_{-}))\oplus C([0,1],C^{}(S_{+})),\quad\text{and}\quad$
	$\displaystyle\tau_{1}(f)$	$\displaystyle:=f\|_{[-1,0]}\oplus f\|_{[0,1]},\quad f\in{\mathcal{C}}_{1}.$

Since $({\mathcal{C}}_{2},{\mathcal{D}}_{2})$ is already a Cartan inclusion, a Cartan envelope for $({\mathcal{C}}_{2},{\mathcal{D}}_{2})$ is just $({\mathcal{C}}_{2},{\mathcal{D}}_{2}\hbox{\,:\hskip-1.0pt:\,}\operatorname{Id}|_{{\mathcal{C}}_{2}})$ . However, it is not hard to see that there is no embedding $\breve{\alpha}:{\mathcal{A}}_{1}\rightarrow{\mathcal{C}}_{2}$ such that $\breve{\alpha}\circ\tau_{1}=\operatorname{Id}|_{{\mathcal{C}}_{2}}\circ\alpha$ . Thus the conclusions of Theorem 6.1 do not hold.

The uniqueness and minimality statements from Theorem 3 give the following rigidity result for the Cartan envelopes of a pseudo-Cartan inclusion.

Proposition \the\numberby.

Let $({\mathcal{C}},{\mathcal{D}})$ be a pseudo-Cartan inclusion, with Cartan envelope $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\tau)$ . If $\theta:{\mathcal{C}}\rightarrow{\mathcal{C}}$ is a $*$ -automorphism such that $\theta({\mathcal{N}}({\mathcal{C}},{\mathcal{D}}))\subseteq{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ , then $\theta$ uniquely extends to a regular $*$ -automorphism $\breve{\theta}:{\mathcal{A}}\rightarrow{\mathcal{A}}$ such that

(6.1.12)

\breve{\theta}\circ\tau=\tau\circ\theta.

Proof.

We first show that $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\tau\circ\theta)$ is a Cartan package for $({\mathcal{C}},{\mathcal{D}})$ . By definition of the Cartan envelope, $\tau$ is a regular map, and as $\theta$ is assumed regular, we find $\tau\circ\theta$ is a regular map. Let $\Delta:{\mathcal{A}}\rightarrow{\mathcal{B}}$ be the conditional expectation. Since $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\tau)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ ,

	$\displaystyle{\mathcal{B}}$	$\displaystyle=C^{}(\Delta(\tau({\mathcal{C}})))=C^{}(\Delta(\tau(\theta({\mathcal{C}}))))$
and
	$\displaystyle{\mathcal{A}}$	$\displaystyle=C^{}(\tau({\mathcal{C}})\cup{\mathcal{B}})=C^{}(\tau(\theta({\mathcal{C}}))\cup{\mathcal{B}}).$

An examination of Definition 3(a) shows $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\tau\circ\theta)$ is a Cartan package for $({\mathcal{C}},{\mathcal{D}})$ .

The ideal ${\mathfrak{J}}\unlhd{\mathcal{A}}$ obtained from the minimality statement of Theorem 3 satisfies

\{0\}={\mathfrak{J}}\cap\tau(\theta({\mathcal{C}}))={\mathfrak{J}}\cap\tau({\mathcal{C}}).

By Observation 4.3, $({\mathcal{A}},{\mathcal{C}},\tau)$ has the ideal intersection property, so ${\mathfrak{J}}=\{0\}$ . Therefore, $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\tau\circ\theta)$ is a Cartan envelope for $({\mathcal{C}},{\mathcal{D}})$ .

Apply the uniqueness statement of Theorem 3 to obtain a unique regular $*$ -automorphism $\breve{\theta}$ of ${\mathcal{A}}$ satisfying (6.1.12). ∎

Remark \the\numberby. Let $({\mathcal{A}},{\mathcal{B}})$ be a Cartan inclusion, or more generally, a regular MASA inclusion, and suppose $\breve{\theta}:{\mathcal{A}}\rightarrow{\mathcal{A}}$ is a regular $*$ -automorphism. Then $\breve{\theta}^{-1}$ is also regular, because

(6.1.14)

\breve{\theta}({\mathcal{B}})={\mathcal{B}}.

(To see (6.1.14) holds, recall that [PittsNoApUnInC*Al, Theorem 2.5] shows $({\mathcal{A}},{\mathcal{B}})$ has the AUP, so by Lemma 3(b) $\breve{\theta}({\mathcal{B}})\subseteq{\mathcal{B}}^{c}={\mathcal{B}}$ ; as $\breve{\theta}({\mathcal{B}})$ is a MASA in ${\mathcal{A}}$ , $\breve{\theta}({\mathcal{B}})={\mathcal{B}}$ .) In particular, the family of regular automorphisms of a Cartan inclusion or of a regular MASA inclusion is a group.

Automorphisms of certain Cartan inclusions and virtual Cartan inclusions satisfying (6.1.14) have been studied, see [Komura*HoBeGrC*Al, Corollary 2.2] and [TaylorEsCoCaSuC*Al, Theorem 6.10]. We expect it is possible to obtain a description of regular $*$ -automorphisms of a pseudo-Cartan inclusion by combining these results with Proposition 6.1.

If the MASA hypothesis in Remark 6.1 is dropped, the inverse of a regular automorphism need not be regular. Here is an example of a pseudo-Cartan inclusion where the family of regular $*$ -automorphisms is not a group.

Example \the\numberby.

Let

	$\displaystyle{\mathcal{C}}$	$\displaystyle=\{h\in C({\mathbb{R}}):\lim_{t\rightarrow-\infty}h(t)=0\text{ and }\lim_{t\rightarrow+\infty}h(t)\text{ exists}\}$
and
	$\displaystyle{\mathcal{D}}$	$\displaystyle=\{h\in{\mathcal{C}}:h\|_{\mathbb{N}}\text{ is constant}\}.$

Since ${\mathbb{N}}\subseteq{\mathbb{R}}$ has empty interior, $({\mathcal{C}},{\mathcal{D}})$ has the ideal intersection property. Also,

(6.1.15)

{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})=\{h\in{\mathcal{C}}:|h|\text{ is constant on }{\mathbb{N}}\},

N:=\{h\in{\mathcal{C}}:h(n)\in{\mathbb{T}}\text{ for all }n\in{\mathbb{N}}\}

is a $*$ -semigroup of normalizers. The Stone-Weiererstrauß theorem shows $\operatorname{span}N$ is a dense subalgebra of ${\mathcal{C}}$ . Thus $({\mathcal{C}},{\mathcal{D}})$ is regular, and hence a pseudo-Cartan inclusion.

For $f\in{\mathcal{C}}$ , define $\theta(f)(t)=f(t+1)$ . Using (6.1.15) we see $\theta$ is regular, but $\theta^{-1}$ is not.

6.2. Inductive Limits

In [KumjianOnC*Di], Kumjian introduced $C^{*}$ -diagonals, a class of inclusions subsequently broadened by Renault to the class of Cartan inclusions. An inclusion $({\mathcal{C}},{\mathcal{D}})$ is a $C^{*}$ -diagonal if it is a Cartan inclusion such that every pure state on ${\mathcal{D}}$ extends uniquely to a state on ${\mathcal{C}}$ . (This definition is equivalent to Kumjian’s original definition, see [PittsNoApUnInC*Al, Proposition 2.10].) The class of unital $C^{*}$ -diagonals is closed under inductive limits provided the connecting maps are unital, regular, and one-to-one ([DonsigPittsCoSyBoIs, Theorem 4.23]), and it can be shown that the same is true in the non-unital setting (with connecting maps again being regular $*$ -monomorphisms). The unique extension property implies that the connecting mappings behave well with respect to the conditional expectations: if for $i=1,2$ , $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ are $C^{*}$ -diagonals and $E_{i}:{\mathcal{C}}_{i}\rightarrow{\mathcal{D}}_{i}$ is the conditional expectation, then for any regular $*$ -monomorphism $\alpha_{21}:{\mathcal{C}}_{1}\rightarrow{\mathcal{C}}_{2}$ ,

(6.2.1)

\alpha_{21}\circ E_{1}=E_{2}\circ\alpha_{21}.

Turning to the context of Cartan inclusions, when the connecting maps are regular and satisfy (6.2.1), Li showed the inductive limit of a sequence of Cartan inclusions is again a Cartan inclusion, see [LiXinEvClSiC*AlCaSu, Theorem 1.10]. Li’s result was extended for non-commutative Cartan inclusions by Meyer, Raad and Taylor, see [MeyerRaadTaylorInLiNoCaIn, Theorem 3.9].

For Cartan inclusions, it follows from Lemma 6.1 and Remark 3(b) that (6.2.1) holds provided the inclusion $({\mathcal{D}}_{2},{\mathcal{D}}_{1},\alpha_{21})$ has the ideal intersection property and $\alpha_{21}$ is a regular map. However, when this assumption on $({\mathcal{D}}_{2},{\mathcal{D}}_{1},\alpha_{21})$ is dropped, examples show (6.2.1) need not hold. Such examples suggest that when the connecting maps are regular, but fail to be essential, an inductive limit of Cartan inclusions may not be a Cartan inclusion.

In this subsection, we consider inductive limits of pseudo-Cartan inclusions. We show that when the connecting maps in an inductive system are essential and regular $*$ -monomorphisms, then the inductive limit of pseudo-Cartan inclusions is again a pseudo-Cartan inclusion. Further, we show that the Cartan envelope of the inductive limit of such a system is the inductive limits of the Cartan envelopes.

The following gives the notion of a directed system of pseudo-Cartan inclusions suitable for our purposes.

Definition \the\numberby.

Let $\Lambda$ be a directed set. Suppose

•

for each $\lambda\in\Lambda$ , $({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda})$ is a pseudo-Cartan inclusion;
•

for every $(\lambda,\mu)\in\Lambda\times\Lambda$ with $\lambda\geq\mu$ , $({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda}\hbox{\,:\hskip-1.0pt:\,}\alpha_{\lambda\mu})$ is an essential and regular expansion of $({\mathcal{C}}_{\mu},{\mathcal{D}}_{\mu})$ ; and
•

whenever $\lambda\geq\mu\geq\nu$ , $\alpha_{\lambda\nu}=\alpha_{\lambda\mu}\circ\alpha_{\mu\nu}$ .

(a)

We will call the collection $\{({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda}\hbox{\,:\hskip-1.0pt:\,}\alpha_{\lambda\mu}):\lambda\geq\mu\}$ a system of pseudo-Cartan inclusions directed by $\Lambda$ . When the directed set is clear from context or it is not necessary to specify it, we will simplify terminology and say $\{({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda}\hbox{\,:\hskip-1.0pt:\,}\alpha_{\lambda\mu}):\lambda\geq\mu\}$ is a system of pseudo-Cartan inclusions.
(b)

Given a system $\{({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda}\hbox{\,:\hskip-1.0pt:\,}\alpha_{\lambda\mu}):\lambda\geq\mu\}$ , recall from the definition of expansion (see Definition 2.1(b)) that $\alpha_{\lambda\mu}({\mathcal{D}}_{\mu})\subseteq{\mathcal{D}}_{\lambda}$ . Letting $\varinjlim{\mathcal{C}}_{\lambda}$ and $\varinjlim{\mathcal{D}}_{\lambda}$ be the inductive limit $C^{*}$ -algebras, we obtain the inclusion $(\varinjlim{\mathcal{C}}_{\lambda},\varinjlim{\mathcal{D}}_{\lambda})$ . We will call this inclusion the inductive limit of the system $\{({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda}\hbox{\,:\hskip-1.0pt:\,}\alpha_{\lambda\mu}):\lambda\geq\mu\}$ .
(c)

Finally, for each $\lambda\in\Lambda$ , we will let $\underrightarrow{\alpha_{\lambda}}:{\mathcal{C}}_{\lambda}\rightarrow\varinjlim{\mathcal{C}}_{\lambda}$ be the canonical embedding: it satisfies

(6.2.2) $\underrightarrow{\alpha_{\lambda}}\circ\alpha_{\lambda\mu}=\underrightarrow{\alpha_{\mu}}\quad\text{for every}\quad\lambda\geq\mu.$

Remark \the\numberby. By Observation 4.3, if $\{({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda}\hbox{\,:\hskip-1.0pt:\,}\alpha_{\lambda\mu}):\lambda\geq\mu\}$ is a system of pseudo-Cartan inclusions, then whenever $\lambda\geq\mu$ , $({\mathcal{C}}_{\lambda},{\mathcal{C}}_{\mu},\alpha_{\lambda\mu})$ has the ideal intersection property.

We now show that the inductive limit of a system of pseudo-Cartan inclusions is again a pseudo-Cartan inclusion, and when the system consists of Cartan inclusions, the inductive limit is Cartan. (In the latter case, we do not apply [LiXinEvClSiC*AlCaSu, Theorem 1.10], preferring instead to use Theorem 6.2(a).)

Theorem \the\numberby.

Suppose $\Lambda$ is a directed set and $\{({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda}\hbox{\,:\hskip-1.0pt:\,}\alpha_{\lambda\mu}):\lambda\geq\mu\}$ is a system of pseudo-Cartan inclusions. Let

{\mathcal{C}}:=\varinjlim{\mathcal{C}}_{\lambda}\quad\text{and}\quad{\mathcal{D}}:=\varinjlim{\mathcal{D}}_{\lambda}.

The following statements hold.

(a)

$({\mathcal{C}},{\mathcal{D}})$ is a pseudo-Cartan inclusion, and for each $\lambda\in\Lambda$ , $({\mathcal{C}},{\mathcal{D}}\hbox{\,:\hskip-1.0pt:\,}\underrightarrow{\alpha_{\lambda}})$ is a regular and essential expansion for $({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda})$ .
(b)

Suppose each $({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda})$ is a Cartan inclusion and $\Delta_{\lambda}:{\mathcal{C}}_{\lambda}\rightarrow{\mathcal{D}}_{\lambda}$ is the conditional expectation. Then $({\mathcal{C}},{\mathcal{D}})$ is a Cartan inclusion and letting $\Delta:{\mathcal{C}}\rightarrow{\mathcal{D}}$ be the conditional expectation, we have $\Delta\circ\underrightarrow{\alpha_{\lambda}}=\underrightarrow{\alpha_{\lambda}}\circ\Delta_{\lambda}$ .

Proof.

(a) Let us show $\underrightarrow{\alpha_{\lambda}}$ is a regular map of $({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda})$ into $({\mathcal{C}},{\mathcal{D}})$ . Let $v\in{\mathcal{N}}({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda})$ and $d\in{\mathcal{D}}$ . For $\varepsilon>0$ , we may find $\mu\geq\lambda$ and $d_{\mu}\in{\mathcal{D}}_{\mu}$ such that $\left\lVert v\right\rVert^{2}\left\lVert d-\underrightarrow{\alpha_{\mu}}(d_{\mu})\right\rVert<\varepsilon$ . Since $\underrightarrow{\alpha_{\mu}}\circ\alpha_{\mu\lambda}=\underrightarrow{\alpha_{\lambda}}$ and $\alpha_{\mu\lambda}$ is a regular map,

\underrightarrow{\alpha_{\lambda}}(v)\underrightarrow{\alpha_{\mu}}(d_{\mu})\underrightarrow{\alpha_{\lambda}}(v^{*})=\underrightarrow{\alpha_{\mu}}(\alpha_{\mu\lambda}(v)d_{\mu}\alpha_{\mu\lambda}(v)^{*})\in\underrightarrow{\alpha_{\mu}}({\mathcal{D}}_{\mu})\subseteq{\mathcal{D}}.

But

\left\lVert\underrightarrow{\alpha_{\lambda}}(v)d\underrightarrow{\alpha_{\lambda}}(v^{*})-\underrightarrow{\alpha_{\lambda}}(v)\underrightarrow{\alpha_{\mu}}(d_{\mu})\underrightarrow{\alpha_{\lambda}}(v^{*})\right\rVert\leq\left\lVert v\right\rVert^{2}\left\lVert d-\underrightarrow{\alpha_{\mu}}(d_{\mu})\right\rVert<\varepsilon.

It follows that $\underrightarrow{\alpha_{\lambda}}(v)d\underrightarrow{\alpha_{\lambda}}(v)^{*}\in{\mathcal{D}}$ . Likewise, $\underrightarrow{\alpha_{\lambda}}(v)^{*}d\underrightarrow{\alpha_{\lambda}}(v)\in{\mathcal{D}}$ . Thus $\underrightarrow{\alpha_{\lambda}}$ is a regular map.

We now show $({\mathcal{C}},{\mathcal{D}})$ is a regular inclusion. Since each $({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda})$ is a regular inclusion we find that

{\mathcal{C}}=\overline{\bigcup_{\lambda}\underrightarrow{\alpha_{\lambda}}({\mathcal{C}}_{\lambda})}=\overline{\bigcup_{\lambda}\operatorname{span}\underrightarrow{\alpha_{\lambda}}({\mathcal{N}}({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda}))}\subseteq\overline{\operatorname{span}}{\mathcal{N}}({\mathcal{C}},{\mathcal{D}}).

Therefore $({\mathcal{C}},{\mathcal{D}})$ is a regular inclusion.

