Covering entropy for types in tracial $\mathrm{W}^{*}$-algebras

The study of $\mathrm{W}^{*}$-algebras or von Neumann algebras is a deep and challenging subject with many connections to fields as diverse as ergodic theory, geometric group theory, random matrix theory, quantum information, and model theory. Our present goal is to bring together two of these facets—the model theory of tracial $\mathrm{W}^{*}$-algebras studied in [7, 8, 9, 1] and Voiculescu’s free entropy theory which, roughly speaking, quantifies the amount of matrix approximations for the generators of $\mathrm{W}^{*}$-algebra (see e.g. [32, 33, 12, 25, 16]). On the free entropy side, we will work in the framework of Hayes’ $1$-bounded entropy $h$ [16] which arose out of the work of Jung [25]; for history and motivation, refer to [18, §2]. The sibling paper [23] develops the analog of Voiculescu’s free microstate entropy for the setting of model-theoretic types.

We adapt the framework of $1$-bounded entropy [25, 16] to capture data about the generators’ model-theoretic type and not only their non-commutative law. The non-commutative law of a tuple $(X_{j})_{j\in\mathbb{N}}$ from $\mathcal{M}=(M,\tau)$ encodes the joint moments $\tau(p(X))$ for non-commutative $*$-polynomials $p$; laws thus describe tracial von Neumann algebras with chosen generators up to generator-preserving isomorphism. The type includes the values of more complicated formulas that involve not only the addition, adjoint, product, and trace operations, but also taking suprema and infima in auxiliary variables over an operator norm ball in $\mathcal{M}$. For instance, the type would include the value of the formula

where $D_{1}^{\mathcal{M}}$ denotes the closed unit ball with respect to operator norm. The entropy $\operatorname{Ent}^{\mathcal{U}}(\mu)$ of a type $\mu$ is defined in terms of the exponential growth rates of the covering numbers of microstate spaces (spaces of matrix tuples with approximately the same type as our chosen generators, as in Voiculescu’s work), just like Jung and Hayes’ $1$-bounded entropy except with types instead of laws. However, we prefer the term “covering entropy” rather than “$1$-bounded entropy” as a more intrinsic description of the definition. The superscript $\mathcal{U}$ denotes the fact that we take limits with respect to a fixed non-principal ultrafilter $\mathcal{U}$ on $\mathbb{N}$.

Just as in the original definition of the $1$-bounded entropy, a key property of the covering entropy $\operatorname{Ent}^{\mathcal{U}}(\mu)$ is that it is invariant under change of coordinates (see §4.3). More precisely, if $\mathbf{X}$ and $\mathbf{Y}$ are tuples from $\mathcal{M}$ with $\mathrm{W}^{*}(\mathbf{X})=\mathrm{W}^{*}(\mathbf{Y})$, then their types $\operatorname{tp}^{\mathcal{M}}(\mathbf{X})$ and $\operatorname{tp}^{\mathcal{M}}(\mathbf{Y})$ have the same covering entropy (Corollary 4.10). This allows us to define the entropy $\operatorname{Ent}^{\mathcal{U}}(\mathcal{N}:\mathcal{M})$ of a separable tracial $\mathrm{W}^{*}$-algebra $\mathcal{N}\subseteq\mathcal{M}$ as the entropy of the type of any generating set. As suggested in [18], we streamline the proof of this invariance property using the result that every tuple $\mathbf{Y}$ from $\mathrm{W}^{*}(\mathbf{X})$ can be expressed as $\mathbf{f}(\mathbf{X})$ for some quantifier-free definable function (see [22, §13]). More generally, we can extend the definition of $\operatorname{Ent}^{\mathcal{U}}(\mathcal{N}:\mathcal{M})$ to the case where $\mathcal{N}\subseteq\mathcal{M}$ is not separable by setting it to be the supremum of $\operatorname{Ent}^{\mathcal{U}}(\mathcal{N}_{0}:\mathcal{M})$ over separable $\mathrm{W}^{*}$-subalgebras $\mathcal{N}_{0}\subseteq\mathcal{N}$ or equivalently, the supremum of $\operatorname{Ent}^{\mathcal{U}}(\operatorname{tp}^{\mathcal{M}}(\mathbf{X}))$ over all tuples $\mathbf{X}\in L^{\infty}(\mathcal{N})^{\mathbb{N}}$ (see Definition 4.11).

The covering entropy $\operatorname{Ent}^{\mathcal{U}}(\mathcal{N}:\mathcal{M})$ can be viewed intuitively as a measurement of the amount of tracial $\mathrm{W}^{*}$-embeddings of $\mathcal{N}$ into the matrix ultraproduct $\mathcal{Q}=\prod_{n\to\mathcal{U}}M_{n}(\mathbb{C})$ that extend to elementary embeddings of $\mathcal{M}$ (compare §4.4). This is the analog of the idea that the $1$-bounded entropy $h(\mathcal{N}:\mathcal{M})$ of $\mathcal{N}$ in the presence of $\mathcal{M}$ quantifies the amount of $\mathrm{W}^{*}$-embeddings of $\mathcal{N}$ into $\mathcal{Q}$ that extend to any embedding of $\mathcal{M}$. Thus, our work is motivated in part by the study of embeddings into ultraproducts, which is one theme of recent work on von Neumann algebras [28, 14, 20, 2, 1, 11].

