Rigidity for geometric ideals in uniform Roe algebras

Baojie Jiang and Jiawen Zhang School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China. jiangbaojie@cqnu.edu.cn School of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai, 200433, China. jiawenzhang@fudan.edu.cn

Abstract.

In this paper, we investigate the rigidity problems for geometric ideals in uniform Roe algebras associated to discrete metric spaces of bounded geometry. These ideals were introduced by Chen and Wang, and can be fully characterised in terms of ideals in the associated coarse structures. Our main result is that if two geometric ideals in uniform Roe algebras are stably isomorphic, then the coarse spaces associated to these ideals are coarsely equivalent. We also discuss the case of ghostly ideals and pose some open questions.

Mathematics Subject Classification (2020): 47L20, 46L80, 51F30, 47L40.
Keywords: Uniform Roe algebras, Geometric ideals, Rigidity, Coarse equivalences

1. Introduction

Roe algebras are $C^{*}$ -algebras associated to metric spaces, which encode the coarse geometry of the underlying spaces. They were introduced by Roe in his pioneering work [16] to study higher indices of differential operators on open manifolds. There is also a uniform version of the Roe algebra, which has found applications in index theory (e.g.,[20]), $C^{*}$ -algebra theory (e.g., [14]), single operator theory (e.g., [22]) and even mathematical physics (e.g., [9]).

To provide a formal definition, consider a discrete metric space $(X,d)$ of bounded geometry (see Section 2.2). Thinking of an operator $T$ on $\ell^{2}(X)$ as an $X$ -by- $X$ matrix $(T(x,y))_{x,y\in X}$ , we say that $T$ has finite propagation if $\sup\{d(x,y)~{}|~{}T(x,y)\neq 0\}<\infty$ . The set of all finite propagation operators forms a $\ast$ -subalgebra of $\mathfrak{B}(\ell^{2}(X))$ , and its norm closure is called the uniform Roe algebra of $X$ and denoted by $C^{*}_{u}(X)$ .

Uniform Roe algebras have nice behaviour in coarse geometry. More precisely, recall that two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$ are coarsely equivalent if there exist a map $f:X\to Y$ and functions $\rho_{+},\rho_{-}:[0,\infty)\to\mathbb{R}$ with $\lim\limits_{t\to+\infty}\rho_{\pm}(t)=+\infty$ such that for any $x_{1},x_{2}\in X$ , we have

\rho_{-}(d_{X}(x_{1},x_{2}))\leq d_{Y}(f(x_{1}),f(x_{2}))\leq\rho_{+}(d_{X}(x_{1},x_{2})),

and there exists $R>0$ such that the $R$ -neighbourhood of $f(X)$ equals $Y$ . It is known (see, e.g., [4, Theorem 4]) that if $(X,d_{X})$ and $(Y,d_{Y})$ are coarsely equivalent, then their uniform Roe algebras are stably isomorphic in the sense that $C^{*}_{u}(X)\otimes\mathfrak{K}(\mathcal{H})\cong C^{*}_{u}(Y)\otimes\mathfrak{K}(\mathcal{H})$ (where $\mathfrak{K}(\mathcal{H})$ is the $C^{*}$ -algebra of compact operators on a separable infinite dimensional Hilbert space $\mathcal{H}$ ).

Conversely, the rigidity problem concerns whether the coarse geometry of a metric space can be fully determined by the associated uniform Roe algebra, i.e., whether two metric spaces are coarsely equivalent if their uniform Roe algebras are (stably) isomorphic. This problem was initially studied by Špakula and Willett in [21], followed by a series of works in the last decade (e.g., [2, 3, 11]), and recently is completely solved by the profound work [1].

Due to the importance of uniform Roe algebras, Chen and Wang initiated the study of their ideal structures ([6, 7, 23]). They managed to obtain a full description for the ideal structure of the uniform Roe algebra when the underlying space has Yu’s Property A (from [25]). More precisely, they introduced a notion of geometric ideal and provided a detailed picture for these ideals. Let us recall the definition:

Definition A ([6, 23]).