To complete the proof that $({\mathcal{C}},{\mathcal{D}})$ is a pseudo-Cartan inclusion, we show that $({\mathcal{C}},{\mathcal{D}})$ has the faithful unique pseudo-expectation property. Let $(I({\mathcal{D}}),\iota)$ be an injective envelope for ${\mathcal{D}}$ and for $\lambda\in\Lambda$ , define $\iota_{\lambda}:{\mathcal{D}}_{\lambda}\rightarrow I({\mathcal{D}})$ by

\iota_{\lambda}:=\iota\circ\underrightarrow{\alpha_{\lambda}}|_{{\mathcal{D}}_{\lambda}}.

Note that for $\mu\geq\lambda$ ,

\iota_{\mu}\circ\alpha_{\mu\lambda}|_{{\mathcal{D}}_{\lambda}}=\iota_{\lambda}.

We first claim that for every $\lambda\in\Lambda$ , $(I({\mathcal{D}}),\iota_{\lambda})$ is an injective envelope for ${\mathcal{D}}_{\lambda}$ . It suffices to show the inclusion $(I({\mathcal{D}}),{\mathcal{D}}_{\lambda},\iota_{\lambda})$ has the ideal intersection property. Let $J\unlhd I({\mathcal{D}})$ satisfy $\iota_{\lambda}({\mathcal{D}}_{\lambda})\cap J=\{0\}$ . Choose $\mu\geq\lambda$ . Since $\alpha_{\mu\lambda}$ is an essential map, $({\mathcal{D}}_{\mu},{\mathcal{D}}_{\lambda},\alpha_{\mu\lambda}|_{{\mathcal{D}}_{\lambda}})$ has the ideal intersection property. Therefore,

(\iota_{\mu}({\mathcal{D}}_{\mu}),{\mathcal{D}}_{\lambda},\iota_{\mu}\circ\alpha_{\mu\lambda}|_{{\mathcal{D}}_{\lambda}})=(\iota_{\mu}({\mathcal{D}}_{\mu}),{\mathcal{D}}_{\lambda},\iota_{\lambda})

has the ideal intersection property as well. As $J\cap\iota_{\mu}({\mathcal{D}}_{\mu})\cap\iota_{\lambda}({\mathcal{D}}_{\lambda})=J\cap\iota_{\lambda}({\mathcal{D}}_{\lambda})=\{0\}$ , we see $J\cap\iota_{\mu}({\mathcal{D}}_{\mu})=\{0\}$ for every $\mu\geq\lambda$ . But $\bigcup_{\mu\geq\lambda}\alpha_{\mu}({\mathcal{D}}_{\mu})$ is dense in ${\mathcal{D}}$ , so $\bigcup_{\mu\geq\lambda}\iota_{\mu}({\mathcal{D}}_{\mu})$ is dense in $\iota({\mathcal{D}})$ . By [BlackadarOpAl, Proposition II.8.2.4], $J\cap\iota({\mathcal{D}})=\{0\}$ . Since $(I({\mathcal{D}}),{\mathcal{D}},\iota)$ has the ideal intersection property, we conclude $J=\{0\}$ . Thus our claim holds.

We are now ready to show $({\mathcal{C}},{\mathcal{D}})$ has a unique pseudo-expectation. For $i=1,2$ , let $E_{i}:{\mathcal{C}}\rightarrow I({\mathcal{D}})$ be a pseudo-expectation. Then $E_{i}\circ\underrightarrow{\alpha_{\lambda}}$ is a pseudo-expectation for $({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda})$ relative to $(I({\mathcal{D}}),\iota_{\lambda})$ . Since $({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda})$ has the faithful unique pseudo-expectation property, $E_{1}\circ\underrightarrow{\alpha_{\lambda}}=E_{2}\circ\underrightarrow{\alpha_{\lambda}}$ . As this holds for every $\lambda$ , we conclude that $E_{1}=E_{2}$ . Thus, $({\mathcal{C}},{\mathcal{D}})$ has a unique pseudo-expectation $E:{\mathcal{C}}\rightarrow I({\mathcal{D}})$ .

We now show $E$ is faithful. Let ${\mathcal{L}}:=\{x\in{\mathcal{C}}:E(x^{*}x)=0\}$ be the left kernel of $E$ . It follows from [PittsStReInII, Theorem 6.5] and Lemma 2.3 that ${\mathcal{L}}$ is an ideal in ${\mathcal{C}}$ . Then $\underrightarrow{\alpha_{\lambda}}^{-1}({\mathcal{L}})$ is an ideal of ${\mathcal{C}}_{\lambda}$ contained in ${\mathcal{L}}_{\lambda}:=\{x\in{\mathcal{C}}_{\lambda}:E(\underrightarrow{\alpha_{\lambda}}(x^{*}x))=0\}$ . Since $({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda})$ has the faithful unique pseudo-expectation property and $E_{\lambda}=E\circ\underrightarrow{\alpha_{\lambda}}$ is the pseudo-expectation for $({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda})$ , ${\mathcal{L}}_{\lambda}=0$ . We conclude that ${\mathcal{L}}\cap\underrightarrow{\alpha_{\lambda}}({\mathcal{C}}_{\lambda})=\{0\}$ . Thus, using [BlackadarOpAl, Proposition II.8.2.4] yet again, we find ${\mathcal{L}}=\{0\}$ . Therefore, $E$ is faithful. Since $({\mathcal{C}},{\mathcal{D}})$ has the faithful unique pseudo-expectation property, it is a pseudo-Cartan inclusion.

As $(I({\mathcal{D}}),\iota_{\lambda})$ is an injective envelope for ${\mathcal{D}}_{\lambda}$ , $(I({\mathcal{D}}),{\mathcal{D}}_{\lambda},\iota_{\lambda})$ is an essential inclusion. Since

I({\mathcal{D}})\supseteq\iota({\mathcal{D}})\supseteq\iota_{\lambda}({\mathcal{D}}_{\lambda})=\iota(\underrightarrow{\alpha_{\lambda}}({\mathcal{D}})),

Lemma 2.4 shows $({\mathcal{D}},{\mathcal{D}}_{\lambda},\underrightarrow{\alpha_{\lambda}})$ is also an essential inclusion. Since we have already shown that $\underrightarrow{\alpha_{\lambda}}:({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda})\rightarrow({\mathcal{C}},{\mathcal{D}})$ is a regular map, the proof of (a) is complete.

(b) Part (a) shows $({\mathcal{C}},{\mathcal{D}})$ is a pseudo-Cartan inclusion. Clearly, for each $\lambda$ , $({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda}\hbox{\,:\hskip-1.0pt:\,}{\operatorname{id}}|_{{\mathcal{C}}_{\lambda}})$ is a Cartan envelope for $({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda})$ ; let $\Delta_{\lambda}:{\mathcal{C}}_{\lambda}\rightarrow{\mathcal{D}}_{\lambda}$ be the conditional expectation. For $\mu,\lambda\in\Lambda$ with $\lambda\geq\mu$ , Lemma 6.1 gives

\alpha_{\lambda\mu}\circ\Delta_{\mu}=\Delta_{\lambda}\circ\alpha_{\lambda\mu}.

This means that there exists $\Delta:\bigcup_{\lambda}\underrightarrow{\alpha_{\lambda}}({\mathcal{C}}_{\lambda})\rightarrow\bigcup_{\lambda}\underrightarrow{\alpha_{\lambda}}({\mathcal{D}}_{\lambda})$ such that $\Delta\circ\underrightarrow{\alpha_{\lambda}}=\underrightarrow{\alpha_{\lambda}}\circ\Delta_{\lambda}$ . Since each $\Delta_{\lambda}$ is surjective, contractive and idempotent, $\Delta$ also has these properties. Therefore, $\Delta$ extends to a conditional expectation $\Delta:{\mathcal{C}}\rightarrow{\mathcal{D}}$ . Proposition 2.3 shows $({\mathcal{C}},{\mathcal{D}})$ is a Cartan inclusion. Lemma 6.1 gives $\Delta\circ\underrightarrow{\alpha_{\lambda}}=\underrightarrow{\alpha_{\lambda}}\circ\Delta_{\lambda}$ . ∎

We next show that the Cartan envelope of an inductive limit of a system of pseudo-Cartan inclusions is the inductive limit of the Cartan envelopes.

Theorem \the\numberby.

Suppose $\Lambda$ is a directed set and $\{({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda}\hbox{\,:\hskip-1.0pt:\,}\alpha_{\lambda\mu}):\lambda\geq\mu\}$ is a system of pseudo-Cartan inclusions. By Theorem 6.2, $(\varinjlim{\mathcal{C}}_{\lambda},\varinjlim{\mathcal{D}}_{\lambda})$ is a pseudo-Cartan inclusion.

For each $\lambda$ , let $({\mathcal{A}}_{\lambda},{\mathcal{B}}_{\lambda}\hbox{\,:\hskip-1.0pt:\,}\tau_{\lambda})$ be a Cartan envelope for $({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda})$ . Theorem 6.1 provides unique regular $*$ -monomorphisms $\breve{\alpha}_{\lambda\mu}:({\mathcal{A}}_{\mu},{\mathcal{B}}_{\mu})\rightarrow({\mathcal{A}}_{\lambda},{\mathcal{B}}_{\lambda})$ ( $\lambda\geq\mu$ ) satisfying

(6.2.4)

\breve{\alpha}_{\lambda\mu}\circ\tau_{\mu}=\tau_{\lambda}\circ\alpha_{\lambda\mu}.

The following statements hold.

(a)

$\{({\mathcal{A}}_{\lambda},{\mathcal{B}}_{\lambda}\hbox{\,:\hskip-1.0pt:\,}\breve{\alpha}_{\lambda\mu}):\lambda\geq\mu\}$ is a system of Cartan inclusions and

$(\varinjlim{\mathcal{A}}_{\lambda},\varinjlim{\mathcal{B}}_{\lambda})$

is a Cartan inclusion.
(b)

There is a regular $*$ -monomorphism $\tau:\varinjlim{\mathcal{C}}_{\lambda}\rightarrow\varinjlim{\mathcal{A}}_{\lambda}$ such that for every $\lambda\in\Lambda$ ,

(6.2.5) $\underrightarrow{\breve{\alpha}_{\lambda}}\circ\tau_{\lambda}=\tau\circ\underrightarrow{\alpha_{\lambda}}$

and $(\varinjlim{\mathcal{A}}_{\lambda},\varinjlim{\mathcal{B}}_{\lambda}\hbox{\,:\hskip-1.0pt:\,}\tau)$ is a Cartan envelope for $(\varinjlim{\mathcal{C}}_{\lambda},\varinjlim{\mathcal{D}}_{\lambda})$ .

Proof.

Throughout the proof we will sometimes simplify notation and write,

	$\displaystyle{\mathcal{C}}$	$\displaystyle:=\varinjlim{\mathcal{C}}_{\lambda}\quad\text{and}\quad{\mathcal{D}}:=\varinjlim{\mathcal{D}}_{\lambda}.$
Similarly, once part (a) is established, we will sometimes write
	$\displaystyle{\mathcal{A}}$	$\displaystyle:=\varinjlim{\mathcal{A}}_{\lambda}\quad\text{and}\quad{\mathcal{B}}:=\varinjlim{\mathcal{B}}_{\lambda}.$

(a) Theorem 6.1(b) shows that for $\lambda\geq\mu$ , $({\mathcal{A}}_{\lambda},{\mathcal{B}}_{\lambda}\hbox{\,:\hskip-1.0pt:\,}\breve{\alpha}_{\lambda\mu})$ is a regular and essential expansion of $({\mathcal{A}}_{\mu},{\mathcal{B}}_{\mu})$ . Now suppose $\lambda\geq\mu\geq\nu$ . We have

\breve{\alpha}_{\lambda\nu}\circ\tau_{\nu}=\tau_{\lambda}\circ\alpha_{\lambda\nu}

and

(\breve{\alpha}_{\lambda\mu}\circ\breve{\alpha}_{\mu\nu})\circ\tau_{\nu}=\breve{\alpha}_{\lambda\mu}\circ\tau_{\mu}\circ\alpha_{\mu\nu}=\tau_{\lambda}\circ\alpha_{\lambda\mu}\circ\alpha_{\mu\nu}=\tau_{\lambda}\circ\alpha_{\lambda\nu}

By the uniqueness part of Theorem 6.1(a), we find $\breve{\alpha}_{\lambda\nu}=\breve{\alpha}_{\lambda\mu}\circ\breve{\alpha}_{\mu\nu}$ , so

\{({\mathcal{A}}_{\lambda},{\mathcal{B}}_{\lambda}\hbox{\,:\hskip-1.0pt:\,}\breve{\alpha}_{\lambda\mu}):\lambda\geq\nu\}

is a system of Cartan inclusions. Theorem 6.2(b) shows $(\varinjlim{\mathcal{A}}_{\lambda},\varinjlim{\mathcal{B}}_{\lambda})$ is a Cartan inclusion.

(b) By properties of inductive limits and (6.2.4), there is a uniquely determined $*$ -monomorphism

\tau:{\mathcal{C}}\rightarrow{\mathcal{A}}

satisfying (6.2.5). While $\tau\circ\underrightarrow{\alpha_{\lambda}}$ is a regular map, we have been unable to directly show that this forces $\tau$ to be a regular map, because we lack a description of ${\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ . (We expect ${\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ is the closure of $\bigcup_{\lambda}\underrightarrow{\alpha_{\lambda}}({\mathcal{N}}({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda}))$ , but we have not found a proof.)

Therefore, we proceed as follows. Fix a Cartan envelope

({\mathcal{A}}^{\prime},{\mathcal{B}}^{\prime}\hbox{\,:\hskip-1.0pt:\,}\tau^{\prime})

for ${\mathcal{C}},{\mathcal{D}})$ , with conditional expectation $\Delta^{\prime}:{\mathcal{A}}^{\prime}\rightarrow{\mathcal{B}}^{\prime}$ . We shall show $({\mathcal{A}}^{\prime},{\mathcal{B}}^{\prime})$ is an inductive limit of a system

\{({\mathcal{A}}_{\lambda}^{\prime},{\mathcal{B}}_{\lambda}^{\prime}\hbox{\,:\hskip-1.0pt:\,}\tau_{\lambda}^{\prime}):\lambda\geq\mu\}

of Cartan envelopes for $({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda})$ , where each ${\mathcal{A}}_{\lambda}^{\prime}\subseteq{\mathcal{A}}^{\prime}$ . We will then use uniqueness of Cartan envelopes to produce an isomorphism $\psi^{\prime}:\varinjlim{\mathcal{A}}_{\lambda}^{\prime}\rightarrow\varinjlim{\mathcal{A}}_{\lambda}$ which carries $\varinjlim{\mathcal{B}}_{\lambda}^{\prime}$ onto $\varinjlim{\mathcal{B}}_{\lambda}$ . The regularity of $\tau$ will then follow from the regularity of $\tau^{\prime}$ . As we construct and discuss the various spaces and maps involved, the reader might find it helpful to consult Figure 1.

For $\lambda\in\Lambda$ , define $\tau_{\lambda}^{\prime}:{\mathcal{C}}_{\lambda}\rightarrow{\mathcal{A}}^{\prime}$ by

\tau_{\lambda}^{\prime}:=\tau^{\prime}\circ\underrightarrow{\alpha_{\lambda}},

and put

{\mathcal{B}}_{\lambda}^{\prime}:=C^{*}(\Delta^{\prime}(\tau_{\lambda}^{\prime}({\mathcal{C}}_{\lambda})),\quad{\mathcal{A}}_{\lambda}^{\prime}:=C^{*}(\tau_{\lambda}^{\prime}({\mathcal{C}}_{\lambda})\cup{\mathcal{B}}_{\lambda}^{\prime}),\quad\text{and}\quad\Delta_{\lambda}^{\prime}:=\Delta^{\prime}|_{{\mathcal{A}}_{\lambda}^{\prime}}.

For future use, we note that by construction,

(6.2.6)

{\mathcal{B}}_{\lambda}^{\prime}\subseteq{\mathcal{B}}^{\prime}\quad\text{and}\quad{\mathcal{A}}_{\lambda}^{\prime}\subseteq{\mathcal{A}}^{\prime}.

Theorem 6.2(a) shows $({\mathcal{C}},{\mathcal{D}}\hbox{\,:\hskip-1.0pt:\,}\underrightarrow{\alpha_{\lambda}})$ is a regular and essential expansion of $({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda})$ and $\tau^{\prime}$ is regular by hypothesis. Therefore $\tau_{\lambda}^{\prime}$ is a regular map. Also,

\tau_{\lambda}^{\prime}({\mathcal{D}}_{\lambda})=\tau^{\prime}(\underrightarrow{\alpha_{\lambda}}({\mathcal{D}}_{\lambda}))\subseteq\tau^{\prime}({\mathcal{D}})\subseteq{\mathcal{B}}^{\prime},

so $({\mathcal{B}}^{\prime},\tau^{\prime}({\mathcal{D}}_{\lambda}))$ has the ideal intersection property by Lemma 2.4. As $\tau_{\lambda}^{\prime}({\mathcal{D}}_{\lambda})\subseteq{\mathcal{B}}_{\lambda}^{\prime}\subseteq{\mathcal{B}}^{\prime}$ we see that $({\mathcal{A}}_{\lambda}^{\prime},{\mathcal{B}}_{\lambda}^{\prime}\hbox{\,:\hskip-1.0pt:\,}\tau_{\lambda}^{\prime})$ is a regular and essential expansion for $({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda})$ . Since $({\mathcal{A}}^{\prime},{\mathcal{B}}^{\prime}\hbox{\,:\hskip-1.0pt:\,}\tau_{\lambda}^{\prime})$ is an essential, regular and Cartan expansion for $({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda})$ , Proposition 3 shows $({\mathcal{A}}_{\lambda}^{\prime},{\mathcal{B}}_{\lambda}^{\prime}\hbox{\,:\hskip-1.0pt:\,}\tau^{\prime}_{\lambda})$ is a Cartan envelope for $({\mathcal{C}}_{\lambda},{\mathcal{D}}_{\lambda})$ and $\Delta_{\lambda}^{\prime}$ is the conditional expectation of ${\mathcal{A}}_{\lambda}^{\prime}$ onto ${\mathcal{B}}_{\lambda}^{\prime}$ .