We make a precise connection between $\operatorname{Ent}^{\mathcal{U}}(\mathcal{N}:\mathcal{M})$ and $1$-bounded entropy as follows. There is a canonical projection $\pi_{\operatorname{qf}}$ from the space of types to the space of non-commutative laws, since a non-commutative law describes the evaluation of quantifier-free formulas (rather than all logical formulas) in a tuple $\mathbf{X}$. Given a non-commutative law (or quantifier-free type) $\mu$, the $1$-bounded entropy $h^{\mathcal{U}}(\mu)$ can be expressed through the following variational principle (Corollary 5.4):

Thus, the $1$-bounded entropy is the quantifier-free version of the entropy for types.

In a similar way, the $1$-bounded entropy of $\mathcal{N}$ in the presence of $\mathcal{M}$ is the version using existential types. Entropy in the presence is described using microstates for a tuple $\mathbf{X}$ in $\mathcal{N}$ such that there exist compatible microstates for a tuple $\mathbf{Y}$ that generates $\mathcal{M}$. In the model-theoretic framework, the existence of such microstates for $\mathcal{M}$ is described through the evaluation of existential formulas in the original generators and their microstates (see §5.4). Similar to the quantifier-free setting, there is a projection $\pi_{\exists}$ from the space of types into the space of existential types, and a similar variational principle expressing the covering entropy of an existential type $\mu$ as the supremum of $\operatorname{Ent}^{\mathcal{U}}(\nu)$ over full types $\nu\in\pi_{\exists}^{-1}(\mu)$ (Lemma 5.13).

Altogether these ingredients allow us to prove the following result about ultraproduct embeddings, which is restated and proved in Theorem 5.24:

The hypotheses of the theorem hold for instance when $\mathcal{N}=\mathcal{M}$ is a nontrivial free product by [33, Proposition 6.8] and [25, Corollary 3.5] and [16, Proposition A.16] since $h^{\mathcal{U}}(\mathcal{M}:\mathcal{M})=\infty$. They also hold for the von Neumann algebras of groups with non-approximately-inner cocycles by [30, Theorem 3] and [25, Corollary 3.5] and [16, Proposition A.16].

In particular, since there exists $\mathcal{M}$ with $h^{\mathcal{U}}(\mathcal{M}:\mathcal{M})=\infty$, the theorem implies that there exist types in $\mathcal{Q}$ with arbitrarily large covering entropy, and therefore, $h^{\mathcal{U}}(\mathcal{Q}:\mathcal{Q})=\infty$. Similarly, the entropy $\operatorname{Ent}^{\mathcal{U}}(\mathcal{Q}:\mathcal{Q})$ given by Definition 4.11 is infinite.

The following corollary of Theorem 1.1 was communicated to me by Ben Hayes.

As shown in [23, Theorem 1.2], the analogous result holds for free entropy rather than $1$-bounded entropy without having to change the ultrafilter $\mathcal{U}$ to the ultrafilter $\mathcal{V}$.

Our results relate to recent work and questions about embeddings into ultraproducts. Jung [24] used the study of microstates to show that a separable tracial $\mathrm{W}^{*}$-algebra $\mathcal{A}$ is amenable if and only if all embeddings of $\mathcal{A}$ into $\mathcal{R}^{\mathcal{U}}$ are unitarily conjugate. Atkinson and Kunnawalkam Elayavalli [2] strengthened this result by showing that $\mathcal{A}$ is amenable if and only if all embeddings of $\mathcal{A}$ into $\mathcal{R}^{\mathcal{U}}$ are ucp-conjugate (meaning they are conjugate by an automorphism of $\mathcal{R}^{\mathcal{U}}$ that lifts to a sequence of unital completely positive maps $\mathcal{R}\to\mathcal{R}$). Atkinson, Goldbring, and Kunnawalkam Elayavalli [1] later showed that if a separable $\mathrm{II}_{1}$ factor $\mathcal{M}$ is Connes-embeddable and all embeddings of $\mathcal{M}$ into $\mathcal{M}^{\mathcal{U}}$ are automorphically conjugate, then $\mathcal{M}\cong\mathcal{R}$.

One can ask similar questions for embeddings into the ultraproduct $\mathcal{Q}=\prod_{n\to\mathcal{U}}M_{n}(\mathbb{C})$ for some fixed free ultrafilter $\mathcal{U}$. In [2], the authors showed that if $\mathcal{A}$ is a separable Connes-embeddable tracial $\mathrm{W}^{*}$-algebra and the space of unitary orbits of embeddings $\mathcal{A}\to\mathcal{Q}$ is separable in a certain metric, then $\mathcal{A}$ must be amenable. In particular, if all embeddings $\mathcal{A}\to\mathcal{Q}$ are unitarily conjugate, then $\mathcal{A}$ is amenable. It is an open question whether this result still holds when “unitarily conjugate” is replaced by “automorphically conjugate.” However, Theorem 1.1 implies the following result, which was pointed out to me by Srivatsav Kunnawalkam Elayavalli:

Intuitively, the corollary says that if the space of embeddings modulo automorphic conjugacy is trivial, then the space of embeddings modulo unitary conjugacy is not too large, since $h^{\mathcal{U}}(\mathcal{A})$ quantifies the “amount” of embeddings $\mathcal{A}\to\mathcal{Q}$ up to unitary conjugacy. The conclusion that $h^{\mathcal{U}}(\mathcal{A})=0$ is a weakening of amenability since by Jung’s theorem [24] amenability is equivalent to the space of embeddings modulo unitary conjugacy being trivial.