An ideal $I$ in the uniform Roe algebra $C^{*}_{u}(X)$ of a discrete metric space $(X,d)$ of bounded geometry is called geometric if the set of finite propagation operators in $I$ is dense in $I$ .

To state the characterisations for geometric ideals in [6], it is convenient to consult the notion of coarse spaces (see, e.g., [17, 18]). Recall that a coarse structure on a set $X$ is a collection $\mathscr{E}$ of subsets of $X\times X$ which is closed under the formation of subsets, inverses, products and finite unions (see Definition 2.1). For $(X,\mathscr{E})$ , we can also define the uniform Roe algebra $C^{*}_{u}(X,\mathscr{E})$ similarly as in the metric space case while replacing finite propagation operators with those whose supports belong to $\mathscr{E}$ (see Definition 2.5). In the case of a metric space $(X,d)$ , there is an associated coarse structure $\mathscr{E}_{d}$ which is the smallest one containing the sets $E_{R}:=\{(x,y)~{}|~{}d(x,y)\leq R\}$ for all $R\geq 0$ , and clearly we have $C^{*}_{u}(X,\mathscr{E}_{d})=C^{*}_{u}(X)$ .

In [6], the authors discovered that geometric ideals in $C^{*}_{u}(X)$ for a metric space $(X,d)$ can be described in terms of ideals in the coarse structure $\mathscr{E}_{d}$ . Recall that an ideal in $\mathscr{E}_{d}$ is a coarse structure $\mathscr{I}\subseteq\mathscr{E}_{d}$ on $X$ which is closed under products by elements in $\mathscr{E}_{d}$ (see Definition 2.6). The geometric ideal in $C^{*}_{u}(X)$ associated to $\mathscr{I}$ is $C^{*}_{u}(X,\mathscr{I})$ , and this procedure provides an isomorphism between the lattice of geometric ideals in $C^{*}_{u}(X)$ and the lattice of ideals in $\mathscr{E}_{d}$ ([6, Theorem 6.3]). For convenience, we denote $\mathscr{I}(I)$ the ideal in $\mathscr{E}_{d}$ associated to a geometric ideal $I$ in $C^{*}_{u}(X)$ . Moreover, [6, Theorem 6.3] also shows that ideals in $\mathscr{E}_{d}$ can be described using ideals in $(X,d)$ (see Definition 2.8). The ideal in $(X,d)$ associated to an ideal $\mathscr{I}$ in $\mathscr{E}_{d}$ is $\mathbf{L}(\mathscr{I}):=\{\mathrm{r}(E)~{}|~{}E\in\mathscr{I}\}$ , where $\mathrm{r}:X\times X\to X$ is the projection onto the first coordinate.

The main focus of this paper is to study the rigidity problem for geometric ideals in uniform Roe algebras of metric spaces. More precisely, we ask the following:

Question B (Rigidity for geometric ideals).

Let $(X,d_{X}),(Y,d_{Y})$ be discrete metric spaces of bounded geometry, and $I_{X},I_{Y}$ be geometric ideals in the uniform Roe algebras $C^{*}_{u}(X)$ and $C^{*}_{u}(Y)$ , respectively. If $I_{X}$ and $I_{Y}$ are (stably) isomorphic, do $(X,\mathscr{I}(I_{X}))$ and $(Y,\mathscr{I}(I_{Y}))$ have the same structure?

Our first task is to make the phrase “have the same structure” in a more precise way. Readers might wonder whether it is possible to use a similar notion of coarse equivalence as in the metric space case recalled above. This works well if the coarse structures contain the diagonals (see, e.g., Definition 2.4 and the paragraph thereafter). (Note that in [18, Definition 2.3], one requires that a coarse structure always contains the diagonal. While in [6, 19], it is necessary to consider the more general case.) However in the general case, there would be some issue if we only consider a single map (as in Definition 2.4) due to the lack of units in geometric ideals. More precisely, note that a nontrivial geometric ideal $I$ in $C^{*}_{u}(X)$ does not bare a unit, and hence the associated ideal $\mathscr{I}(I)$ does not contain the diagonal.