For $\lambda\geq\mu$ , let $\breve{\alpha}_{\lambda\mu}^{\prime}:{\mathcal{A}}_{\mu}^{\prime}\rightarrow{\mathcal{A}}_{\lambda}^{\prime}$ be the $*$ -monomorphism satisfying

(6.2.7)

\breve{\alpha}_{\lambda\mu}^{\prime}\circ\tau_{\mu}^{\prime}=\tau_{\lambda}^{\prime}\circ\alpha_{\lambda\mu}

obtained from Theorem 6.1. Since

\tau_{\mu}^{\prime}=\tau^{\prime}\circ\underrightarrow{\alpha_{\mu}}=\tau^{\prime}\circ\underrightarrow{\alpha_{\lambda}}\circ\alpha_{\lambda\mu}=\tau_{\lambda}^{\prime}\circ\alpha_{\lambda\mu},

the uniqueness portion of Theorem 6.1 shows $\breve{\alpha}_{\lambda\mu}^{\prime}$ is the inclusion map. Thus, $\underrightarrow{\breve{\alpha}_{\lambda}}:{\mathcal{A}}_{\lambda}^{\prime}\rightarrow{\mathcal{A}}^{\prime}$ is also the inclusion map (see (6.2.6)). Theorem 6.1 shows $({\mathcal{A}}_{\lambda}^{\prime},{\mathcal{B}}_{\lambda}^{\prime}\hbox{\,:\hskip-1.0pt:\,}\breve{\alpha}_{\lambda\mu}^{\prime})$ is a regular and essential expansion of $({\mathcal{A}}_{\mu}^{\prime},{\mathcal{B}}_{\mu}^{\prime})$ .

Since $\bigcup_{\lambda}\underrightarrow{\alpha_{\lambda}}({\mathcal{C}}_{\lambda})$ is dense in ${\mathcal{C}}$ , we find $\bigcup_{\lambda}\Delta^{\prime}(\tau^{\prime}(\underrightarrow{\alpha_{\lambda}}({\mathcal{C}}_{\lambda})))=\Delta^{\prime}(\tau^{\prime}(\bigcup_{\lambda}\underrightarrow{\alpha_{\lambda}}({\mathcal{C}}_{\lambda})))$ is dense in $\Delta^{\prime}(\tau^{\prime}({\mathcal{C}}))$ . Therefore since ${\mathcal{B}}^{\prime}=C^{*}(\Delta^{\prime}(\tau^{\prime}({\mathcal{C}})))$ ,

\bigcup_{\lambda}{\mathcal{B}}_{\lambda}^{\prime}=\bigcup_{\lambda}C^{*}(\Delta^{\prime}(\tau^{\prime}(\underrightarrow{\alpha_{\lambda}}({\mathcal{C}}_{\lambda}))))

is dense in ${\mathcal{B}}^{\prime}$ . Similarly, $\bigcup_{\lambda}(\tau^{\prime}(\underrightarrow{\alpha_{\lambda}}({\mathcal{C}}_{\lambda}))\cup{\mathcal{B}}_{\lambda}^{\prime})=\left(\bigcup_{\lambda}\tau^{\prime}(\underrightarrow{\alpha_{\lambda}}({\mathcal{C}}_{\lambda}))\right)\cup(\bigcup_{\lambda}{\mathcal{B}}_{\lambda}^{\prime})$ is dense in $\tau^{\prime}({\mathcal{C}})\cup{\mathcal{B}}^{\prime}$ , so $\bigcup_{\lambda}{\mathcal{A}}_{\lambda}^{\prime}$ is dense in ${\mathcal{A}}^{\prime}$ . We conclude that

{\mathcal{A}}^{\prime}=\varinjlim{\mathcal{A}}_{\lambda}^{\prime}\quad\text{and}\quad{\mathcal{B}}^{\prime}=\varinjlim{\mathcal{B}}_{\lambda}^{\prime}.

Figure 1. The squares on the left side having vertical sides labeled with a subscripted

\tau

or subscripted

\tau^{\prime}

commute by Theorem 6.1. The horizontal maps on the bottom row are inclusion maps.

We now use uniqueness of Cartan envelopes to show $\varinjlim{\mathcal{A}}_{\lambda}^{\prime}$ is isomorphic to $\varinjlim{\mathcal{A}}_{\lambda}$ . The uniqueness part of Theorem 3 gives the existence of a unique $*$ -isomorphism $\psi^{\prime}_{\lambda}:{\mathcal{A}}_{\lambda}^{\prime}\rightarrow{\mathcal{A}}_{\lambda}$ satisfying

\psi_{\lambda}^{\prime}\circ\tau_{\lambda}^{\prime}=\tau_{\lambda}.

By properties of inductive limits, to show the existence of an isomorphism $\psi^{\prime}:\varinjlim{\mathcal{A}}_{\lambda}^{\prime}\rightarrow\varinjlim{\mathcal{A}}_{\lambda}$ it suffices to show

(6.2.8)

\breve{\alpha}_{\lambda\mu}\circ\psi_{\mu}^{\prime}=\psi_{\lambda}^{\prime}\circ\breve{\alpha}_{\lambda\mu}^{\prime};

(that $\psi^{\prime}$ is invertible will follow from rewriting (6.2.8) as $(\psi_{\lambda}^{\prime})^{-1}\circ\breve{\alpha}_{\lambda\mu}=\breve{\alpha}^{\prime}_{\lambda\mu}\circ(\psi_{\mu}^{\prime})^{-1}$ ). Recalling that ${\mathcal{A}}_{\lambda}^{\prime}$ is generated by $\tau_{\lambda}^{\prime}({\mathcal{C}}_{\lambda})\cup\Delta_{\lambda}^{\prime}(\tau_{\lambda}^{\prime}({\mathcal{C}}_{\lambda}))$ , it is enough to show

(6.2.9)	$\displaystyle\breve{\alpha}_{\lambda\mu}\circ\psi_{\mu}^{\prime}\circ\tau_{\mu}^{\prime}$	$\displaystyle=\psi_{\lambda}^{\prime}\circ\breve{\alpha}_{\lambda\mu}^{\prime}\circ\tau_{\mu}^{\prime}$
and
(6.2.10)	$\displaystyle\breve{\alpha}_{\lambda\mu}\circ\psi_{\mu}^{\prime}\circ\Delta_{\mu}^{\prime}\circ\tau_{\mu}^{\prime}$	$\displaystyle=\psi_{\lambda}^{\prime}\circ\breve{\alpha}_{\lambda\mu}^{\prime}\circ\Delta_{\mu}^{\prime}\circ\tau_{\mu}^{\prime}.$

For (6.2.9), the uniqueness part of Theorem 3 gives,

\breve{\alpha}_{\lambda\mu}\circ\psi_{\mu}^{\prime}\circ\tau_{\mu}^{\prime}\stackrel{{\scriptstyle\eqref{!pschar}}}{{=}}\breve{\alpha}_{\lambda\mu}\circ\tau_{\mu}\stackrel{{\scriptstyle\eqref{cmap}}}{{=}}\tau_{\lambda}\circ\alpha_{\lambda\mu}\stackrel{{\scriptstyle\eqref{!pschar}}}{{=}}\psi_{\lambda}^{\prime}\circ\tau_{\lambda}^{\prime}\circ\alpha_{\lambda\mu}\stackrel{{\scriptstyle\eqref{cmap}}}{{=}}\psi_{\lambda}^{\prime}\circ\breve{\alpha}_{\lambda\mu}^{\prime}\circ\tau_{\mu}^{\prime}.

Turning to (6.2.10), $\psi_{\mu}^{\prime}\circ\Delta_{\mu}^{\prime}\circ(\psi_{\mu}^{\prime})^{-1}$ is a conditional expectation from ${\mathcal{A}}_{\mu}$ onto ${\mathcal{B}}_{\mu}$ . By uniqueness of the conditional expectation for Cartan inclusions, we obtain $\psi_{\mu}^{\prime}\circ\Delta_{\mu}^{\prime}\circ(\psi_{\mu}^{\prime})^{-1}=\Delta_{\mu}$ . Thus

(6.2.11)

\psi_{\mu}^{\prime}\circ\Delta_{\mu}^{\prime}=\Delta_{\mu}\circ\psi_{\mu}^{\prime};\quad\text{similarly}\quad\psi_{\lambda}^{\prime}\circ\Delta_{\lambda}^{\prime}=\Delta_{\lambda}\circ\psi_{\lambda}^{\prime}.

Hence,

	$\displaystyle\breve{\alpha}_{\lambda\mu}\circ\psi_{\mu}^{\prime}\circ\Delta_{\mu}^{\prime}\circ\tau_{\mu}^{\prime}$	$\displaystyle\stackrel{{\scriptstyle\eqref{indlim3.b}}}{{=}}\breve{\alpha}_{\lambda\mu}\circ\Delta_{\mu}\circ\psi_{\mu}^{\prime}\circ\tau_{\mu}^{\prime}$
		$\displaystyle\stackrel{{\scriptstyle\eqref{!pschar}}}{{=}}\breve{\alpha}_{\lambda\mu}\circ\Delta_{\mu}\circ\tau_{\mu}\stackrel{{\scriptstyle\eqref{cmap1.1}}}{{=}}\Delta_{\lambda}\circ\tau_{\lambda}\circ\alpha_{\lambda\mu}$
		$\displaystyle\stackrel{{\scriptstyle\eqref{!pschar}}}{{=}}\Delta_{\lambda}\circ\psi_{\lambda}^{\prime}\circ\tau_{\lambda}^{\prime}\circ\alpha_{\lambda\mu}\stackrel{{\scriptstyle\eqref{cmap1}}}{{=}}\Delta_{\lambda}\circ\psi_{\lambda}^{\prime}\circ\breve{\alpha}_{\lambda\mu}^{\prime}\circ\tau_{\mu}^{\prime}$
		$\displaystyle\stackrel{{\scriptstyle\eqref{indlim3.b}}}{{=}}\psi_{\lambda}^{\prime}\circ\Delta_{\lambda}^{\prime}\circ\breve{\alpha}_{\lambda\mu}^{\prime}\circ\tau_{\mu}^{\prime}\stackrel{{\scriptstyle\eqref{cmap1.05}}}{{=}}\psi_{\lambda}^{\prime}\circ\breve{\alpha}_{\lambda\mu}^{\prime}\circ\Delta_{\mu}^{\prime}\circ\tau_{\mu}^{\prime},$

so (6.2.10) holds. This completes the proof of (6.2.8), and the existence of the isomorphism $\psi^{\prime}:\varinjlim{\mathcal{A}}_{\lambda}^{\prime}\rightarrow\varinjlim{\mathcal{A}}_{\lambda}$ .

For each $\mu\in\Lambda$ , $\psi_{\mu}^{\prime}({\mathcal{B}}_{\mu}^{\prime})={\mathcal{B}}_{\mu}$ , and $\breve{\alpha}_{\lambda\mu}\circ(\psi_{\mu}^{\prime}|_{{\mathcal{B}}_{\lambda}^{\prime}})=\psi_{\lambda}^{\prime}\circ(\breve{\alpha}_{\lambda\mu}^{\prime}|_{{\mathcal{B}}_{\mu}^{\prime}})$ , so

\psi^{\prime}(\varinjlim{\mathcal{B}}_{\lambda}^{\prime})=\varinjlim{\mathcal{B}}_{\lambda}.

This implies that $\psi^{\prime}$ is regular. Since $\tau_{\lambda}=\psi_{\lambda}^{\prime}\circ\tau_{\lambda}^{\prime}$ we obtain

\tau=\psi^{\prime}\circ\tau^{\prime}.

As $\psi$ and $\tau^{\prime}$ are regular, we conclude $\tau$ is regular as well. Hence $(\varinjlim{\mathcal{A}}_{\lambda},\varinjlim{\mathcal{B}}_{\lambda},\tau)$ is a Cartan envelope for $(\varinjlim{\mathcal{C}}_{\lambda},\varinjlim{\mathcal{D}}_{\lambda})$ . This completes the proof. ∎

6.3. Minimal Tensor Products

The purpose of this subsection is to show that the minimal tensor product of two pseudo-Cartan inclusions is again a pseudo-Cartan inclusion. Throughout, for $i=1,2$ , let $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ be pseudo-Cartan inclusions. For algebras ${\mathcal{A}}$ and ${\mathcal{B}}$ , we use the notation ${\mathcal{A}}\odot{\mathcal{B}}$ for the linear span of $\{a\otimes b:a\in{\mathcal{A}},b\in{\mathcal{B}}\}$ .

Regularity of the tensor product inclusion is straightforward, and recorded in the following lemma.

Lemma \the\numberby.

Let ${\mathcal{S}}:=\{v_{1}\otimes v_{2}:v_{i}\in{\mathcal{N}}({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ . Then ${\mathcal{S}}\subseteq{\mathcal{N}}({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ and $\operatorname{span}{\mathcal{S}}$ is dense in ${\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2}$ . In particular, $({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ is a regular inclusion.

Proof.

Let ${\mathfrak{A}}:=\operatorname{span}{\mathcal{S}}$ .

That ${\mathcal{S}}\subseteq{\mathcal{N}}({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ is evident. For $i=1,2$ , suppose $x_{i}\in{\mathcal{C}}_{i}$ . We wish to show that $x_{1}\otimes x_{2}\in\overline{\mathfrak{A}}$ . Let $\varepsilon>0$ . By regularity of $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ , there exists $N\in{\mathbb{N}}$ , $\{v_{j}\}_{j=1}^{N}\subseteq{\mathcal{N}}({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ and $\{w_{j}\}_{j=1}^{N}\subseteq{\mathcal{N}}({\mathcal{C}}_{2},{\mathcal{D}}_{2})$ such that

\left\lVert x_{1}-\sum_{j=1}^{N}v_{j}\right\rVert<\varepsilon\quad\text{and}\quad\left\lVert x_{2}-\sum_{j=1}^{N}w_{j}\right\rVert<\varepsilon.

Then

	$\displaystyle\left\lVert x_{1}\otimes x_{2}-\left(\sum_{j=1}^{N}v_{j}\right)\otimes\left(\sum_{j=1}^{N}w_{j}\right)\right\rVert$	$\displaystyle\leq\left\lVert\left(x_{1}-\sum_{j=1}^{N}v_{j}\right)\otimes x_{2}\right\rVert$
		$\displaystyle+\left\lVert\left(\sum_{j=1}^{N}v_{j}\right)\otimes\left(x_{2}-\sum_{j=1}^{N}w_{j}\right)\right\rVert$
		$\displaystyle<\varepsilon\left\lVert x_{2}\right\rVert+(\left\lVert x_{1}\right\rVert+\varepsilon)\varepsilon.$

Therefore, $x_{1}\otimes x_{2}\in\overline{\mathfrak{A}}$ . As the span of elementary tensors is dense in ${\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2}$ , the lemma follows. ∎

Our next goal is Proposition 6.3, which shows that when $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ and $({\mathcal{C}}_{2},{\mathcal{D}}_{2})$ are unital pseudo-Cartan inclusions, then $({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ has the unique pseudo-expectation property. On first glance, one might expect that if $E_{i}:{\mathcal{C}}_{i}\rightarrow I({\mathcal{D}}_{i})$ are the unique pseudo-expectations for $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ , then $E_{1}\otimes E_{2}$ will be the unique pseudo-expectation for $({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ . However, the codomain of $E_{1}\otimes E_{2}$ is $I({\mathcal{D}}_{1})\otimes I({\mathcal{D}}_{2})$ , rather than $I({\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ and these $C^{*}$ -algebras need not be the same [WillardGeTo, Exercises 15G and 19I.2]. Nevertheless, Proposition 6.3 below implies there is an essential embedding $g$ of $I({\mathcal{D}}_{1})\otimes I({\mathcal{D}}_{2})$ into $I({\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ . Using $g$ to identify $I({\mathcal{D}}_{1})\otimes I({\mathcal{D}}_{2})$ with its image in $I({\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ will allow us to view $E_{1}\otimes E_{2}$ as a pseudo-expectation for $({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ .

Dual to the notion of an essential embedding of $C(X)$ into $C(Y)$ is the notion of essential surjection of $Y$ onto $X$ : a continuous surjection $\pi:Y\twoheadrightarrow X$ between compact Hausdorff spaces is called an essential surjection if whenever $F\subseteq Y$ is a closed set and $\pi(F)=X$ , then $F=Y$ . (Essential surjections are also called irreducible maps.)

Lemma \the\numberby.