We remark that the free entropy techniques used here to study embeddings into $\mathcal{Q}$ cannot be directly applied to study embeddings into $\mathcal{R}^{\mathcal{U}}$. For instance, Theorem 1.1 does not make sense with $\mathcal{Q}$ replaced by $\mathcal{R}^{\mathcal{U}}$. Indeed, $\mathcal{R}^{\mathcal{U}}$ has property Gamma by [9, §3.2.2], and every tracial $\mathrm{W}^{*}$-algebra with property Gamma has $1$-bounded entropy zero (this is a special case of [16, Corollary 4.6] and it is shown explicitly in [18, §1.2, Example 4]). Thus, $h(\mathcal{R}^{\mathcal{U}})=0$ and therefore, for any subalgebra $\mathcal{M}$ of $\mathcal{R}^{\mathcal{U}}$, we also have $h^{\mathcal{U}}(\mathcal{M}:\mathcal{R}^{\mathcal{U}})=0$ by [16, §2, Property 1]. Hence, Theorem 1.1 would not hold with $\mathcal{R}^{\mathcal{U}}$ instead of $\mathcal{Q}$. By contrast, many other operator-theoretic and model-theoretic techniques are more easily applied to $\mathcal{R}^{\mathcal{U}}$ than to $\mathcal{Q}$ since $\mathcal{R}^{\mathcal{U}}$ is an ultrapower; see for instance [14, 15, 13].

In large part, our goal is to establish communication between the free probabilistic and model theoretic subgroups of operator algebras, and to show that many of the notions in free probability (such as non-commutative laws, microstates spaces in the presence, and relative microstate spaces) arise naturally from the model-theoretic framework. Therefore, we strive to make the exposition largely self-contained and use model-theoretic language throughout.

We start out by explaining the model-theoretic framework for operator algebras in §2. In particular, we give a more detailed explanation than current literature of the languages and structures for multiple sorts and multiple domains of quantification for each. Next, in §3, we give a self-contained development of definable predicates and functions of infinite tuples, including the result that every element of a tracial $\mathrm{W}^{*}$-algebra can be realized by applying a quantifier-free definable function to the generators which was observed in [22, 18].

§4 develops the framework of covering entropy for types. We show the invariance of entropy under change of coordinates in §4.3, describe the relationship with ultraproduct embeddings in §4.4, and finally show that adding variables in the (model-theoretic) algebraic closure of given tuple $\mathbf{X}$ does not change its entropy in §4.5.

In §5, we describe the quantifier-free and existential versions of entropy, showing that they agree with the $1$-bounded entropy of Hayes. We conclude the proof Theorem 1.1 there.

In the appendix §6, we describe a generalization to conditional (or “relative”) entropy, which focuses on quantifying the embeddings $\mathcal{N}\to\mathcal{Q}$ which restrict to a fixed embedding $\iota:\mathcal{A}\to\mathcal{Q}$ on a given $\mathrm{W}^{*}$-subalgebra $\mathcal{A}$. The existential version of the conditional covering entropy was studied by Hayes explicitly for $\mathcal{A}$ diffuse abelian and implicitly for $\mathcal{A}$ diffuse amenable [16], in which case it agrees with the unconditional version. However, the conditional covering entropy (for full, quantifier-free, or existential types) makes sense for any diffuse $\mathcal{A}$ with a specified embedding $\alpha:\mathcal{A}\to\mathcal{Q}$ (though, as far as we know, it may depend on the embedding $\alpha$). Moreover, conditional entropy is natural from the model-theoretic perspective, since it arises from replacing formulas in the original language with formulas that have coefficients from the subalgebra $\mathcal{A}$.

This was work was partially funded by the NSF postdoc grant DMS-2002826. I thank the organizers and hosts of the 2017 workshop on the model theory of operator algebras at University of California, Irvine, and of the 2018 long program Quantitative Linear Algebra at the Institute of Pure and Applied Mathematics at UCLA for conferences that greatly contributed to my knowledge of operator algebras, entropy, and model theory. Special thanks to Ben Hayes for pointing out Corollary 1.3, Srivatsav Kunnawalkam Elayavalli for pointing out Corollary 1.4, and Jennifer Pi for reading the paper in detail and finding many typos. Thanks to Isaac Goldbring, Ben Hayes, and Adrian Ioana for their comments on the paper and advice on references. Finally, thanks to the anonymous referees for numerous emendations to the paper.

We start by giving some basic terminology and background on operator algebras. For further detail and history, we suggest consulting the references [26, 6, 29, 31, 5, 34].

$\mathrm{C}^{*}$-algebras:

1. (1)

   A (unital) algebra over $\mathbb{C}$ is a unital ring $A$ with a unital inclusion map $\mathbb{C}\to A$.

2. (2)

   A (unital) $*$-algebra is an algebra $A$ equipped with a conjugate linear involution $*$ such that $(ab)^{*}=b^{*}a^{*}$.