To overcome this issue, we need to consult the notion of coarse equivalence for general coarse spaces introduced by Skandalis, Tu and Yu:

Definition C ([19, Definition 2.2]).

Let $(X,\mathscr{E}_{X})$ and $(Y,\mathscr{E}_{Y})$ be coarse spaces. A coarse correspondence from $(X,\mathscr{E}_{X})$ to $(Y,\mathscr{E}_{Y})$ is a coarse structure $\mathscr{E}$ on $X\sqcup Y$ which restricts to $\mathscr{E}_{Y}$ on $Y$ , contains $\mathscr{E}_{X}$ , and is generated by the elements contained in $Y\times(X\sqcup Y)$ . A coarse equivalence between $(X,\mathscr{E}_{X})$ and $(Y,\mathscr{E}_{Y})$ is a coarse structure on $X\sqcup Y$ which is a coarse correspondence from $X$ to $Y$ and from $Y$ to $X$ .

As noted in [19, Proposition 2.3] (see also Proposition 3.6 and Corollary 3.7), Definition C coincides with the notion of coarse equivalence recalled above in the metric space case. However, it seems inconvenient to use directly the language of coarse correspondence to treat the rigidity problem. Hence we unpack Definition C by means of families of maps and prove the following:

Proposition D (Corollary 3.24).

Two coarse spaces $(X,\mathscr{E}_{X})$ and $(Y,\mathscr{E}_{Y})$ are coarsely equivalent if and only if there exist coarse families (see Definition 3.14) of maps

\mathcal{F}=\{f_{L}:L\to Y~{}|~{}L\in\mathbf{L}(\mathscr{E}_{X})\}\quad\text{and}\quad\mathcal{G}=\{g_{L^{\prime}}:L^{\prime}\to X~{}|~{}L^{\prime}\in\mathbf{L}(\mathscr{E}_{Y})\}

such that $\{(g_{f_{L}(L)}\circ f_{L}(x),x)~{}|~{}x\in L\}\in\mathscr{E}_{X}$ and $\{(f_{g_{L^{\prime}}(L^{\prime})}\circ g_{L^{\prime}}(y),y)~{}|~{}y\in L^{\prime}\}\in\mathscr{E}_{Y}$ for any $L\in\mathbf{L}(\mathscr{E}_{X})$ and $L^{\prime}\in\mathbf{L}(\mathscr{E}_{Y})$ .

Having established Proposition D, we manage to answer Question B completely. The following is the main result of this paper (see Section 4 and Section 6 for the missing definitions):

Theorem E.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be discrete metric spaces of bounded geometry, and $I_{X},I_{Y}$ be geometric ideals in the uniform Roe algebras $C^{*}_{u}(X)$ and $C^{*}_{u}(Y)$ , respectively. Consider the following conditions:

(1)

$I_{X}$ and $I_{Y}$ are stably isomorphic;
(2)

$(X,\mathscr{I}(I_{X}))$ and $(Y,\mathscr{I}(I_{Y}))$ are coarsely equivalent;
(3)

$I_{X}$ and $I_{Y}$ are Morita equivalent.

Then we have (1) $\Rightarrow$ (2) $\Rightarrow$ (3). Additionally if the associated ideals $\mathbf{L}(\mathscr{I}(I_{X}))$ and $\mathbf{L}(\mathscr{I}(I_{Y}))$ are countably generated, then the three conditions above are all equivalent.

Note that “(2) $\Rightarrow$ (3)” is known to experts, and the key step is to prove “(1) $\Rightarrow$ (2)”. The strategy follows from that for [1, Theorem 1.4], which is the standard approach originated in [21]. However, there are technical issues due to the lack of units in geometric ideals. To overcome this, we discover concrete approximate units in geometric ideals (see Lemma 2.12), which have close relations to the coarse geometry of the underlying spaces and the associated ideals. Hence we can approximate a geometric ideal by a net of uniform Roe algebras of subspaces, and therefore manage to consult the proof in the unital case.