Suppose for $i=1,2$ , $Y_{i}$ and $X_{i}$ are compact Hausdorff spaces, and $\pi_{i}:Y_{i}\twoheadrightarrow X_{i}$ are continuous and essential surjections. Then $\pi_{1}\times\pi_{2}:Y_{1}\times Y_{2}\twoheadrightarrow X_{1}\times X_{2}$ , given by $(y_{1},y_{2})\mapsto(\pi_{1}(y_{1}),\pi_{2}(y_{2}))$ , is an essential surjection.

Proof.

We start by proving a special case: assume that $Y_{2}=X_{2}$ and $\pi_{2}$ is the identity map.

For typographical ease, let $\pi:=\pi_{1}\times{\operatorname{id}}_{X_{2}}$ . Let $F\subseteq Y_{1}\times X_{2}$ be a closed set such that $\pi(F)=X_{1}\times X_{2}$ . For $t\in X_{2}$ , let

F_{t}:=\{y_{1}\in Y_{1}:(y_{1},t)\in F\}.

Then $F_{t}$ is a closed subset of $Y_{1}$ . Given $s\in X_{1}$ , since $\pi(F)=X_{1}\times X_{2}$ , there is $(y_{1},t)\in F$ such that $\pi(y_{1},t)=(s,t)$ . Thus $\pi_{1}(F_{t})=X_{1}$ . Since $\pi_{1}$ is essential, we obtain $F_{t}=Y_{1}$ . Hence for $t\in X_{2}$ , $Y_{1}\times\{t\}\subseteq F$ . We conclude that $F=Y_{1}\times X_{2}$ , showing that the lemma holds in this special case.

For the general case, consider the composition,

Y_{1}\times Y_{2}\stackrel{{\scriptstyle\pi_{1}\times{\operatorname{id}}_{Y_{2}}}}{{\longrightarrow}}X_{1}\times Y_{2}\stackrel{{\scriptstyle{\operatorname{id}}_{X_{1}}\times\pi_{2}}}{{\longrightarrow}}X_{1}\times X_{2}.

By the special case, each of the maps in the composition are essential surjections. As the composition of essential surjections is an essential surjection, the proof is complete. ∎

Proposition \the\numberby.

For $i=1,2$ , let ${\mathcal{B}}_{i}$ be abelian $C^{*}$ -algebras (perhaps not unital) and suppose $({\mathcal{B}}_{i},{\mathcal{D}}_{i},\alpha_{i})$ are essential inclusions. Then $({\mathcal{B}}_{1}\otimes{\mathcal{B}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2},\alpha_{1}\otimes\alpha_{2})$ is an essential inclusion.

Proof.

First assume the inclusions $({\mathcal{B}}_{i},{\mathcal{D}}_{i},\alpha_{i})$ are unital. For $i=1,2$ , let $Y_{i}:=\widehat{\mathcal{B}}_{i}$ and $X_{i}:=\widehat{\mathcal{D}}_{i}$ . The maps $\alpha_{i}$ dualize to continuous surjections $\pi_{i}:Y_{i}\rightarrow X_{i}$ (thus for $d\in{\mathcal{D}}_{i}$ , $\pi_{i}(y_{i})(d)=y_{i}(\alpha_{i}(d))$ ). The proof in this case now follows from Lemma 6.3 after noting that the inclusion $({\mathcal{B}}_{i},{\mathcal{D}}_{i},\alpha_{i})$ is essential if and only if $\pi_{i}$ is an essential surjection of $Y_{i}$ onto $X_{i}$ , see [PittsIrMaIsBoReOpSeReId, Lemma 4.9].

For the general case, note that $(\tilde{\mathcal{D}}_{1}\otimes\tilde{\mathcal{D}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2},u_{{\mathcal{D}}_{1}}\otimes u_{{\mathcal{D}}_{2}})$ is an essential inclusion because the image of ${\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2}$ under $u_{{\mathcal{D}}_{1}}\otimes u_{{\mathcal{D}}_{2}}$ is an essential ideal in $\tilde{\mathcal{D}}_{1}\otimes\tilde{\mathcal{D}}_{2}$ . (This can be shown directly or one can use the facts that $\tilde{\mathcal{D}}_{1}\otimes\tilde{\mathcal{D}}_{2}\subseteq M({\mathcal{D}}_{1})\otimes M({\mathcal{D}}_{2})\subseteq M({\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ and any $C^{*}$ -algebra is an essential ideal in its multiplier algebra. A proof of the inclusion of the multiplier algebras can be found at https://math.stackexchange.com/a/4458451.) This fact, together with the unital case show that both of the inclusions,

\alpha_{1}({\mathcal{D}}_{1})\otimes\alpha_{2}({\mathcal{D}}_{2})\subseteq\tilde{\alpha}_{1}(\tilde{\mathcal{D}}_{1})\otimes\tilde{\alpha}_{2}(\tilde{\mathcal{D}}_{2}))\subseteq\tilde{\mathcal{B}}_{1}\otimes\tilde{\mathcal{B}}_{2}

have the ideal intersection property. By Lemma 2.4, $\alpha_{1}({\mathcal{D}}_{1})\otimes\alpha_{2}({\mathcal{D}}_{2})\subseteq\tilde{\mathcal{B}}_{1}\otimes\tilde{\mathcal{B}}_{2}$ has the ideal intersection property. Finally, since $\alpha_{1}({\mathcal{D}}_{1})\otimes\alpha_{2}({\mathcal{D}}_{2})\subseteq{\mathcal{B}}_{1}\otimes{\mathcal{B}}_{2}\subseteq\tilde{\mathcal{B}}_{1}\otimes\tilde{\mathcal{B}}_{2}$ , another application of Lemma 2.4 shows $\alpha_{1}({\mathcal{D}}_{1})\otimes\alpha_{2}({\mathcal{D}}_{2})\subseteq{\mathcal{B}}_{1}\otimes{\mathcal{B}}_{2}$ has the ideal intersection property, as desired. ∎

Assume now that $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ are unital pseudo-Cartan inclusions, let $(I({\mathcal{D}}_{i}),\iota_{i})$ be injective envelopes for ${\mathcal{D}}_{i}$ and let $(I({\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2}),\iota)$ be an injective envelope for ${\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2}$ . By Proposition 6.3,

(I({\mathcal{D}}_{1})\otimes I({\mathcal{D}}_{2}),{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2},\iota_{1}\otimes\iota_{2})

is an essential inclusion. Thus [PittsZarikianUnPsExC*In, Corollary 3.22] gives a unique $*$ -monomorphism $g:I({\mathcal{D}}_{1})\otimes I({\mathcal{D}}_{2})\rightarrow I({\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ such that

\iota=g\circ(\iota_{1}\otimes\iota_{2}).

Let $E_{i}:{\mathcal{C}}_{i}\rightarrow I({\mathcal{D}}_{i})$ be the pseudo-expectations for $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ (relative to the envelopes $(I({\mathcal{D}}_{i}),\iota_{i})$ ). By [BrownOzawaC*AlFiDiAp, Theorem 3.5.3], there is a unique unital completely positive map $E_{1}\otimes E_{2}:{\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2}\rightarrow I({\mathcal{D}}_{1})\otimes I(D_{2})$ such that for every elementary tensor $x_{1}\otimes x_{2}\in{\mathcal{C}}_{1}\odot{\mathcal{C}}_{2}$ , $(E_{1}\otimes E_{2})(x_{1}\otimes x_{2})=E_{1}(x_{1})\otimes E_{2}(x_{2})$ . Let

(6.3.1)

E:=g\circ(E_{1}\otimes E_{2}).

Then $E:{\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2}\rightarrow I({\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ is a pseudo-expectation for $({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ relative to $(I({\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2}),\iota)$ . The following commuting diagram illustrates these maps; (the existence of the map labeled “inclusion” follows from [BrownOzawaC*AlFiDiAp, Proposition 3.6.1]).

We are now ready to show $({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ has the unique pseudo-expectation property.

Proposition \the\numberby.

For $i=1,2$ , let $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ be unital pseudo-Cartan inclusions. The map $E$ defined in (6.3.1) is the unique pseudo-expectation for the inclusion $({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ (relative to $(I({\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2}),\iota)$ ).

Proof.

Since $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ are pseudo-Cartan inclusions, ${\mathcal{D}}_{i}^{c}$ is abelian and $({\mathcal{D}}_{i}^{c},{\mathcal{D}}_{i})$ has the ideal intersection property, so by Proposition 2.4, $({\mathcal{D}}_{i}^{c},{\mathcal{D}}_{i})$ has the faithful unique pseudo-expectation property. In particular, $(I({\mathcal{D}}_{i}),E_{i}|_{{\mathcal{D}}_{i}})$ is an injective envelope for ${\mathcal{D}}_{i}^{c}$ (see [PittsZarikianUnPsExC*In, Corollary 3.22]). Thus, $E_{i}$ is also a pseudo-expectation for $({\mathcal{C}}_{i},{\mathcal{D}}_{i}^{c})$ . Observation 4.1 shows $({\mathcal{C}}_{i},{\mathcal{D}}_{i}^{c})$ is a virtual Cartan inclusion, and hence $({\mathcal{C}}_{i},{\mathcal{D}}_{i}^{c})$ has the unique pseudo-expectation property. Therefore, $E_{i}$ is the unique pseudo-expectation for $({\mathcal{C}}_{i},{\mathcal{D}}_{i}^{c})$ .

Now let $\Delta:{\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2}\rightarrow I({\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ be a pseudo-expectation. Proposition 6.3 shows $({\mathcal{D}}_{1}^{c}\otimes{\mathcal{D}}_{2}^{c},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ has the ideal intersection property, so another application of [PittsZarikianUnPsExC*In, Corollary 3.22] gives

(6.3.2)

\Delta|_{{\mathcal{D}}_{1}^{c}\otimes{\mathcal{D}}_{2}^{c}}=E|_{{\mathcal{D}}_{1}^{c}\otimes{\mathcal{D}}_{2}^{c}}.

Clearly ${\mathcal{D}}_{1}^{c}\otimes{\mathcal{D}}_{2}^{c}\subseteq({\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})^{c}$ . If equality held, an application of [PittsStReInII, Proposition 6.11] would complete the proof. However, such formulae for tensor products of relative commutants do not hold in general, see [ArchboldCoCoTePrC*Al]. Since we do not know whether ${\mathcal{D}}_{1}^{c}\otimes{\mathcal{D}}_{2}^{c}=({\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})^{c}$ , we adapt the arguments found in the proof of [PittsStReInII, Proposition 6.11] and [PittsStReInI, Proposition 3.4] to show $\Delta=E$ .

We claim that for any $v_{i}\in{\mathcal{N}}({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ ( $i=1,2$ ),

(6.3.3)

\Delta(v_{1}\otimes v_{2})=E(v_{1}\otimes v_{2}).

Let

J:=\{d\in{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2}:(\Delta(v_{1}\otimes v_{2})-E(v_{1}\otimes v_{2}))\iota(d)=0\},

and let $\{K_{j}(v_{i})\}_{j=0}^{4}$ be a right Frolík family of ideals for $v_{i}$ ( $i=1,2$ ), see [PittsStReInI, Definition 2.13]. As noted there, for $j=1,2,3$ (and $i=1,2)$ ),

(6.3.4)

\theta_{v_{i}}(K_{j}(v_{i}))K_{j}(v_{i})=0.

Since $K_{i}:=\bigvee_{j=0}^{4}K_{j}(v_{i})$ is an essential ideal in ${\mathcal{D}}_{i}$ (see [PittsStReInI, Definition 2.13]), it follows that $K:=K_{1}\otimes K_{2}$ is an essential ideal of ${\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2}$ . Our goal is to show that $J$ is an essential ideal of ${\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2}$ , which we do by showing $K\subseteq J$ .

Let $s,t\in\{0,1,2,3,4\}$ . We shall show that

(6.3.5)

K_{s}(v_{1})\otimes K_{t}(v_{2})\subseteq J.

Starting with the case $s=t=0$ , let $h_{i}\in K_{0}(v_{i})$ . By [PittsStReInI, Lemma 2.15],

(h_{1}\otimes h_{2})(v_{1}\otimes v_{2})=(v_{1}\otimes v_{2})(h_{1}\otimes h_{2})\in{\mathcal{D}}_{1}^{c}\otimes{\mathcal{D}}_{2}^{c}.

Then

	$\displaystyle E(v_{1}\otimes v_{2})\iota(h_{1}\otimes h_{2})$	$\displaystyle=E((v_{1}\otimes v_{2})(h_{1}\otimes h_{2}))$
		$\displaystyle\stackrel{{\scriptstyle\eqref{upse0}}}{{=}}\Delta((v_{1}\otimes v_{2})(h_{1}\otimes h_{2}))=\Delta(v_{1}\otimes v_{2})\iota(h_{1}\otimes h_{2}),$

where the first and third equalities follow from [PaulsenCoBoMaOpAl, Corollary 3.19]. Thus $h_{1}\otimes h_{2}\in J$ , so (6.3.5) holds when $s=t=0$ .

The remaining cases are similar to those found in the proof of [PittsStReInI, Proposition 3.4]. For example, if $h_{1}\in K_{1}(v_{1})$ and $h_{2}\in K_{t}(v_{2})$ , where $0\leq t\leq 4$ , let $x\in K_{1}(v_{1})$ . Then using [PittsStReInI, Lemma 2.1],

	$\displaystyle E(v_{1}\otimes v_{2})\iota(h_{1}x\otimes h_{2}))$	$\displaystyle=E((v_{1}h_{1}x)\otimes(v_{2}h_{2}))=E(\theta_{v_{1}^{*}}(h_{1})v_{1}x\otimes v_{2}h_{2})$
		$\displaystyle=\iota(\theta_{v_{1}^{*}}(h_{1})\otimes I)E(v_{1}\otimes v_{2})\iota(x\otimes h_{2})$
		$\displaystyle=\iota((\theta_{v_{1}^{*}}(h_{1})\otimes I))(x\otimes h_{2}))E(v_{1}\otimes v_{2})$
		$\displaystyle=0,$

because $\theta_{v_{1}^{*}}(h_{1})\,x=0$ by (6.3.4). Taking $x$ from an approximate unit for $K_{1}(v_{1})$ , we obtain

E(v_{1}\otimes v_{2})\iota(h_{1}\otimes h_{2})=0

Similarly, $\Delta(v_{1}\otimes v_{2})\iota(h_{1}\otimes h_{2})=0$ , so that $h_{1}\otimes h_{2}\in J$ . Thus $K_{1}(v_{1})\otimes K_{t}(v_{2})\subseteq J$ . As the other combinations of $s$ and $t$ are obtained in the same way, we obtain (6.3.5).

We now have $K\subseteq J$ , so $J$ is an essential ideal of ${\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2}$ . An application of [PittsStReInI, Lemma 3.3] yields $E=\Delta$ . ∎

Our next goal is to show $E$ is faithful. This requires some preparation, and we first deal with some generalities involving unital regular inclusions having the unique pseudo-expectation property.

Let $({\mathcal{C}},{\mathcal{D}})$ be a regular inclusion having the unique (but not necessarily faithful) pseudo-expectation property and let $\Phi$ be the pseudo-expectation. We will use notation and terminology from [PittsStReInII] as reprised in Section 5.1(a)–(i).

Lemma \the\numberby.

Assume $({\mathcal{C}},{\mathcal{D}})$ is a unital regular inclusion having the unique pseudo-expectation $\Phi$ . The following statements hold.

(a)

${\mathcal{L}}({\mathcal{C}},{\mathcal{D}})=\{x\in{\mathcal{C}}:\phi(x)=0\text{ for all }\phi\in\Sigma({\mathcal{C}},{\mathcal{D}})\}$ .
(b)

Suppose in addition that $\Phi$ is faithful. Then $\operatorname{span}\Sigma({\mathcal{C}},{\mathcal{D}})$ is weak- $*$ dense in the dual space, ${\mathcal{C}}^{\#}$ , of ${\mathcal{C}}$ .

Proof.

(a) Suppose $x\in{\mathcal{C}}$ and $\Phi(x^{*}x)=0$ . Given $\phi\in\Sigma({\mathcal{C}},{\mathcal{D}})$ we may find $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ and $\rho\in{\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})$ so that for every $y\in{\mathcal{C}}$ ,

\phi(y)=[v,\rho](y)=\frac{\rho(v^{*}y)}{\rho(v^{*}v)^{1/2}}.

As $\rho=\sigma\circ\Phi$ for some $\sigma\in\widehat{I({\mathcal{D}})}$ , the Cauchy-Schwartz inequality shows $|\phi(x)|^{2}\leq\rho(x^{*}x)=\sigma(\Phi(x^{*}x))=0$ .

On the other hand, suppose $x\in{\mathcal{C}}$ satisfies $\phi(x)=0$ for every $\phi\in\Sigma({\mathcal{C}},{\mathcal{D}})$ . Fix $\rho\in{\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})$ and let $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ . If $\rho(v^{*}v)\neq 0$ , then $[v,\rho](x)=0$ , so $\rho(v^{*}x)=0$ . On the other hand, if $\rho(v^{*}v)=0$ , the Cauchy-Schwartz inequality gives $\rho(v^{*}x)=0$ . Thus, for every $v\in{\mathcal{N}}(C,{\mathcal{D}})$ ,

\rho(v^{*}x)=0.

By regularity of $({\mathcal{C}},{\mathcal{D}})$ , we find $\rho(x^{*}x)=0$ .

Allowing $\rho$ to vary throughout ${\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})$ , we conclude that $\sigma(\Phi(x^{*}x))=0$ for every $\sigma\in\widehat{I({\mathcal{D}})}$ , that is, $\Phi(x^{*}x)=0$ .