3. (3)

   A unital $\mathrm{C}^{*}$-algebra is a $*$-algebra $A$ equipped with a complete norm $\lVert\cdot\rVert$ such that $\lVert ab\rVert\leq\lVert a\rVert\lVert b\rVert$ and $\lVert a^{*}a\rVert=\lVert a\rVert^{2}$ for $a,b\in A$.

A collection of fundamental results in $\mathrm{C}^{*}$-algebra theory establishes that $\mathrm{C}^{*}$-algebras can always be represented as algebras of operators on Hilbert spaces. If $H$ is a Hilbert space, the algebra of bounded operators $B(H)$ is a $\mathrm{C}^{*}$-algebra. Conversely, every unital $\mathrm{C}^{*}$-algebra can be embedded into $B(H)$ by some unital and isometric $*$-homomorphism $\rho$. By isometric, we mean that $\lVert\rho(a)\rVert=\lVert a\rVert$, where $\lVert\rho(a)\rVert$ is the operator norm on $B(H)$ and $\lVert a\rVert$ is the given norm on the $\mathrm{C}^{*}$-algebra $A$.

$\mathrm{W}^{*}$-algebras: A von Neumann algebra is a $*$-subalgebra of $B(H)$ (for some Hilbert space $H$) that is closed in the strong operator topology, the topology of pointwise convergence as functions on $H$. A $\mathrm{W}^{*}$-algebra is a $\mathrm{C}^{*}$-algebra that admits a predual (that is, it is the dual of some Banach space). A deep result of Sakai showed that for a $\mathrm{C}^{*}$-algebra $A$ is a $\mathrm{W}^{*}$-algebra if and only if it is isomorphic to a von Neumann algebra; moreover, the weak-$\star$ topology on a $\mathrm{W}^{*}$-algebra $A$ is uniquely determined by its $\mathrm{C}^{*}$-algebra structure [29, Corollary 1.13.3].

Tracial $\mathrm{W}^{*}$-algebras: A tracial $\mathrm{W}^{*}$-algebra is a $\mathrm{W}^{*}$-algebra $M$ together with a linear map $\tau:M\to\mathbb{C}$ satisfying:

• •

  positivity: $\tau(x^{*}x)\geq 0$ for all $x\in M$

• •

  unitality: $\tau(1)=1$

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  traciality: $\tau(xy)=\tau(yx)$ for $x,y\in A$

• •

  faithfulness: $\tau(x^{*}x)=0$ implies $x=0$ for $x\in A$.

• •

  weak-$\star$ continuity: $\tau:M\to\mathbb{C}$ is weak-$\star$ continuous.

We call $\tau$ a faithful normal tracial state.

The standard representation: Given a tracial $\mathrm{W}^{*}$-algebra $(M,\tau)$, we can form a Hilbert space $L^{2}(M,\tau)$ as the completion of $M$ with respect to the inner product $\langle x,y\rangle_{\tau}=\tau(x^{*}y)$; if $x\in M$, then we denote the corresponding element of $L^{2}(M,\tau)$ by $\widehat{x}$. There is a unique unital $*$-homomorphism $\pi_{\tau}:M\to B(L^{2}(M,\tau))$ satisfying $\pi_{\tau}(x)\widehat{y}=\widehat{xy}$ for $x,y\in M$. Now $\pi_{\tau}$ is a $*$-homomorphism isometric with respect to the operator norm, and its image is a von Neumann algebra. The construction of $L^{2}(M,\tau)$ and $\pi_{\tau}$ is a special case of the GNS (Gelfand-Naimark-Segal) construction and $\pi_{\tau}$ is also known as the standard representation of $(M,\tau)$. Note the convergence of a net $x_{i}$ to $x$ in $M$ with respect to the strong operator topology in $B(L^{2}(M,\tau))$ implies convergence of $\widehat{x}_{i}=\pi_{\tau}(x_{i})\widehat{1}$ to $\widehat{x}=\pi_{\tau}(x)\widehat{1}$ in $L^{2}(M,\tau)$. (It turns out that the converse is true if $(x_{i})_{i\in I}$ is bounded in operator norm, but we will not need to use this fact directly.)

$*$-polynomials and generators: Given an index set $I$, we denote by $\mathbb{C}\langle x_{i},x_{i}^{*}:i\in I\rangle$ the free unital algebra (or non-commutative polynomial algebra) generated by indeterminates $x_{i}$ and $x_{i}^{*}$ for $i\in I$. We equip $\mathbb{C}\langle x_{i},x_{i}^{*}:i\in I\rangle$ with the unique $*$-operation sending $x_{i}$ to $x_{i}^{*}$, thus making it into a $*$-algebra. If $A$ is a unital $*$-algebra and $(a_{i})_{i\in I}$ a collection of elements, there is a unique unital $*$-homomorphism $\rho:\mathbb{C}\langle x_{i},x_{i}^{*}:i\in I\rangle\to A$ mapping $x_{i}$ to $a_{i}$ for each $i\in I$. We refer to the elements of $\mathbb{C}\langle x_{i},x_{i}^{*}:i\in I\rangle$ as non-commutative $*$-polynomials, and if $p\in\mathbb{C}\langle x_{i},x_{i}^{*}:i\in I\rangle$ and $\rho:\mathbb{C}\langle x_{i},x_{i}^{*}:i\in I\rangle\to A$ is as above, we denote $\rho(p)$ by $p(a_{i}:i\in I)$. Moreover, the image of $\rho$ is the $*$-algebra generated by $(a_{i})_{i\in I}$.