(b) Let ${\mathcal{S}}:=\operatorname{span}\Sigma({\mathcal{C}},{\mathcal{D}})$ . Suppose $u:{\mathcal{C}}^{\#}\rightarrow{\mathbb{C}}$ is a weak $*$ -continuous linear functional which annihilates ${\mathcal{S}}$ . As the weak $*$ -continuous linear functionals on ${\mathcal{C}}^{\#}$ may be identified with ${\mathcal{C}}$ , there exists $x\in{\mathcal{C}}$ such that $u(f)=f(x)$ for every $f\in{\mathcal{C}}^{\#}$ . Since $\phi(x)=u(\phi)=0$ for every $\phi\in\Sigma({\mathcal{C}},{\mathcal{D}})$ , part (a) shows $x\in{\mathcal{L}}({\mathcal{C}},{\mathcal{D}})$ . Since $\Phi$ is faithful by assumption, $x=0$ . This gives $u=0$ , whence ${\mathcal{S}}$ is weak- $*$ dense in ${\mathcal{C}}^{\#}$ . ∎

Now that these preliminaries have been completed, we return to the context of Proposition 6.3 and the task of showing $E$ is faithful. Since the map $g$ appearing in (6.3.1) is a $*$ -monomorphism, we may use $g$ to identify $I({\mathcal{D}}_{1})\otimes I({\mathcal{D}}_{2})$ with a subalgebra of $I({\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ ; upon doing so, we have

E=E_{1}\otimes E_{2}.

This enables us to describe $\Sigma({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ . The notation in the next lemma is discussed in Section 5.

Lemma \the\numberby.

For unital pseudo-Cartan inclusions $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ and $({\mathcal{C}}_{2},{\mathcal{D}}_{2})$ , the following statements hold.

(a)

${\mathfrak{S}}_{s}({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})=\{\rho_{1}\otimes\rho_{2}:\rho_{i}\in{\mathfrak{S}}_{s}({\mathcal{C}}_{i},{\mathcal{D}}_{i})\}$ ; and
(b)

$\Sigma({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})=\{\phi_{1}\otimes\phi_{2}:\phi_{i}\in\Sigma({\mathcal{C}}_{i},{\mathcal{D}}_{i})\}$ .

Proof.

(a) For $i=1,2$ , let $\rho_{i}\in{\mathfrak{S}}_{s}({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ and find $\sigma_{i}\in\widehat{I({\mathcal{D}}_{i})}$ so that $\rho_{i}=\sigma_{i}\circ E_{i}$ . As $E=E_{1}\otimes E_{2}$ and $\sigma_{1}\otimes\sigma_{2}$ extends from a multiplicative linear functional on $I({\mathcal{D}}_{1})\otimes I({\mathcal{D}}_{2})$ to a multiplicative linear functional $\sigma$ on $I({\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ . we see that

\rho_{1}\otimes\rho_{2}=(\sigma_{1}\circ E_{1})\otimes(\sigma_{2}\circ E_{2})=(\sigma_{1}\otimes\sigma_{2})\circ(E_{1}\otimes E_{2})=\sigma\circ E\in{\mathfrak{S}}_{s}({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2}).

On the other hand, if $\rho\in{\mathfrak{S}}_{s}({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ , there exists a multiplicative linear functional $\sigma$ on $I({\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ so that $\rho=\sigma\circ E=\sigma\circ(E_{1}\otimes E_{2})$ . Since the maximal ideal space of $I({\mathcal{D}}_{1})\otimes I({\mathcal{D}}_{2})$ is homeomorphic to $\widehat{I({\mathcal{D}}_{1})}\times\widehat{I({\mathcal{D}}_{2})}$ , the restriction of $\sigma$ to $I({\mathcal{D}}_{1})\otimes I({\mathcal{D}}_{2})$ has the form $\sigma_{1}\otimes\sigma_{2}$ where $\sigma_{i}\in\widehat{I({\mathcal{D}}_{i})}$ . Therefore $\rho=(\sigma_{1}\circ E_{1})\otimes(\sigma_{2}\circ E_{2})$ , as desired.

(b) Let $\phi_{i}\in\Sigma({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ . Then there are $v_{i}\in{\mathcal{N}}({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ and $\rho_{i}\in{\mathfrak{S}}_{s}({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ so that $\phi_{i}=[v_{i},\rho_{i}]$ . Since $v_{1}\otimes v_{2}\in{\mathcal{N}}({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ , we get $\phi_{1}\otimes\phi_{2}=[v_{1}\otimes v_{2},\rho_{1}\otimes\rho_{2}]\in\Sigma({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ .

For the reverse inclusion, suppose $\phi\in\Sigma({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ , and let $\rho={\mathfrak{s}}(\phi)$ . By part (a), $\rho=\rho_{1}\otimes\rho_{2}$ , where $\rho_{i}\in{\mathfrak{S}}_{s}({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ . Since $\operatorname{span}\{v_{1}\otimes v_{2}:v_{i}\in{\mathcal{N}}({\mathcal{C}}_{i},{\mathcal{D}}_{i})\}$ is dense in ${\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2}$ , we may choose $v_{i}\in{\mathcal{N}}({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ such that $\phi(v_{1}\otimes v_{2})>0$ . Then

\phi=[v_{1}\otimes v_{2},\rho]=[v_{1}\otimes v_{2},\rho_{1}\otimes\rho_{2}]=[v_{1},\rho_{1}]\otimes[v_{2},\rho_{2}],

with the first equality following from [PittsStReInII, Theorem 7.9(f)]. ∎

Proposition \the\numberby.

For $i=1,2$ , let $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ be unital pseudo-Cartan inclusions. The following statements hold.

(a)

$({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ is a pseudo-Cartan inclusion.
(b)

If $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ are Cartan inclusions, so is $({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ .

Before giving the proof, we remark that part (b) is probably known, but we do not have a reference.

Proof.

(a) We already know $({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ is a regular inclusion having the unique pseudo-expectation property. It remains to show the pseudo-expectation $E$ (see (6.3.1)) is faithful.

For typographical reasons, let us write ${\mathcal{L}}:=\{z\in{\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2}:E(z^{*}z)=0\}$ for the left kernel of $E$ . Recall that ${\mathcal{L}}$ is an ideal in ${\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2}$ by [PittsStReInII, Theorem 6.5].

We claim

(6.3.6)

{\mathcal{L}}\cap({\mathcal{C}}_{1}\odot{\mathcal{C}}_{2})=\{0\}.

Suppose $z\in{\mathcal{L}}\cap({\mathcal{C}}_{1}\odot{\mathcal{C}}_{2})$ . Write

z=\sum_{j=1}^{n}x_{j}\otimes y_{j},

where $x_{j}\in{\mathcal{C}}_{1}$ , $y_{j}\in{\mathcal{C}}_{2}$ and $\{y_{1},\dots,y_{n}\}$ is a linearly independent set. For $\phi_{1}\otimes\phi_{2}\in\Sigma({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ ,

0=(\phi_{1}\otimes\phi_{2})(z)=\sum_{j=1}^{n}\phi_{1}(x_{j})\phi_{2}(y_{j}),

with Lemma 6.3(a) giving the first equality. Holding $\phi_{1}$ fixed, this equality persists if $\phi_{2}$ is replaced by $\psi\in\operatorname{span}\Sigma({\mathcal{C}}_{2},{\mathcal{D}}_{2})$ , and hence also for any $\psi\in{\mathcal{C}}_{2}^{\#}$ by Lemma 6.3(b) applied to $({\mathcal{C}}_{2},{\mathcal{D}}_{2})$ and $E_{2}$ . Therefore, setting

s:=\sum_{j=1}^{n}\phi_{1}(x_{j})y_{j},

we find $\psi(s)=0$ for every $\psi\in{\mathcal{C}}_{2}^{\#}$ , whence $s=0$ . By the linear independence of $\{y_{1},\dots,y_{n}\}$ , we conclude that $\phi_{1}(x_{j})=0$ for each $j$ . Varying $\phi_{1}$ and applying Lemma 6.3(b) to $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ and $E_{1}$ , we see that $x_{j}=0$ for every $j$ , so $z=0$ . Thus (6.3.6) holds.

For $z\in{\mathcal{C}}_{1}\odot{\mathcal{C}}_{2}$ , define

\eta(z):=\left\lVert z+{\mathcal{L}}\right\rVert_{({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2})/{\mathcal{L}}}.

By the claim, $\eta$ is a $C^{*}$ -norm on ${\mathcal{C}}_{1}\odot{\mathcal{C}}_{2}$ . But $\left\lVert\cdot\right\rVert_{{\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2}}$ is the smallest $C^{*}$ -norm on ${\mathcal{C}}_{1}\odot{\mathcal{C}}_{2}$ , so for any $z\in{\mathcal{C}}_{1}\odot C_{2}$ ,

\eta(z)\leq\left\lVert z\right\rVert_{{\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2}}\leq\eta(z).

Thus for $x\in{\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2}$ ,

\left\lVert x+{\mathcal{L}}\right\rVert_{({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2})/{\mathcal{L}}}=\left\lVert x\right\rVert_{{\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2}}.

It follows that ${\mathcal{L}}=\{0\}$ , completing the proof of (a).

(b) When $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ are Cartan inclusions, $\iota_{i}^{-1}\circ E_{i}$ and $(\iota_{1}\otimes\iota_{2})^{-1}\circ(E_{1}\otimes E_{2})$ are faithful conditional expectations. Thus part (b) follows from part (a) and the characterization of Cartan inclusions given following Observation 4.1. ∎

Proposition \the\numberby.

Suppose $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ and $({\mathcal{C}}_{2},{\mathcal{D}}_{2})$ are unital pseudo-Cartan inclusions with Cartan envelopes $({\mathcal{A}}_{1},{\mathcal{B}}_{1}\hbox{\,:\hskip-1.0pt:\,}\alpha_{1})$ and $({\mathcal{A}}_{2},{\mathcal{B}}_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha_{2})$ respectively. Then $({\mathcal{A}}_{1}\otimes_{\min}{\mathcal{A}}_{2},{\mathcal{B}}_{1}\otimes{\mathcal{B}}_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha_{1}\otimes\alpha_{2})$ is a Cartan envelope for $({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ .

Proof.

Proposition 6.3(b) shows that $({\mathcal{A}}_{1}\otimes_{\min}{\mathcal{A}}_{2},{\mathcal{B}}_{1}\otimes{\mathcal{B}}_{2})$ is a Cartan inclusion. Let $\Delta_{i}:{\mathcal{A}}_{i}\rightarrow{\mathcal{B}}_{i}$ be the conditional expectations and put

\Delta:=\Delta_{1}\otimes\Delta_{2}\quad\text{and}\quad\alpha:=\alpha_{1}\otimes\alpha_{2}.

Then $\Delta:{\mathcal{A}}_{1}\otimes_{\min}{\mathcal{A}}_{2}\rightarrow{\mathcal{B}}_{1}\otimes{\mathcal{B}}_{2}$ is the (faithful) conditional expectation. Also, since $({\mathcal{A}}_{1},{\mathcal{B}}_{1},\alpha)$ is a Cartan envelope for $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ ,

{\mathcal{B}}_{1}\otimes I=C^{*}(\Delta(\alpha({\mathcal{C}}_{1}\otimes I)))\quad\text{and}\quad{\mathcal{A}}_{1}\otimes I=C^{*}((\alpha({\mathcal{C}}_{1}\otimes I)\cup({\mathcal{B}}_{1}\otimes I)).

Likewise, $I\otimes{\mathcal{A}}_{2}$ is generated by $\alpha(I\otimes{\mathcal{C}}_{2})\cup\Delta(\alpha(I\otimes{\mathcal{C}}_{2}))$ . It follows that $({\mathcal{A}}_{1}\otimes_{\min}{\mathcal{A}}_{2},{\mathcal{B}}_{1}\otimes{\mathcal{B}}_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha_{1}\otimes\alpha_{2})$ is a Cartan package for $({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ . Proposition 6.3 shows that $({\mathcal{B}}_{1}\otimes{\mathcal{B}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2},(\alpha_{1}\otimes\alpha_{2})|_{{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2}})$ is an essential inclusion. Therefore $({\mathcal{A}}_{1}\otimes_{\min}{\mathcal{A}}_{2},{\mathcal{B}}_{1}\otimes{\mathcal{B}}_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha_{1}\otimes\alpha_{2})$ is a Cartan envelope for $({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ . ∎

Having completed the unital case, we are ready for the main theorem of this section.

Theorem \the\numberby.

Suppose for $i=1,2$ that $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ are pseudo-Cartan inclusions (not assumed unital) and let $({\mathcal{A}}_{1},{\mathcal{B}}_{i}\hbox{\,:\hskip-1.0pt:\,}\alpha_{i})$ be Cartan envelopes for $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ . Then $({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{D}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ is a pseudo-Cartan inclusion, and $({\mathcal{A}}_{1}\otimes_{\min}{\mathcal{A}}_{2},{\mathcal{B}}_{1}\otimes{\mathcal{B}}_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha_{1}\otimes\alpha_{2})$ is a Cartan envelope for $({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{D}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ .

Proof.

We begin by assuming that $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ and $({\mathcal{C}}_{2},{\mathcal{D}}_{2})$ have the AUP. Observation 4.2 shows $(\tilde{\mathcal{C}}_{i},\tilde{\mathcal{D}}_{i})$ is a unital pseudo-Cartan inclusion, and Proposition 6.3 shows that $(\tilde{\mathcal{C}}_{1}\otimes_{\min}\tilde{\mathcal{C}}_{2},\tilde{\mathcal{D}}_{1}\otimes\tilde{\mathcal{D}}_{2})$ is a unital pseudo-Cartan inclusion.

Since $(\tilde{\mathcal{D}}_{1},{\mathcal{D}}_{1})$ and $(\tilde{\mathcal{D}}_{2},{\mathcal{D}}_{2})$ have the ideal intersection property, Proposition 6.3 shows $(\tilde{\mathcal{D}}_{1}\otimes\tilde{\mathcal{D}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ has the ideal intersection property. Let

{\mathcal{M}}:=\{v_{1}\otimes v_{2}:v_{i}\in{\mathcal{N}}({\mathcal{C}}_{i},{\mathcal{D}}_{i})\}\subseteq{\mathcal{N}}(\tilde{\mathcal{C}}_{1}\otimes_{\min}\tilde{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2}).

Then ${\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2}\subseteq\overline{\operatorname{span}}({\mathcal{M}})={\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2}$ . Applying Proposition 4.4(a) to $(\tilde{\mathcal{C}}_{1}\otimes_{\min}\tilde{\mathcal{C}}_{2},\tilde{\mathcal{D}}_{1}\otimes\tilde{\mathcal{D}}_{2})$ with this choice for ${\mathcal{M}}$ , we conclude $({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ is a pseudo-Cartan inclusion.

Next, Proposition 6.3 and Theorem 4.2(a) show $(\tilde{\mathcal{A}}_{1}\otimes_{\min}\tilde{\mathcal{A}}_{2},\tilde{\mathcal{B}}_{1}\otimes\tilde{\mathcal{B}}_{2}\hbox{\,:\hskip-1.0pt:\,}\tilde{\alpha}_{1}\otimes\tilde{\alpha}_{2})$ is a Cartan envelope for $(\tilde{\mathcal{C}}_{1}\otimes_{\min}\tilde{\mathcal{C}}_{2},\tilde{\mathcal{D}}_{1}\otimes\tilde{\mathcal{D}}_{2})$ . Let $\Delta_{i}:{\mathcal{A}}_{i}\rightarrow{\mathcal{B}}_{i}$ be the conditional expectations (for the Cartan inclusions $({\mathcal{A}}_{i},{\mathcal{B}}_{i})$ ). Write $\Delta:=\Delta_{1}\otimes\Delta_{2}$ and $\alpha:=\alpha_{1}\otimes\alpha_{2}$ . Notice that

{\mathcal{B}}_{1}\otimes{\mathcal{B}}_{2}=C^{*}(\tilde{\Delta}(\tilde{\alpha}({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2})))\quad\text{and}\quad{\mathcal{A}}_{1}\otimes_{\min}{\mathcal{A}}_{2}=C^{*}(({\mathcal{B}}_{1}\otimes{\mathcal{B}}_{2})\cup\tilde{\alpha}({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2}))

That $({\mathcal{A}}_{1}\otimes_{\min}{\mathcal{A}}_{2},{\mathcal{B}}_{1}\otimes{\mathcal{B}}_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha_{1}\otimes\alpha_{2})$ is a Cartan envelope for $({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ now follows from Proposition 4.4(b).

Now suppose $({\mathcal{C}}_{1},{\mathcal{D}}_{1})$ and $({\mathcal{C}}_{2},{\mathcal{D}}_{2})$ are pseudo-Cartan inclusions, where neither is assumed to have the AUP. Then $({\mathcal{C}}_{i},{\mathcal{D}}_{i}^{c})$ are pseudo-Cartan inclusions with the AUP, whence $({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}^{c}\otimes{\mathcal{D}}_{2}^{c})$ is a pseudo-Cartan inclusion. Proposition 6.3 shows $({\mathcal{D}}_{1}^{c}\otimes{\mathcal{D}}_{2}^{c},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ has the ideal intersection property, so an application of Proposition 4.4(a) (with ${\mathcal{M}}=\{v_{1}\otimes v_{2}:v_{i}\in{\mathcal{N}}({\mathcal{C}}_{i},{\mathcal{D}}_{i})\}$ ) shows $({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ is a pseudo-Cartan inclusion. Let $({\mathcal{A}}_{i},{\mathcal{B}}_{i}\hbox{\,:\hskip-1.0pt:\,}\alpha_{i})$ be a Cartan envelope for $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ . By Proposition 4.2, $({\mathcal{A}}_{i},{\mathcal{B}}_{i}\hbox{\,:\hskip-1.0pt:\,}\alpha_{i})$ is a Cartan envelope for $({\mathcal{C}}_{i},{\mathcal{D}}_{i}^{c})$ , whence $({\mathcal{A}}_{1}\otimes_{\min}{\mathcal{A}}_{2},{\mathcal{B}}_{1}\otimes{\mathcal{B}}_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha_{1}\otimes\alpha_{2})$ is a Cartan envelope for $({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}^{c}\otimes{\mathcal{D}}_{2}^{c})$ . Finally, Proposition 4.4(b) shows $({\mathcal{A}}_{1}\otimes_{\min}{\mathcal{A}}_{2},{\mathcal{B}}_{1}\otimes{\mathcal{B}}_{2}\hbox{\,:\hskip-1.0pt:\,}\alpha_{1}\otimes\alpha_{2})$ is also a Cartan envelope for $({\mathcal{C}}_{1}\otimes_{\min}{\mathcal{C}}_{2},{\mathcal{D}}_{1}\otimes{\mathcal{D}}_{2})$ . ∎

7. Applications

In this section we give a few applications of the results presented so far. We begin with an application of Theorem 3 which describes the $C^{*}$ -envelope for an intermediate Banach algebras. Several of the results in this section apply to unital inclusions, see Definition 2.1(d).

Theorem \the\numberby.

Suppose $({\mathcal{C}},{\mathcal{D}})$ is a unital pseudo-Cartan inclusion and ${\mathcal{A}}\subseteq{\mathcal{C}}$ is a closed subalgebra satisfying ${\mathcal{D}}\subseteq{\mathcal{A}}\subseteq{\mathcal{C}}$ (we do not assume ${\mathcal{A}}={\mathcal{A}}^{*})$ . Let $C^{*}({\mathcal{A}})$ be the $C^{*}$ -subalgebra of ${\mathcal{C}}$ generated by ${\mathcal{A}}$ and let $C^{*}_{e}({\mathcal{A}})$ be the $C^{*}$ -envelope for ${\mathcal{A}}$ . Then

C^{*}({\mathcal{A}})=C^{*}_{e}({\mathcal{A}}).

Proof.

Since $({\mathcal{C}},{\mathcal{D}})$ has the faithful unique pseudo-expectation property, the result follows from [PittsStReInI, Theorem 8.3]. ∎

For unital $C^{*}$ -algebras ${\mathcal{A}}$ and ${\mathcal{B}}$ with ${\mathcal{B}}\subseteq{\mathcal{A}}$ , when the assumption that ${\mathcal{B}}$ is abelian and regular is dropped, [PittsZarikianUnPsExC*In, Example 6.9] shows that the faithful unique pseudo-expectation property for ${\mathcal{B}}\subseteq{\mathcal{A}}$ need not imply that ${\mathcal{B}}$ norms ${\mathcal{A}}$ in the sense of [PopSinclairSmithNoC*Al]. Nevertheless, [PittsZarikianUnPsExC*In, Section 6.2] argues that the faithful unique pseudo-expectation property is “conducive” to norming. Further evidence for this statement is Theorem 7 below: for unital regular inclusions satisfying Standing Assumption 3, the faithful unique pseudo-expectation property implies norming.

The proof of Theorem 7 requires two preparatory results which we now present.

Lemma \the\numberby.

Let $({\mathcal{C}},{\mathcal{D}})$ be a unital pseudo-Cartan inclusion. Suppose $\sigma\in\hat{\mathcal{D}}$ is free and let $f$ be the unique state extension of $\sigma$ to ${\mathcal{C}}$ . If $(\pi_{f},{\mathcal{H}}_{f},\xi_{f})$ is the GNS triple associated to $f$ , then $\pi_{f}({\mathcal{D}})^{\prime\prime}$ is an atomic MASA in ${\mathcal{B}}({\mathcal{H}}_{f})$ .

Furthermore, if

{\mathcal{O}}:=\{\tau\in\hat{\mathcal{D}}:\exists\,v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})\text{ such that }\tau=\beta_{v}(\sigma)\}

is the orbit of $\sigma$ under ${\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ , then

(7.1)

\ker(\pi_{f}|_{\mathcal{D}})=\bigcap_{\tau\in{\mathcal{O}}}\ker\tau.

Proof.

Our first task is to show that $f$ is a compatible state on ${\mathcal{C}}$ ([PittsStReInI, Definition 4.1]). Let $(I({\mathcal{D}}),\iota)$ be an injective envelope for ${\mathcal{D}}$ and let $E:{\mathcal{C}}\rightarrow I({\mathcal{D}})$ be the pseudo-expectation for $({\mathcal{C}},{\mathcal{D}})$ . Choose $\sigma^{\prime}\in\widehat{I({\mathcal{D}})}$ such that $\sigma^{\prime}\circ\iota=\sigma$ . Note that $\sigma^{\prime}\circ E|_{\mathcal{D}}=\sigma^{\prime}\circ\iota=\sigma$ . Since $\sigma$ is free, we conclude that $f=\sigma^{\prime}\circ E$ . By [PittsStReInII, Theorem 6.9] (see Theorem A below) $f$ is a compatible state.

The Cauchy-Schwartz inequality implies that for any $d\in{\mathcal{D}}$ and $x\in{\mathcal{C}}$ , $f(dx)=\sigma(d)f(x)=f(xd)$ ([PittsNoApUnInC*Al, Fact 1.2] has the details), so $f\in\text{Mod}({\mathcal{C}},{\mathcal{D}})$ (see [PittsStReInI, Definition 2.4]).

We claim that if $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ satisfies $\sigma(v^{*}v)=1$ and $\sigma(v^{*}dv)=\sigma(d)$ for every $d\in{\mathcal{D}}$ , then

f(v^{*}xv)=f(x)\quad\text{for all}\quad x\in{\mathcal{C}}.

Define a positive linear functional $f_{v}$ on ${\mathcal{C}}$ by $f_{v}(x)=f(v^{*}xv)$ . Note that $1\leq\left\lVert f_{v}\right\rVert$ because $f_{v}|_{\mathcal{D}}=\sigma$ . Also for $x\in{\mathcal{C}}$ and $d\in{\mathcal{D}}$ with $\sigma(d)=1$ ,

|f(v^{*}xv)|=|f(d^{*}v^{*}xvd)|\leq\left\lVert vd\right\rVert^{2}\left\lVert x\right\rVert\left\lVert f\right\rVert=\left\lVert d^{*}v^{*}vd\right\rVert\left\lVert x\right\rVert.

As $\inf\{\left\lVert d^{*}v^{*}vd\right\rVert:d\in{\mathcal{D}}\text{ and }\sigma(d)=1\}=\sigma(v^{*}v)=1$ , we conclude $\left\lVert f_{v}\right\rVert=1$ . Thus, $f_{v}$ is also an extension of $\sigma$ to a state on ${\mathcal{C}}$ . We conclude that $f_{v}=f$ , so the claim holds.

It follows that $f$ is a pure and ${\mathcal{D}}$ -rigid state (see [PittsStReInI, Definition 4.11]). By first applying [PittsStReInI, Proposition 4.12] and then [PittsStReInI, Proposition 4.8(iv)], we find $\pi_{f}({\mathcal{D}})^{\prime\prime}$ is an atomic MASA in ${\mathcal{B}}({\mathcal{H}}_{f})$ .

Let $N_{f}=\{x\in{\mathcal{C}}:f(x^{*}x)=0\}$ be the left kernel of $f$ . Suppose $d\in{\mathcal{D}}$ and $\pi_{f}(d)\neq 0$ . Since $({\mathcal{C}},{\mathcal{D}})$ is a regular inclusion, there exists $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ such that $\pi_{f}(d)(v+N_{f})\neq 0$ . Then $f(v^{*}v)\neq 0$ because $0\neq f(v^{*}d^{*}dv)\leq\left\lVert d\right\rVert^{2}f(v^{*}v)$ . As $f|_{\mathcal{D}}=\sigma$ , [PittsStReInI, Proposition 4.4(v)], shows $\beta_{v}(\sigma)(d)\neq 0$ . As $\beta_{v}(\sigma)\in{\mathcal{O}}$ , we find

\bigcap_{\tau\in{\mathcal{O}}}\ker\tau\subseteq\ker(\pi_{f}|_{\mathcal{D}}).

On the other hand, if $d\in\ker(\pi_{f}|_{\mathcal{D}})$ , then for every $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ with $\sigma(v^{*}v)\neq 0$ , we have $0=\pi_{f}(d)(v+N)$ , so applying [PittsStReInI, Proposition 4.4(v)] again, $\beta_{v}(\sigma)(d)=0$ . Therefore, $d\in\bigcap_{\tau\in{\mathcal{O}}}\ker\tau$ . Thus (7.1) holds and the proof is complete. ∎

Our second preparatory result, of independent interest, shows that a natural class of pseudo-Cartan inclusions is topologically free.

Proposition \the\numberby.

Suppose $({\mathcal{C}},{\mathcal{D}})$ is a unital pseudo-Cartan inclusion such that there exists a countable subset ${\mathfrak{C}}\subseteq{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ satisfying ${\mathcal{C}}=C^{*}({\mathfrak{C}}\cup{\mathcal{D}})$ . Then $({\mathcal{C}},{\mathcal{D}})$ is a topologically free inclusion.

Proof.

Let ${\mathfrak{U}}:=\{\sigma\in\hat{\mathcal{D}}:\sigma\text{ is free for $({\mathcal{C}},{\mathcal{D}})$}\}$ . Our task is to show ${\mathfrak{U}}$ is dense in $\hat{\mathcal{D}}$ . For convenience, set

{\mathcal{B}}:={\mathcal{D}}^{c}

and let $\pi:\hat{\mathcal{B}}\rightarrow\hat{\mathcal{D}}$ be the restriction map,

\pi(\sigma_{1})=\sigma_{1}|_{\mathcal{D}}\qquad(\sigma_{1}\in\hat{\mathcal{B}}).

By [ExelPittsChGrC*AlNoHaEtGr, Theorem 2.11.16], the set of free points for the inclusion $({\mathcal{B}},{\mathcal{D}})$ contains a dense $G_{\delta}$ set. We may therefore choose dense open sets $U_{n}\subseteq\hat{\mathcal{D}}$ such that

\bigcap_{n}U_{n}

is contained in the set of free points for $({\mathcal{B}},{\mathcal{D}})$ . As ${\mathcal{D}}\subseteq{\mathcal{B}}$ has the ideal intersection property, $\pi$ is an essential surjection, so for each $n$ , $\pi^{-1}(U_{n})$ is a dense open subset of $\hat{\mathcal{B}}$ .

Let ${\mathcal{S}}$ be the $*$ -semigroup generated by ${\mathfrak{C}}\cup\{I\}$ , that is, ${\mathcal{S}}$ is the collection of all finite products of elements of ${\mathfrak{C}}\cup{\mathfrak{C}}^{*}\cup\{I\}$ . Then ${\mathcal{S}}$ is countable, $\{wd:w\in{\mathcal{S}}\text{ and }d\in{\mathcal{D}}\}\subseteq{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ , and the linear span,

{\mathfrak{S}}:=\operatorname{span}\{wd:w\in{\mathcal{S}}\text{ and }d\in{\mathcal{D}}\},

is dense in ${\mathcal{C}}$ .

Let $v\in{\mathcal{S}}$ . By Lemma 3(a), $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{B}})$ . Moreover, as $({\mathcal{C}},{\mathcal{B}})$ is a regular MASA inclusion, the proof of [PittsStReInI, Theorem 3.10] shows there is a dense open set $X_{v}\subseteq\hat{\mathcal{B}}$ such that each $\sigma_{1}\in X_{v}$ is free relative to $v$ . Let

{\mathcal{Y}}:=\left(\bigcap_{v\in{\mathcal{S}}}X_{v}\right)\cap\left(\bigcap_{n}\pi^{-1}(U_{n})\right).

As ${\mathcal{B}}$ is unital, $\hat{\mathcal{B}}$ is compact, so Baire’s theorem implies ${\mathcal{Y}}\subseteq\hat{\mathcal{B}}$ is a dense $G_{\delta}$ set.

We claim that if $y\in{\mathcal{Y}}$ , then $\pi(y)$ is a free point for $({\mathcal{C}},{\mathcal{D}})$ . To see this, suppose $\rho_{1}$ and $\rho_{2}$ are states on ${\mathcal{C}}$ such that $\rho_{1}|_{\mathcal{D}}=\rho_{2}|_{\mathcal{D}}=\pi(y)$ . Since $y\in\bigcap_{n}\pi^{-1}(U_{n})$ , we find $\pi(y)\in\bigcap_{n}U_{n}$ . Thus $\pi(y)$ is a free point for the inclusion $({\mathcal{B}},{\mathcal{D}})$ . Next, since $y\in\bigcap_{v\in{\mathcal{S}}}X_{v}$ , $\rho_{1}(v)=\rho_{2}(v)$ for every $v\in{\mathcal{S}}$ . Using [PittsNoApUnInC*Al, Fact 1.2], for every $v\in{\mathcal{S}}$ and $d\in{\mathcal{D}}$ , $\rho_{1}(vd)=\rho_{2}(vd)$ . Therefore, for every $x\in{\mathfrak{S}}$ , $\rho_{1}(x)=\rho_{2}(x)$ . As ${\mathfrak{S}}$ is dense in ${\mathcal{C}}$ , we conclude $\rho_{1}=\rho_{2}$ . Thus $\pi(y)$ is a free point for $({\mathcal{C}},{\mathcal{D}})$ , so the claim holds.

Finally we show that $\pi({\mathcal{Y}})$ is dense in $\hat{\mathcal{D}}$ . Let $G\subseteq\hat{\mathcal{D}}$ be an open set. Then $\pi^{-1}(G)$ is open in $\hat{\mathcal{B}}$ , so there exists $y\in{\mathcal{Y}}\cap\pi^{-1}(G)$ , whence $\pi(y)\in G$ , as desired. As $\pi({\mathcal{Y}})\subseteq{\mathfrak{U}}$ , the proof is complete. ∎

We now come to our norming result.

Theorem \the\numberby.

Suppose $({\mathcal{C}},{\mathcal{D}})$ is a unital pseudo-Cartan inclusion. Then ${\mathcal{D}}$ norms ${\mathcal{C}}$ .

Proof.

The argument is mostly the same as the proof of [PittsStReInI, Theorem 8.2], except we use Proposition 7 instead of [PittsStReInI, Theorem 3.10] and Lemma 7 instead of [PittsStReInI, Proposition 4.12]. Therefore, we shall only outline the proof, leaving the reader to consult the proof of [PittsStReInI, Theorem 8.2] for additional details as desired. (The notation here differs somewhat from that used in [PittsStReInI, Theorem 8.2], but this will present no difficulty.)

Arguing as in the last paragraph of the proof of [PittsStReInI, Theorem 8.2], it suffices to show ${\mathcal{D}}$ norms ${\mathcal{C}}$ under the additional assumption that there is a countable set ${\mathfrak{C}}\subseteq{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ such that ${\mathcal{C}}=C^{*}({\mathfrak{C}}\cup{\mathcal{D}})$ . For the remainder of the proof, we assume this.

Let ${\mathcal{F}}\subseteq{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ be the $*$ -semigroup generated by ${\mathfrak{C}}\cup{\mathcal{D}}$ . By the additional assumption, $\operatorname{span}{\mathcal{F}}$ is a dense $*$ -subalgebra of ${\mathcal{C}}$ .

Let $Y\subseteq\hat{\mathcal{D}}$ be the set of free points for $({\mathcal{C}},{\mathcal{D}})$ . Proposition 7 shows $Y$ is dense in $\hat{\mathcal{D}}$ . For $y\in Y$ , denote by $f_{y}$ the unique state extension of $y$ to ${\mathcal{C}}$ , and let $(\pi_{y},{\mathcal{H}}_{y},\xi_{y})$ be the GNS triple for $f_{y}$ . Since $f_{y}$ is a pure state, $\pi_{y}$ is an irreducible representation, and Lemma 7 shows that $\pi_{y}({\mathcal{D}})^{\prime\prime}$ is a MASA in ${\mathcal{B}}({\mathcal{H}}_{y})$ .

For $y_{1},y_{2}\in Y$ , define $y_{1}\sim y_{2}$ if and only if there exists $v\in{\mathcal{F}}$ such that $\beta_{v}(y_{1})=y_{2}$ . As in the proof of [PittsStReInI, Theorem 8.2], $\sim$ is an equivalence relation, and if $\pi_{y_{1}}$ is unitarily equivalent to $\pi_{y_{2}}$ , then $y_{1}\sim y_{2}$ . Thus, if $y_{1}\not\sim y_{2}$ , $\pi_{y_{1}}$ and $\pi_{y_{2}}$ are disjoint representations.