If $A$ is a $\mathrm{C}^{*}$-algebra and $(a_{i})_{i\in I}$ is a collection of elements of $I$, then the $\mathrm{C}^{*}$-algebra generated by $(a_{i})_{i\in I}$ is the norm-closure of the $*$-algebra generated by $(a_{i})_{i\in I}$. Similarly, if $M$ is a von Neumann algebra and $(a_{i})_{i\in I}$ is a collection of elements of $M$, then the von Neumann subalgebra or $\mathrm{W}^{*}$-subalgebra generated by $(a_{i})_{i\in I}$ is the strong operator topology closure of the $*$-algebra generated by $(a_{i})_{i\in I}$. In particular, we say that $(a_{i})_{i\in I}$ generates $M$ if the strong operator topology closure is all of $M$.

Next, let us sketch the setup of continuous model theory, or model theory for metric structures [3, 4]. We will follow the treatment in [8] which introduces “domains of quantification” to cut down on the number of “sorts” neeeded.

A language $\mathcal{L}$ consists of:

• •

  A set $\mathcal{S}$ whose elements are called sorts.

• •

  For each $S\in\mathcal{S}$, a privileged relation symbol $d_{S}$ (which will represent a metric) and a set $\mathcal{D}_{S}$ whose elements are called domains of quantification for $S$.

• •

  For each $S\in\mathcal{S}$ and $D,D^{\prime}\in\mathcal{D}_{S}$ an assigned constant $C_{D,D^{\prime}}$.

• •

  A countably infinite set of variable symbols for each sort $S$. We denote the variables by $(x_{i})_{i\in\mathbb{N}}$.

• •

  A set of function symbols.

• •

  For each function symbol $f$, an assigned tuple $(S_{1},\dots,S_{n})$ of sorts called the domain, another sort $S$ called the codomain. We call $n$ the arity of $f$.

• •

  For each function symbol $f$ with domain $(S_{1},\dots,S_{n})$ and codomain $S$, and for every $\mathbf{D}=(D_{1},\dots,D_{n})\in\mathcal{D}_{S_{1}}\times\dots\times\mathcal{D}_{S_{n}}$, there is an assigned $D_{f,\mathbf{D}}\in\mathcal{D}_{S}$ (representing a range bound), and assigned moduli of continuity $\omega_{f,\mathbf{D},1}$, …, $\omega_{f,\mathcal{D},n}$. (Here “modulus of coninuity” means a continuous increasing, zero-preserving function $[0,\infty)\to[0,\infty)$).

• •

  A set of relation symbols.

• •

  For each relation symbol $R$, an assigned domain $(S_{1},\dots,S_{n})$ as in the case of function symbols.

• •

  For each relation symbol $R$ and for every $\mathbf{D}=(D_{1},\dots,D_{n})\in\mathcal{D}_{S_{1}}\times\dots\times\mathcal{D}_{S_{n}}$, an assigned bound $N_{R,\mathbf{D}}\in[0,\infty)$ and assigned moduli of continuity $\omega_{R,\mathbf{D},1}$, …, $\omega_{R,\mathcal{D},n}$.

Given a language $\mathcal{L}$, an $\mathcal{L}$-structure $\mathcal{M}$ assigns an object to each symbol in $\mathcal{L}$, called the interpretation of that symbol, in the following manner:

• •

  Each sort $S\in\mathcal{S}$ is assigned a metric space $S^{\mathcal{M}}$, and the symbol $d_{S}$ is interpreted as the metric $d_{S}^{\mathcal{M}}$ on $S^{\mathcal{M}}$.

• •

  Each domain of quantification $D\in\mathcal{D}_{S}$ is assigned a subset $D^{\mathcal{M}}\subseteq S^{\mathcal{M}}$, such that $D^{\mathcal{M}}$ is complete for each $D$, $S^{\mathcal{M}}=\bigcup_{D\in\mathcal{D}_{S}}D^{\mathcal{M}}$, and $\sup_{X\in D,Y\in D^{\prime}}d_{S}^{\mathcal{M}}(X,Y)\leq C_{D,D^{\prime}}$.

• •

  Each function symbol $f$ with domain $(S_{1},\dots,S_{n})$ and codomain $S$ is interpreted as a function $f^{\mathcal{M}}:S_{1}^{\mathcal{M}}\times\dots\times S_{n}^{\mathcal{M}}\to S^{\mathcal{M}}$. Moreover, for each $\mathbf{D}=(D_{1},\dots,D_{n})\in\mathcal{D}_{S_{1}}\times\dots\times\mathcal{D}_{S_{n}}$, the function $f^{\mathcal{M}}$ maps $D_{1}^{\mathcal{M}}\times\dots\times D_{n}^{\mathcal{M}}$ into $D_{f,\mathbf{D}}^{\mathcal{M}}$. Finally, $f^{\mathcal{M}}$ restricted to $D_{1}^{\mathcal{M}}\times\dots\times D_{n}^{\mathcal{M}}$ is uniformly continuous in the $i$th variable with modulus of continuity of $\omega_{f,\mathbf{D},i}$.

• •

  Each relation symbol $R$ with domain $(S_{1},\dots,S_{n})$ is interpreted as a function $R^{\mathcal{M}}:S_{1}^{\mathcal{M}}\times\dots\times S_{n}^{\mathcal{M}}\to\mathbb{R}$. Moreover, for each $\mathbf{D}=(D_{1},\dots,D_{n})\in\mathcal{D}_{S_{1}}\times\dots\times\mathcal{D}_{S_{n}}$, $f^{\mathcal{M}}$ is bounded by $N_{R,\mathbf{D}}$ on $M(D_{1})\times\dots\times M(D_{n})$ and uniformly continuous in the $i$th argument with modulus of continuity of $\omega_{R,\mathbf{D},i}$.

The language $\mathcal{L}_{\operatorname{tr}}$ of tracial $\mathrm{W}^{*}$-algebras can be described as follows. We will also simultaneously describe how a tracial $\mathrm{W}^{*}$-algebra $(M,\tau)$ gives rise to an $\mathcal{L}_{\operatorname{tr}}$-structure $\mathcal{M}$, that is, how each symbol will be interpreted.

• •

  A single sort, to be interpreted as the $\mathrm{W}^{*}$-algebra $M$. If $\mathcal{M}=(M,\tau)$ is a tracial $\mathrm{W}^{*}$-algebra, we denote the interpretation of this sort by $L^{\infty}(\mathcal{M})$ because of the intuition of tracial $\mathrm{W}^{*}$-algebras as non-commutative measure spaces.

• •

  Domains of quantification $\{D_{r}\}_{r\in(0,\infty)}$, to be interpreted as the operator norm balls of radius $r$ in $M$.

• •

  The metric symbol $d$, to be interpreted as the metric induced by $\lVert\cdot\rVert_{2,\tau}$.

• •

  A binary function symbol $+$, to be interpreted as addition.

• •

  A binary function symbol $\cdot$, to be interpreted as multiplication.

• •

  A unary function symbol $*$, to be interpreted as the adjoint operation.

• •

  For each $\lambda\in\mathbb{C}$, a unary function symbol, to be interpreted as multiplication by $\lambda$.

• •

  Function symbols of arity $0$ (in other words constants) $0$ and $1$, to be interpreted as additive and multiplicative identity elements.

• •

  Two unary relation symbols $\operatorname{Re}\operatorname{tr}$ and $\operatorname{Im}\operatorname{tr}$, to be interpreted the real and imaginary parts of the trace $\tau$.

• •

  For technical reasons explained in [8], we also introduce for each $d$-variable non-commutative polynomial $p$ a symbol $t_{p}:L^{\infty}(\mathcal{M})^{d}$ representing the evaluation of $p$, along with the appropriate range bounds $N_{t_{p},\mathbf{r}}$ given by the supremum of $\lVert p(X_{1},\dots,X_{d})\rVert$ over all $(X_{1},\dots,X_{d})$ in a tracial $\mathrm{W}^{*}$-algebra $\mathcal{M}$.

Each function and relation symbol is assigned range bounds and moduli of continuity that one would expect, e.g. multiplication is supposed to map $D_{r}\times D_{r^{\prime}}$ into $D_{rr^{\prime}}$ with $\omega_{(D_{r},D_{r}^{\prime}),1}^{\cdot}(t)=r^{\prime}t$ and $\omega_{(D_{r},D_{r}^{\prime}),2}^{\cdot}=rt$.

Although not every $\mathcal{L}_{\operatorname{tr}}$-structure comes from a tracial $\mathrm{W}^{*}$-algebra, one can formulate axioms in the language such that any structure satisfying these axioms comes from a tracial $\mathrm{W}^{*}$-algebra [8, §3.2]. In order to state this result precisely, we first have to explain formulas and sentences.

Terms in a language $\mathcal{L}$ are expressions obtained by iteratively composing the function symbols and variables. For example, if $x_{1}$, $x_{2}$, …are variables in a sort $S$ and $f:S\times S\to S$ and $g:S\times S\to S$ are function symbols, then $f(g(x_{1},x_{2}),x_{1})$ is a term. Each term has assigned range bounds and moduli of continuity in each variable which are the natural ones computed from those of the individual function symbols making up the composition. Any term $f$ with variables $x_{1}\in S_{1}$, …, $x_{k}\in S_{k}$ and output in $S$ can be interpreted in an $\mathcal{L}$-structure as a function $S_{1}^{\mathcal{M}}\times\dots\times S_{k}^{\mathcal{M}}\to S^{\mathcal{M}}$. For example, in the language $\mathcal{L}_{\operatorname{tr}}$, the terms are expressions obtained from iterating scalar multiplication, addition, multiplication, and the $*$-operation on variables and the unit symbol $1$. If $(M,\tau)$ is a tracial $\mathrm{W}^{*}$-algebra, then the interpretation of a term in $\mathcal{M}$ is a function represented by a $*$-polynomial.