Let ${\mathcal{Y}}\subseteq Y$ be a set containing exactly one element of each $\sim$ equivalence class. Consider the representation

\pi:=\bigoplus_{y\in{\mathcal{Y}}}\pi_{y}

of ${\mathcal{C}}$ on ${\mathcal{H}}_{\pi}:=\bigoplus_{y\in{\mathcal{Y}}}{\mathcal{H}}_{y}$ . Since each $\pi_{y}({\mathcal{D}})^{\prime\prime}$ is an atomic MASA and the representations $\{\pi_{y}:y\in{\mathcal{Y}}\}$ are pairwise disjoint, $\pi({\mathcal{D}})^{\prime\prime}$ is also an atomic MASA in ${\mathcal{B}}({\mathcal{H}}_{\pi})$ .

We now show $\pi$ is faithful. As $Y$ is dense in $\hat{\mathcal{D}}$ and is the union of the ${\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ -orbits of the elements of ${\mathcal{Y}}$ , (7.1) shows that the restriction of $\pi$ to ${\mathcal{D}}$ is faithful. By the ideal intersection property for $({\mathcal{D}}^{c},{\mathcal{D}})$ , $\ker\pi\cap{\mathcal{D}}^{c}=\{0\}$ . Then $\ker\pi=\{0\}$ because $({\mathcal{C}},{\mathcal{D}}^{c})$ also has the ideal intersection property. Therefore $\pi$ is a faithful representation of ${\mathcal{C}}$ .

A direct argument or the argument found on [CameronPittsZarikianBiCaMASAvNAlNoAlMeTh, Page 466] shows $\pi({\mathcal{D}})^{\prime\prime}$ is locally cyclic (as defined in [PopSinclairSmithNoC*Al, Page 173]) for ${\mathcal{B}}({\mathcal{H}}_{\pi})$ . Using [PopSinclairSmithNoC*Al, Lemma 2.3 and Theorem 2.7], $\pi({\mathcal{D}})$ norms ${\mathcal{B}}({\mathcal{H}}_{\pi})$ . As $\pi$ is faithful, it follows that ${\mathcal{D}}$ norms ${\mathcal{C}}$ . ∎

We now extend [PittsStReInII, Theorem 8.4] from the setting of virtual Cartan inclusions considered there to pseudo-Cartan inclusions. This is a significant generalization of [PittsStReInII, Theorem 8.4] because it weakens the hypothesis that ${\mathcal{C}}$ is unital and relaxes the condition that ${\mathcal{D}}$ be a MASA. We remark that [BrownExelFullerPittsReznikoffInC*AlCaEm, Example 5.1] gives an example of a Cartan inclusion $({\mathcal{C}},{\mathcal{D}})$ and an intermediate $C^{*}$ -subalgebra ${\mathcal{D}}\subseteq{\mathcal{B}}\subseteq{\mathcal{C}}$ such that $({\mathcal{B}},{\mathcal{D}})$ is not a regular inclusion. Thus, in the context of Theorem 7, it is possible that the inclusions $(C^{*}({\mathcal{A}}_{i}),{\mathcal{D}}_{i})$ are not regular.

Theorem \the\numberby.

Suppose for $i=1,2$ , $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ are pseudo-Cartan inclusions such that $(\tilde{\mathcal{C}}_{i},\tilde{\mathcal{D}}_{i})$ are pseudo-Cartan inclusions, and ${\mathcal{A}}_{i}$ are Banach algebras satisfying ${\mathcal{D}}_{i}\subseteq{\mathcal{A}}_{i}\subseteq{\mathcal{C}}_{i}$ . Let $C^{*}({\mathcal{A}}_{i})$ be the $C^{*}$ -subalgebra of ${\mathcal{C}}_{i}$ generated by ${\mathcal{A}}_{i}$ . If $u:{\mathcal{A}}_{1}\rightarrow{\mathcal{A}}_{2}$ is an isometric isomorphism, then $u$ uniquely extends to a $*$ -isomorphism of $C^{*}({\mathcal{A}}_{1})$ onto $C^{*}({\mathcal{A}}_{2})$ .

Remark \the\numberby. We have included the hypothesis that $(\tilde{\mathcal{C}}_{i},\tilde{\mathcal{D}}_{i})$ are pseudo-Cartan inclusions because we do not know whether regularity is preserved when units are adjoined. When $(\tilde{\mathcal{C}}_{i},\tilde{\mathcal{D}}_{i})$ are regular, Observation 4.2 shows $(\tilde{\mathcal{C}}_{i},\tilde{\mathcal{D}}_{i})$ are pseudo-Cartan inclusions. As noted earlier, if $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ have the AUP or if ${\mathcal{C}}_{i}$ are abelian, $(\tilde{\mathcal{C}}_{i},\tilde{\mathcal{D}}_{i})$ are pseudo-Cartan inclusions provided that $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ are pseudo-Cartan inclusions.

Proof of Theorem 7.

Suppose first that $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ are unital pseudo-Cartan inclusions. Theorem 7 shows that $C^{*}({\mathcal{A}}_{i})=C^{*}_{e}({\mathcal{A}}_{i})$ . By Theorem 7, ${\mathcal{D}}_{i}$ norms ${\mathcal{C}}_{i}$ ; in particular, ${\mathcal{D}}_{i}$ norms ${\mathcal{A}}_{i}$ . Thus [PittsNoAlAuCoBoIsOpAl, Corollary 1.5] shows $u$ uniquely extends to a $*$ -isomorphism of $C^{*}({\mathcal{A}}_{1})$ onto $C^{*}({\mathcal{A}}_{2})$ .

Now suppose $({\mathcal{C}}_{i},{\mathcal{D}}_{i})$ are not assumed unital and $u:{\mathcal{A}}_{1}\rightarrow{\mathcal{A}}_{2}$ is an isometric isomorphism. Since $({\mathcal{C}},{\mathcal{D}})$ is weakly non-degenerate, $(\tilde{\mathcal{C}}_{i},\tilde{\mathcal{D}}_{i})$ is a unital pseudo-Cartan inclusion, so $I_{\tilde{\mathcal{D}}_{i}}=I_{\tilde{\mathcal{C}}_{i}}$ . Write $\tilde{\mathcal{A}}_{i}:={\mathcal{A}}_{i}+{\mathbb{C}}I_{\tilde{\mathcal{C}}_{i}}$ , so that $\tilde{\mathcal{D}}_{i}\subseteq\tilde{\mathcal{A}}_{i}\subseteq\tilde{\mathcal{C}}_{i}$ . Notice that

C^{*}(\tilde{\mathcal{A}}_{i})=C^{*}({\mathcal{A}}_{i})+{\mathbb{C}}I_{\tilde{\mathcal{C}}_{i}}.

Applying [MeyerAdUnOpAl, Corollary 3.3] with $n=1$ shows $\tilde{u}:\tilde{\mathcal{A}}_{1}\rightarrow\tilde{\mathcal{A}}_{2}$ is an isometric isomorphism. Therefore, $\tilde{u}$ uniquely extends to a $*$ -isomorphism $\theta$ of $C^{*}(\tilde{\mathcal{A}}_{1})$ onto $C^{*}(\tilde{\mathcal{A}}_{2})$ . Thus $\theta|_{C^{*}({\mathcal{A}}_{1})}$ is a $*$ -isomorphism of $C^{*}({\mathcal{A}}_{1})$ onto $C^{*}({\mathcal{A}}_{2})$ extending $u$ .

If $\pi:C^{*}({\mathcal{A}}_{1})\rightarrow C^{*}({\mathcal{A}}_{2})$ is another $*$ -isomorphism such that $\pi|_{{\mathcal{A}}_{1}}=u$ , then $\tilde{\pi}|_{\tilde{\mathcal{A}}_{1}}=\tilde{u}$ . As $\theta$ is the unique extension of $\tilde{u}$ to $C^{*}(\tilde{\mathcal{A}}_{1})$ , $\tilde{\pi}=\theta$ . Therefore, $\pi=\theta|_{C^{*}({\mathcal{A}}_{1})}$ . This gives uniqueness of the extension of $u$ to $C^{*}({\mathcal{A}}_{1})$ and completes the proof.∎

If the hypothesis that $(C({\mathcal{A}}_{1}),{\mathcal{D}}_{1})$ is a regular inclusion is added to the hypotheses of Theorem 7, more can be said.

Corollary \the\numberby.

If $(C^{*}({\mathcal{A}}_{1}),{\mathcal{D}}_{1})$ is regular, then $(C^{*}({\mathcal{A}}_{1}),{\mathcal{D}}_{1})$ and $(C^{*}({\mathcal{A}}_{2}),u({\mathcal{D}}_{1}))$ are pseudo-Cartan inclusions; let $({\mathcal{M}}_{1},{\mathcal{N}}_{1}\hbox{\,:\hskip-1.0pt:\,}\tau_{1})$ and $({\mathcal{M}}_{2},{\mathcal{N}}_{2}\hbox{\,:\hskip-1.0pt:\,}\tau_{2})$ be their Cartan envelopes. Furthermore, there is a unique extension of $u$ to a $*$ -isomorphism $\breve{u}:{\mathcal{M}}_{1}\rightarrow{\mathcal{M}}_{2}$ such that

\breve{u}\circ\tau_{1}=\tau_{2}\circ u.

Proof.

Let $u^{\prime}:C^{*}({\mathcal{A}}_{1})\rightarrow C^{*}({\mathcal{A}}_{2})$ be the unique $*$ -isomorphism extending $u$ provided by Theorem 7. Since $u^{\prime}|_{{\mathcal{D}}_{1}}=u|_{{\mathcal{D}}_{1}}$ , $u({\mathcal{D}}_{1})$ is a $C^{*}$ -algebra. Since $(C^{*}({\mathcal{A}}_{1}),{\mathcal{D}}_{1})$ is a regular inclusion, so is $(C^{*}({\mathcal{A}}_{2}),u({\mathcal{D}}_{1}))$ . By Corollary 2.3, both $(C^{*}({\mathcal{A}}_{1}),{\mathcal{D}}_{1})$ and $(C^{*}({\mathcal{A}}_{2}),u({\mathcal{D}}_{1})$ ) have the faithful unique pseudo-expectation property, so they are pseudo-Cartan inclusions. Also for any $v\in{\mathcal{N}}(C^{*}({\mathcal{A}}_{1}),{\mathcal{D}}_{1})$ and $h\in{\mathcal{D}}_{1}$ ,

u^{\prime}(v)^{*}u(h)u^{\prime}(v)=u^{\prime}(v^{*}hv)\in u^{\prime}({\mathcal{D}}_{1})=u({\mathcal{D}}_{1}).

Thus $u^{\prime}$ is a regular $*$ -isomorphism. An application of Theorem 6.1 completes the proof. ∎

The following is immediate from Theorem 7.

Corollary \the\numberby.

Suppose $({\mathcal{C}},{\mathcal{D}})$ is a pseudo-Cartan inclusion such that $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is a pseudo-Cartan inclusion. If ${\mathcal{A}}$ is an algebra such that ${\mathcal{D}}\subseteq{\mathcal{A}}\subseteq{\mathcal{C}}$ and $C^{*}({\mathcal{A}})={\mathcal{C}}$ , then the group of isometric automorphisms of ${\mathcal{A}}$ is isomorphic to the group of all $*$ -automorphisms of ${\mathcal{C}}$ which leave ${\mathcal{A}}$ invariant.

Theorems 7–7 seem interesting even in the commutative case, for they apply to certain function algebras.

Example \the\numberby.

Let ${\mathcal{C}}$ be a unital and abelian $C^{*}$ -algebra, and suppose $J\unlhd{\mathcal{C}}$ is an essential ideal. Let ${\mathcal{A}}\subseteq{\mathcal{C}}$ be a Banach algebra such that: ${\mathcal{A}}$ separates points of $\hat{\mathcal{C}}$ , ${\mathbb{C}}I_{\mathcal{C}}\subseteq{\mathcal{A}}$ , and ${\mathcal{A}}\cap J=\{0\}$ . Then ${\mathcal{A}}+J$ is a Banach algebra satisfying $J\subseteq J+{\mathcal{A}}\subseteq{\mathcal{C}}$ . By the Stone-Weierstrauß Theorem, $C^{*}({\mathcal{A}})={\mathcal{C}}$ . Since $({\mathcal{C}},J)$ is a pseudo-Cartan inclusion, $C^{*}_{env}({\mathcal{A}}+J)={\mathcal{C}}$ and any isometric automorphism of ${\mathcal{A}}+J$ uniquely extends to a $*$ -isomorphism of ${\mathcal{C}}$ .

8. Questions

In addition to Conjecture 4.3, we now present a few open questions.

Recall that Barlak and Li [BarlakLiCaSuUCTPr, Corollary 1.2] showed that if a separable, nuclear, $C^{*}$ -algebra contains a Cartan MASA, then it satisfies the Universal Coefficient Theorem (UCT). Separable, nuclear $C^{*}$ -algebras which satisfy the UCP are of significant interest in the classification program. This is in part the motivation for the following two questions.

Question \the\numberby. Suppose $({\mathcal{C}},{\mathcal{D}})$ is a pseudo-Cartan inclusion. Must ${\mathcal{C}}$ contain a Cartan MASA?

Here is an example which motivates our next question. Let $X$ be the Cantor set and let $\Gamma$ be a countable discrete group having property $(T)$ . It follows from a result of Elek [ElekFrMiAcCoGrInPrMe, Theorem 1], that there exists a free and minimal action of $\Gamma$ on $X$ which admits an ergodic (regular, non-atomic) invariant Borel probability measure $\mu$ . Let $t\mapsto U_{t}$ be the unitary representation of $\Gamma$ on $L^{2}:=L^{2}(X,\mu)$ : $(U_{t}\xi)(s)=\xi(t^{-1}s)$ and $M$ be the representation of $C(X)$ by multiplication operators on $L^{2}$ .

Define $\mathcal{C}$ to be the $C^{*}$ -algebra generated by the images of $U$ and $M$ , and put $\mathcal{D}=M(C(X))$ . (It turns out $\mathcal{C}$ is an exotic crossed product.) The freeness of the action implies $(\mathcal{C},\mathcal{D})$ is a regular MASA inclusion with the unique state extension property, so there exists a (unique) conditional expectation $E:\mathcal{C}\rightarrow\mathcal{D}$ .

Let $J:=\{x\in\mathcal{C}:E(x^{*}x)=0\}$ be the left kernel of $E$ . By [PittsStReInI, Theorem 3.15], $J$ is a ideal of $\mathcal{C}$ having trivial intersection with $\mathcal{D}$ and [ExelPittsZarikianExIdFrTrGrC*Al, Theorem 3.7] shows $J\neq\{0\}$ . Then $(\mathcal{C}/J,\mathcal{D})$ is a regular inclusion having the unique state extension property by [ArchboldBunceGregsonExStC*AlII, Lemma 3.1]. It follows $(\mathcal{C}/J,\mathcal{D})$ is a regular inclusion with the extension property which has a faithful conditional expectation; thus $(\mathcal{C}/J,\mathcal{D})$ is a $C^{*}$ -diagonal.

Question \the\numberby. Suppose ${\mathcal{C}}$ is a $C^{*}$ -algebra and $J\unlhd{\mathcal{C}}$ . If ${\mathcal{C}}/J$ admits a subalgebra ${\mathcal{E}}$ such that $({\mathcal{C}}/J,{\mathcal{E}})$ is a pseudo-Cartan inclusion (or a Cartan inclusion), must there exist ${\mathcal{D}}\subseteq{\mathcal{C}}$ such that $({\mathcal{C}},{\mathcal{D}})$ is a pseudo-Cartan inclusion (or Cartan inclusion)?

Next, the uniqueness statement of Theorem 3 suggests the possibility of finding a canonical method of replacing certain non-Hausdorff twists with Hausdorff ones.

Question \the\numberby. Suppose $(\Sigma,G)$ is a twist over the second countable, étale and topologically free groupoid $G$ (see [ExelPittsChGrC*AlNoHaEtGr, Definition 3.4.6] for the definition of a topologically free groupoid). We assume $G^{(0)}$ is Hausdorff, but we do not assume $G$ is Hausdorff. Then $(C^{*}_{ess}(\Sigma,G),C_{0}(G^{(0)}))$ is a weak-Cartan inclusion (see [ExelPittsChGrC*AlNoHaEtGr, Corollary 3.9.5]) and hence a pseudo-Cartan inclusion. Let $({\mathcal{A}},{\mathcal{B}}\hbox{\,:\hskip-1.0pt:\,}\alpha)$ be its Cartan envelope, and let $(\Sigma_{{\mathcal{A}}},G_{{\mathcal{A}}})$ be the twist associated to $({\mathcal{A}},{\mathcal{B}})$ . Then $G_{{\mathcal{A}}}$ is Hausdorff. What is the relationship between $(\Sigma_{{\mathcal{A}}},G_{{\mathcal{A}}})$ and $(\Sigma,G)$ ?

We close with a technical question. Several of the results in Section 7 assume that the unitization of a pseudo-Cartan inclusion is again a pseudo-Cartan inclusion and it would be interesting to know if that hypothesis can be removed. The issue is whether $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ is regular.

Question \the\numberby. If $({\mathcal{C}},{\mathcal{D}})$ is a pseudo-Cartan inclusion, must $(\tilde{\mathcal{C}},\tilde{\mathcal{D}})$ be a pseudo-Cartan inclusion?