Basic formulas in a language are obtained by evaluating relation symbols on terms. In other words, if $T_{1}$, …, $T_{k}$ are terms valued in sorts $S_{1}$, …, $S_{k}$, and $R$ is a relation $S_{1}\times\dots\times S_{k}\to\mathbb{R}$, then $R(T_{1},\dots,T_{n})$ is a basic formula. The basic formulas have assigned range bounds and moduli of continuity similar to the function symbols. In an $\mathcal{L}$-structure $\mathcal{M}$, a basic formula $\phi$ is interpreted as a function $\phi^{\mathcal{M}}:S_{1}^{\mathcal{M}}\times\dots\times S_{k}^{\mathcal{M}}\to\mathbb{R}$. In $\mathcal{L}_{\operatorname{tr}}$, a basic formula can take the form $\operatorname{Re}\operatorname{tr}(f)$ or $\operatorname{Im}\operatorname{tr}(f)$ where $f$ is an expression obtained by iterating the algebraic operations. Thus, when evaluated in a tracial $\mathrm{W}^{*}$-algebra, it corresponds to the real or imaginary part of the trace of a non-commutative $*$-polynomial.

Formulas are obtained from basic formulas by iterating several operations:

• •

  Given a formulas $\phi_{1}$, …, $\phi_{n}$ and $F:\mathbb{R}^{n}\to\mathbb{R}$ continuous, $F(\phi_{1},\dots,\phi_{n})$ is a formula.

• •

  If $\phi$ is a formula, $D$ is a domain of quantification for some sort $S$, and $x$ is one of our variables in $S$, then $\inf_{x\in D}\phi$ and $\sup_{x\in D}\phi$ are formulas.

Each occurrence of a variable in $\phi$ is either bound to a quanitifer $\sup_{x\in D}$ or $\inf_{x\in D}$, or else it is free. We will often write $\phi(x_{1},\dots,x_{n})$ for a formula to indicate that the free variables are $x_{1}$, …, $x_{n}$.

All these formulas also have assigned range bounds and moduli of continuity. The moduli of continuity of $F(\phi_{1},\dots,\phi_{n})$ are obtained by composition from the moduli of continuity of $F$ and $\phi_{j}$ as in [3, §2 appendix and Theorem 3.5]. Next, if $\phi:S_{1}\times\dots\times S_{n}\to S$ and $D\in\mathcal{D}_{S_{n}}$

then

Each formula has an interpretation in every $\mathcal{L}$-structure $\mathcal{M}$, defined by induction on the complexity of the formula. If $\phi=F(\phi_{1},\dots,\phi_{n})$, then $\phi^{\mathcal{M}}=F(\phi_{1}^{\mathcal{M}},\dots,\phi_{n}^{\mathcal{M}})$. Similarly, if $\psi(x_{1},\dots,x_{n-1})=\sup_{x_{n}\in D}\psi(x_{1},\dots,x_{n})$, then

Here $X_{1}$, …, $X_{n}$ are elements of the sorts in the $\mathcal{L}$-structure $\mathcal{M}$, rather than formal variables.

For convenience, we will assume that our formulas do not have two copies of the same variable (i.e. if a variable is bound to a quantifier, there is no other variable of the same name that is free or bound to a different quantifier). For instance, in the formula

the first occurrence of $x_{1}$ is free while the latter two occurrences are bound to the quantifier $\sup_{x_{1}\in D_{1}}$, but we can rewrite this formula equivalently as

We will typically denote the free variables by $(x_{i})_{i\in\mathbb{N}}$ and the bound variables by $(y_{i})_{i\in\mathbb{N}}$. Lowercase letters will be used for formal variables while uppercase letters will be used for individual operators in operator algebras (or more generally elements of an $\mathcal{L}$-structure).

A sentence is a formula with no free variables. If $\phi$ is a sentence, then the interpretation $\phi^{\mathcal{M}}$ in an $\mathcal{L}$-structure is simply a real number.

A theory $\mathrm{T}$ in a language $\mathcal{L}$ is a set of sentences. We say that an $\mathcal{L}$-structure $\mathcal{M}$ models the theory $\mathrm{T}$, or $\mathcal{M}\models\mathrm{T}$ if $\phi^{\mathcal{M}}=0$ for all $\phi\in\mathrm{T}$.

If $\mathcal{M}$ is an $\mathcal{L}$-structure, then the theory of $\mathcal{M}$, denoted $\operatorname{Th}(\mathcal{M})$ is the set of sentences $\phi$ such that $\phi^{\mathcal{M}}=0$. As observed in [8], the theory of $\mathcal{M}$ also uniquely determines the values $\phi^{\mathcal{M}}$ of all sentences $\phi$ since $\phi-c$ is a sentence for every constant $c\in\mathbb{R}$.

More generally, if $\mathcal{C}$ is a class of $\mathcal{L}$-structures, then $\operatorname{Th}(\mathcal{C})$ is the set of all sentences $\phi$ such that $\phi^{\mathcal{M}}=0$ for all $\mathcal{M}$ in $\mathcal{C}$. The class $\mathcal{C}$ is said to be axiomatizable if every $\mathcal{L}$-structure that models $\operatorname{Th}(\mathcal{C})$ is actually in $\mathcal{C}$.

It is shown in [8, §3.2] that the class of $\mathcal{L}_{\operatorname{tr}}$-structures that represent actual tracial $\mathrm{W}^{*}$-algebras is axiomatizable. The axioms, roughly speaking, encode the fact that $\mathcal{M}$ is a $*$-algebra, the fact that $\tau$ is a tracial state, the fact that $\lVert xy\rVert_{L^{2}(\mathcal{M})}\leq r\lVert y\rVert_{L^{2}(\mathcal{M})}$ for $x\in D_{r}^{\mathcal{M}}$, the relationship between the distance and the trace, the fact that $D_{r}^{\mathcal{M}}$ is contained in $D_{r^{\prime}}^{\mathcal{M}}$ for $r<r^{\prime}$ (that is, $\sup_{x\in D_{r}}\inf_{y\in D_{r^{\prime}}}d(x,y)=0$), and the fact that $t_{p}$ agrees with the evaluation of the non-commutative polynomial $p$.

The theory of tracial $\mathrm{W}^{*}$-algebras will be denoted $\mathrm{T}_{\operatorname{tr}}$. It is also shown in [8] that $\mathrm{II}_{1}$ factors (infinite-dimensional tracial $\mathrm{W}^{*}$-algebras with trivial center) are axiomatizable by a theory $\mathrm{T}_{\mathrm{II}_{1}}$.

An important construction for continuous model theory and for $\mathrm{W}^{*}$-algebras is the ultraproduct. Ultraproducts are a way of constructing a limiting object out of arbitrary sequences (or more generally indexed families) of objects. In order to force limits to exist, one uses a device called an ultrafilter.

Let $I$ be an index set. An ultrafilter $\mathcal{U}$ on $I$ is a collection of subsets of $I$ such that

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  $\varnothing\not\in\mathcal{U}$.

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  If $A\subseteq B\subseteq I$ and $A\in\mathcal{U}$, then $B\in\mathcal{U}$.

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  If $A$, $B\in\mathcal{U}$, then $A\cap B\in\mathcal{U}$.

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  For each $A\subseteq I$, either $A\in\mathcal{U}$ or $A^{c}\in\mathcal{U}$.

If $\mathcal{U}$ is an ultrafilter on $I$, $\Omega$ is a topological space, and $f:I\to\Omega$ is a function, then we say that

if for every neighborhood $O\ni w$ in $\Omega$, the preimage $f^{-1}(O)$ is an element of $\mathcal{U}$. Now if $i\in I$, there is an ultrafilter $\mathcal{U}_{i}:=\{A\subseteq I:i\in A\}$, which is called a principal ultrafilter. All other ultrafilters are called non-principal or free ultrafilters.

The set of all ultrafilters on $I$ can be identified with the Stone-Čech compactification of $I$, where $I$ is given the discrete topology (see e.g. [19, §3]). The principal ultrafilters correspond to the points of the original space $I$. In particular, this means that if $\Omega$ is a compact Hausdorff space and $f:I\to\Omega$ is a function, then $\lim_{i\to\mathcal{U}}f(i)$ exists in $\Omega$.

Now consider a language $\mathcal{L}$. Let $I$ be an index set, $\mathcal{U}$ an ultrafilter on $I$, and for each $i\in I$, let $\mathcal{M}$ be an $\mathcal{L}$-structure. The ultraproduct $\prod_{i\to\mathcal{U}}\mathcal{M}_{i}$ is the $\mathcal{L}$-structure $\mathcal{M}$ defined as follows (see [3, §5]): For each sort $S$, consider tuples $(X_{i})_{i\in I}$ where $X_{i}\in S^{\mathcal{M}_{i}}$.

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  Let’s call $(X_{i})_{i\in I}$ confined if there exists $D\in\mathcal{D}_{S}$ such that $X_{i}\in D^{\mathcal{M}_{i}}$ for all $i$.

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  Let’s call $(X_{i})_{i\in I}$ and $(Y_{i})_{i\in I}$ equivalent if $\lim_{i\to\mathcal{U}}d_{S}^{\mathcal{M}_{i}}(X_{i},Y_{i})=0$.

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  For a confined tuple $(X_{i})_{i\in I}$, let $[X_{i}]_{i\in I}$ denote its equivalence class.

We define $S^{\mathcal{M}}$ to be the set of equivalence classes of confined tuples $(X_{i})_{i\in I}$. The metric $d_{S}^{\mathcal{M}}$ on $S^{\mathcal{M}}$ is then given by

This is independent of the choice of representative for the equivalence classes because of the triangle inequality, and it is finite because if $X_{i}\in D^{\mathcal{M}_{i}}$ and $Y_{i}\in(D^{\prime})^{\mathcal{M}_{i}}$ for all $i$, then $d_{S}^{\mathcal{M}_{i}}(X_{i},Y_{i})\leq C_{D,D^{\prime}}$. Then $S^{\mathcal{M}}$ is a metric space and $S^{\mathcal{M}}=\bigcup_{D\in\mathcal{D}_{S}}D^{\mathcal{M}}$, where $D^{\mathcal{M}}$ is the set of classes $[X_{i}]_{i\in I}$ with $X_{i}\in D^{\mathcal{M}_{i}}$ for all $i$. Moreover, $D^{\mathcal{M}}$ is automatically complete [3, Proposition 5.3].

Each function symbol $f:S_{1}\times\dots\times S_{n}\to S$ receives its interpretation $f^{\mathcal{M}}$ through

which is well-defined because of the uniform continuity of $f$ on each domain of quantification, and similarly, each relation receives its interpretation in $\mathcal{M}$ through

One can verify by the same reasoning as [3, §5] that $\mathcal{M}$ is indeed an $\mathcal{L}$-structure.

One of the reasons ultraproducts are so important is because of the following result, known as (the continuous analog of) Łos’s theorem. See [3, Theorem 5.4].