Appendix A

Changes in Notation and Terminology. Throughout Appendix A, we depart from the notation and terminology used in Sections 1–7 above; instead we use the notation and terminology found in [PittsStReInII]. In particular, all inclusions below are unital, and for a unital $C^{*}$ -algebra $B$ , an essential extension for ${\mathcal{B}}$ will mean a pair $({\mathcal{A}},\alpha)$ consisting of a unital $C^{*}$ -algebra ${\mathcal{A}}$ and a unital $*$ -monomorphism $\alpha:{\mathcal{B}}\rightarrow{\mathcal{A}}$ such that $\alpha({\mathcal{B}})\subseteq{\mathcal{A}}$ has the ideal intersection property.

Lemma 2.3 of [PittsStReInII] is presented without proof, and there is an error in its statement. The error is the assertion that the map $\operatorname{\textsc{Ropen}}(\hat{\mathcal{A}})\ni G\mapsto r^{-1}(G)$ is a Boolean algebra isomorphism of $\operatorname{\textsc{Ropen}}(\hat{\mathcal{A}})$ onto $\operatorname{\textsc{Ropen}}(\hat{\mathcal{B}})$ : it should have said $\operatorname{\textsc{Ropen}}(\hat{\mathcal{A}})\ni G\mapsto\left(\overline{r^{-1}(G)}\right)^{\circ}\in\operatorname{\textsc{Ropen}}(\hat{\mathcal{B}})$ is a Boolean algebra isomorphism. Here is the full and corrected statement.

Lemma \the\numberby (Corrected [PittsStReInII, Lemma 2.3]).

Suppose ${\mathcal{A}}$ and ${\mathcal{B}}$ are abelian, unital $C^{*}$ -algebras, $({\mathcal{B}},\alpha)$ is an essential extension of ${\mathcal{A}}$ , and $r:\hat{\mathcal{B}}\rightarrow\hat{\mathcal{A}}$ is the continuous surjection, $\rho\in\hat{\mathcal{B}}\mapsto\rho\circ\alpha$ . Then the maps

(A.1)	$\displaystyle\operatorname{\textsc{Rideal}}({\mathcal{B}})\ni J\mapsto\alpha^{-1}(J)$	$\displaystyle\quad\text{and}\quad\operatorname{\textsc{Ropen}}(\hat{\mathcal{A}})\ni G\mapsto\left(\overline{r^{-1}(G)}\right)^{\circ}$
are Boolean algebra isomorphisms of $\operatorname{\textsc{Rideal}}({\mathcal{B}})$ onto $\operatorname{\textsc{Rideal}}({\mathcal{A}})$ and $\operatorname{\textsc{Ropen}}(\hat{\mathcal{A}})$ onto $\operatorname{\textsc{Ropen}}(\hat{\mathcal{B}})$ respectively. The inverses of these maps are
(A.2)	$\displaystyle\operatorname{\textsc{Rideal}}({\mathcal{A}})\ni K\mapsto\alpha(K)^{\perp\perp}$	$\displaystyle\quad\text{and}\quad\operatorname{\textsc{Ropen}}(\hat{\mathcal{B}})\ni H\mapsto(r(\overline{H}))^{\circ}$
respectively. Furthermore, for $J\in\operatorname{\textsc{Rideal}}({\mathcal{B}})$ and $G\in\operatorname{\textsc{Ropen}}(\hat{\mathcal{A}})$ ,
(A.3)	$\displaystyle\operatorname{supp}(\alpha^{-1}(J))=(r(\overline{\operatorname{supp}(J)})^{\circ}$	$\displaystyle\quad\text{and}\quad\operatorname{ideal}(\left(\overline{r^{-1}(G)}\right)^{\circ})=\alpha(\operatorname{ideal}(G))^{\perp\perp}.$

A complete proof of Lemma A may be found in [PittsIrMaIsBoReOpSeReId]. The proofs of (A.1) and (A.2) are found in [PittsIrMaIsBoReOpSeReId, Propositions 3.2 and 4.17] and (A.3) is [PittsIrMaIsBoReOpSeReId, Lemma 4.12] combined with [PittsIrMaIsBoReOpSeReId, Lemma 4.7]. To make the notational transition from [PittsIrMaIsBoReOpSeReId] to the notation of Lemma A, take $Y:=\hat{\mathcal{B}}$ , $X=\hat{A}$ and $\pi:=r$ .

Unfortunately, the misstatement in [PittsStReInII, Lemma 2.3] leads to a gap in the proof of [PittsStReInII, Theorem 6.9]. The statement of [PittsStReInII, Theorem 6.9] is correct, but its proof is insufficient, as we now describe.

Let $({\mathcal{C}},{\mathcal{D}})$ be a regular inclusion with the unique pseudo-expectation property, let $(I({\mathcal{D}}),\iota)$ be an injective envelope for ${\mathcal{D}}$ and let $E:{\mathcal{C}}\rightarrow I({\mathcal{D}})$ be the pseudo-expectation. In the proof of [PittsStReInII, Theorem 6.9] we claimed that if $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ and $\rho\in\widehat{I({\mathcal{D}})}$ satisfies $\rho(E(v))\neq 0$ , then $\rho\circ\iota\in(\operatorname{fix}\beta_{v})^{\circ}$ . That claim is then used show that $\rho\circ E$ is a compatible state. Our proof of this claim is insufficient because it uses the incorrectly stated portion of [PittsStReInII, Lemma 2.3].

We now give a proof of [PittsStReInII, Theorem 6.9], whose statement we have reproduced in Theorem A. The proof uses the correction to [PittsStReInII, Lemma 2.3] given as Lemma A above to show that $\rho\circ E$ is a compatible state.

Theorem \the\numberby ([PittsStReInII, Theorem 6.9]).

Suppose $({\mathcal{C}},{\mathcal{D}})$ is a regular inclusion with the unique pseudo-expectation property. Then $({\mathcal{C}},{\mathcal{D}})$ is a covering inclusion and ${\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})$ is a compatible cover for $\hat{\mathcal{D}}$ . Furthermore, if $F$ is any closed subset of $\text{Mod}({\mathcal{C}},{\mathcal{D}})$ which covers $\hat{\mathcal{D}}$ , then ${\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})\subseteq F$ .

Proof.

Denote by $r$ the “restriction” map, $\widehat{I({\mathcal{D}})}\ni\rho\mapsto\rho\circ\iota\in\hat{\mathcal{D}}$ . Since $(I({\mathcal{D}}),\iota)$ is an injective envelope for ${\mathcal{D}}$ , it is in particular an essential extension of ${\mathcal{D}}$ .

For $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ , let

U_{v}:=\{\rho\in\widehat{I({\mathcal{D}})}:\rho(E(v))\neq 0\}.

If $\rho\in U_{v}$ , the Cauchy-Schwartz inequality gives,

|\rho(E(v))|^{2}\leq\rho(E(v^{*}v))=(\rho\circ\iota)(v^{*}v),

so $r(\rho)\in\operatorname{dom}\beta_{v}$ . By [PittsStReInI, Lemma 2.5], $\rho\circ E|_{\mathcal{D}}=r(\rho)\in\operatorname{fix}\beta_{v}$ . Thus,

(A.4)

r(U_{v})\subseteq\operatorname{fix}(\beta_{v}).

Let us show that every $\tau\in{\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})$ is a compatible state. Fix $\tau\in{\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})$ and suppose $\tau(v)\neq 0$ for some $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ . Our task is to show that $|\tau(v)|^{2}=\tau(v^{*}v)$ . Write $\tau=\rho\circ E$ for some $\rho\in\widehat{I({\mathcal{D}})}$ . Let

X:=\{\rho^{\prime}\in\widehat{I({\mathcal{D}})}:\rho^{\prime}(E(v))>\rho(E(v))/2\}.

By construction, $X$ is an open subset of $\widehat{I({\mathcal{D}})}$ and $\rho\in X$ . Since $\overline{X}\subseteq U_{v}$ , (A.4) gives,

r(\overline{X})^{\circ}\subseteq(\operatorname{fix}\beta_{v})^{\circ}.

Put $G:=r(\overline{X})^{\circ}$ . Since $\widehat{I({\mathcal{D}})}$ is a Stonean space, $\overline{X}$ and $\overline{r^{-1}(G)}$ are clopen sets and hence are regular open subsets of $\widehat{I({\mathcal{D}})}$ . Then

\overline{r^{-1}(G)}=(\overline{r^{-1}(G)})^{\circ}=\left(\overline{r^{-1}(r(\overline{X})^{\circ})}\right)^{\circ}=\overline{X},

with the last equality following from the portions of Lemma A concerning regular open sets. Therefore, we may find a net $(\rho_{\lambda})$ in $r^{-1}(G)$ such that $\rho_{\lambda}\rightarrow\rho$ . Since $r(\rho_{\lambda})=\rho_{\lambda}\circ\iota\in(\operatorname{fix}\beta_{v})^{\circ}$ , we may find $d_{\lambda}\in\operatorname{ideal}((\operatorname{fix}\beta_{v})^{\circ})$ with $\rho_{\lambda}(\iota(d_{\lambda}))=1$ . Since $vd_{\lambda}\in{\mathcal{D}}^{c}$ (see [PittsStReInII, Lemmas 2.14 and 2.15]), and $E|_{{\mathcal{D}}^{c}}$ is a homomorphism (by [PittsStReInII, Lemma 6.8])

|\rho_{\lambda}(E(v))|^{2}=|\rho_{\lambda}(E(v)\iota(d_{\lambda}))|^{2}=|\rho_{\lambda}(E(vd_{\lambda}))|^{2}=|\rho_{\lambda}(E(d_{\lambda}^{*}v^{*}vd_{\lambda}))|=\rho_{\lambda}(E(v^{*}v)).

As $\tau=\rho\circ E$ ,

|\tau(v)|^{2}=\lim_{\lambda}|\rho_{\lambda}(E(v))|^{2}=\lim_{\lambda}\rho_{\lambda}(E(v^{*}v))=\tau(v^{*}v).

It follows that $\tau$ is a compatible state.

The remainder of the proof now follows exactly as in the proof of [PittsStReInII, Theorem 6.9]; we include the details for convenience.

Let us show the invariance of ${\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})$ . Choose $\tau\in{\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})$ and write $\tau=\rho\circ E$ for some $\rho\in\widehat{I({\mathcal{D}})}$ . Suppose $v\in{\mathcal{N}}({\mathcal{C}},{\mathcal{D}})$ is such that $\rho(\iota(v^{*}v))\neq 0$ . Let $P$ and $Q$ be the support projections in $I({\mathcal{D}})$ for the ideals $\overline{vv^{*}{\mathcal{D}}}$ and $\overline{v^{*}v{\mathcal{D}}}$ respectively. Then $\tilde{\theta}_{v}$ is a partial automorphism with domain $PI({\mathcal{D}})$ and range $QI({\mathcal{D}})$ . Define $\tau^{\prime}\in\widehat{I({\mathcal{D}})}$ by

\tau^{\prime}(h)=\rho(\tilde{\theta}_{v}(Ph)),\qquad h\in I({\mathcal{D}}).

For $x\in{\mathcal{C}}$ , [PittsStReInII, Proposition 6.2] gives,

	$\displaystyle\rho(E(v^{*}xv))$	$\displaystyle=\rho(\tilde{\theta}_{v}(E(vv^{}x)))=\rho(\tilde{\theta}_{v}(\iota(vv^{})PE(x)))$
		$\displaystyle=\rho(\iota(v^{}v))\rho(\tilde{\theta}_{v}(PE(x)))=\rho(\iota(v^{}v))(\tau^{\prime}\circ E)(x).$

Thus ${\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})$ is invariant.

If $\sigma\in\hat{\mathcal{D}}$ , choose any $\rho\in\widehat{I({\mathcal{D}})}$ such that $\rho\circ\iota=\sigma$ . Then $\sigma=(\rho\circ E)|_{\mathcal{D}}$ , so ${\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})$ covers $\hat{\mathcal{D}}$ . Thus, ${\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})$ is a compatible cover for $\hat{\mathcal{D}}$ and $({\mathcal{C}},{\mathcal{D}})$ is a covering inclusion.

Finally, if $F\subseteq\text{Mod}({\mathcal{C}},{\mathcal{D}})$ is closed and covers $\hat{\mathcal{D}}$ , then ${\mathfrak{S}}_{s}({\mathcal{C}},{\mathcal{D}})\subseteq F$ by [PittsStReInII, Theorem 6.1(b)].

∎

	$\displaystyle\alpha(v^{*})\>y\Delta(\alpha(x_{n+1}))\alpha(h_{0})\>\alpha(v)$	$\displaystyle=\alpha(v^{*})y\alpha(v)\>\theta_{\alpha(v)}(\Delta(\alpha(x_{n+1}))\alpha(h_{0}))$
		$\displaystyle=\alpha(v^{})y\alpha(v)\>\theta_{\alpha(v)}(\alpha(vv^{})\Delta(\alpha(x_{n+1}k)))$
		$\displaystyle=(\alpha(v^{})y\alpha(v))\>\left(\alpha(v)^{}\Delta(\alpha(x_{n+1}k))\alpha(v)\right)$
		$\displaystyle=(\alpha(v^{})y\alpha(v))\>\Delta(\alpha(v)^{}\alpha(x_{n+1}k)\alpha(v))\in{\mathcal{B}}_{1}.$

	$\displaystyle\theta_{v}^{-1}(\theta_{v}(h)k)$	$\displaystyle=\theta_{v}^{-1}(\theta_{v}((vv^{})^{2}s)k)=\theta_{v}^{-1}(v^{}(vv^{*}s)vk)$
		$\displaystyle=vv^{}svkv^{}=(vv^{})sv(tww^{})v^{*}$
		$\displaystyle=s(vv^{})vtww^{}v^{*}$
		$\displaystyle=s\,(vtv^{})(vw)(vw)^{}\in(vw)(vw)^{*}{\mathcal{D}}\subseteq R(vw).$

	$\displaystyle 0=\sum_{k=1}^{n}\alpha_{1}(v_{k})h_{k}$	$\displaystyle\iff\Delta_{1}\left(\sum_{k,\ell=1}^{n}h_{k}^{}\alpha_{1}(v_{k}^{}v_{\ell})h_{\ell}\right)=0$
		$\displaystyle\iff\psi_{\mathcal{B}}\left(\sum_{k,\ell=1}^{n}h_{k}^{}\Delta_{1}(\alpha_{1}(v_{k}^{}v_{\ell}))h_{\ell}\right)=0$
		$\displaystyle\stackrel{{\scriptstyle\eqref{!pscharu1}}}{{\iff}}\sum_{k,\ell=1}^{n}\psi_{\mathcal{B}}(h_{k}^{})\Delta_{2}(\alpha_{2}(v_{k}^{}v_{\ell}))\psi_{\mathcal{B}}(h_{\ell})=0$
		$\displaystyle\iff\Delta_{2}\left(\sum_{k,\ell=1}^{n}\psi_{\mathcal{B}}(h_{k}^{})\alpha_{2}(v_{k}^{}v_{\ell})\psi_{\mathcal{B}}(h_{\ell})\right)=0$
		$\displaystyle\iff\sum_{k=1}^{n}\alpha_{2}(v_{k})\psi_{\mathcal{B}}(h_{k})=0.$

Pseudo-Cartan Inclusions

Abstract.

2020 Mathematics Subject Classification:

1. Introduction

2. Foundations

2.1. Terminology and Notation

Observation \the\numberby.

Proof.

Definition \the\numberby.

Fact \the\numberby (see [ExelOnKuC*DiOpId]).

2.2. Weakly Non-Degenerate Inclusions

Definition \the\numberby.

Lemma \the\numberby.

Proof.

Lemma \the\numberby.

Proof.

Lemma \the\numberby.

Proof.

Corollary \the\numberby.

Proof.

Proposition \the\numberby.

Proof.

2.3. Pseudo-Expectations

Remark \the\numberby.

Lemma \the\numberby (c.f. [PittsStReInI, Proposition 1.11]).

Lemma \the\numberby.

Proof.

Definition \the\numberby (c.f. [PittsStReInI]).

Definition \the\numberby.

Lemma \the\numberby.

Proof.

Lemma \the\numberby.

Proof.

Corollary \the\numberby.

Proof.

Proposition \the\numberby.

Lemma \the\numberby.

Proof.

Proposition \the\numberby.

Proof.

Proposition \the\numberby.

Proof.

Lemma \the\numberby.

Proof.

Theorem \the\numberby.

Proof.

Corollary \the\numberby.

Proof.

Corollary \the\numberby.

Proof.

Proposition \the\numberby.

Proof.

2.4. The Ideal Intersection Property

Lemma \the\numberby.

Proof.

Lemma \the\numberby.

Proof.

Proposition \the\numberby.

Proof.

3. Cartan Envelopes

Lemma \the\numberby ([PittsNoApUnInC*Al, Corollary 2.2]).

Lemma \the\numberby.

Proof.

Corollary \the\numberby.

Proof.

Example \the\numberby.

Theorem \the\numberby.

Proof.

Definition \the\numberby.

Lemma \the\numberby.

Proof.

Lemma \the\numberby.

Proof.

Definition \the\numberby.

Proposition \the\numberby.

Proof.

Claim \the\numberby.

Proposition \the\numberby.

Proof.

Observation \the\numberby.

(a) $\Rightarrow$ (b).

(b) $\Rightarrow$ (c).

(c) $\Rightarrow$ (a).