At this point, we would also like to highlight that Theorem E is not just an easy combination of results in [3, Section 4] and [1]. Recall that in [3], the rigidity problem was studied for general coarse spaces. Note that there is an extra hypothesis that coarse spaces are small in the sense of [3, Definition 4.2] to prove that rigid isomorphisms induce coarse equivalences. However in the current setting, coarse structures associated to geometric ideals might not be small in general. This obstructs us from directly using the techniques from [3, Section 4]. To overcome the issue, again we make use of the concrete approximate units to reduce to the unital case.

Also note that in a recent work [24], Wang and the second-named author introduced a notion of ghostly ideals and studied the ideal structure of uniform Roe algebras beyond the scope of Property A. We provide some discussions on the rigidity problems for ghostly ideals and pose some open questions.

The paper is organised as follows. In Section 2, we recall necessary background knowledge in coarse geometry and uniform Roe algebras. In Section 3, we study the notion of coarse equivalence for general coarse spaces (Definition C) and prove Proposition D. Section 4 to Section 6 are devoted to the proof for Theorem E. We record the proof for the easy direction “(2) $\Rightarrow$ (3)” in Section 4 and prove “(1) $\Rightarrow$ (2)” in Section 6. To make the argument in Section 6 more transparent, we prove a weak version of “(1) $\Rightarrow$ (2)” in Section 5. Finally, we provide some discussion on ghostly ideals in Section 7.

Acknowledgement

This project began during BJ’s visit to the School of Mathematical Sciences at Fudan University. BJ would like to express his gratitude to Prof. Xiaoman Chen and Yijun Yao for the invitation and coordination, and also to the faculty members for their hospitality and engaging discussions during his stay. We would also like to thank Rufus Willett for his comments after reading an early draft of this paper.

BJ was supported by NSFC12001066 and NSFC12071183. JZ was partly supported by National Key R&D Program of China 2022YFA100700.

2. Preliminaries

2.1. Standard notation

Here we collect the notation used throughout the paper.

For a set $X$ , denote $|X|$ the cardinality of $X$ and $\mathcal{P}(X)$ the set of all subsets in $X$ . For $A\subseteq X$ , denote $\chi_{A}$ the characteristic function of $A$ , and set $\delta_{x}:=\chi_{\{x\}}$ for $x\in X$ .

For a discrete space $X$ , denote $\ell^{\infty}(X)$ the $C^{*}$ -algebra of bounded functions on $X$ with the supremum norm $\|f\|_{\infty}:=\sup_{x\in X}|f(x)|$ . The support of $f\in\ell^{\infty}(X)$ is defined to be $\{x\in X~{}|~{}f(x)\neq 0\}$ , denoted by $\mathrm{supp}(f)$ . Given a Hilbert space $\mathcal{H}$ , denote $\ell^{2}(X;\mathcal{H})$ the Hilbert space of square-summable functions from $X$ to $\mathcal{H}$ . When $\mathcal{H}=\mathbb{C}$ , we denote $\ell^{2}(X):=\ell^{2}(X;\mathbb{C})$ , which has an orthonormal basis $\{\delta_{x}\}_{x\in X}$ .

Given a Hilbert space $\mathcal{H}$ , denote $\mathfrak{B}(\mathcal{H})$ the $C^{*}$ -algebra of all bounded linear operators on $\mathcal{H}$ , and $\mathfrak{K}(\mathcal{H})$ the $C^{*}$ -subalgebra of all compact operators on $\mathcal{H}$ .

2.2. Notions from coarse geometry

Here we collect necessary notions from coarse geometry, and guide readers to [13, 18] for more details.

First recall that for a set $X$ and $E,F\subseteq X\times X$ , denote

	$\displaystyle E^{-1}$	$\displaystyle:=\{(y,x)~{}\|~{}(x,y)\in E\},$
	$\displaystyle E\circ F$	$\displaystyle:=\{(x,z)~{}\|~{}\exists~{}y\in X\text{ such that }(x,y)\in E\text{ and }(y,z)\in F\}.$

Also denote $\mathrm{r},\mathrm{s}:X\times X\to X$ the projection onto the first and the second coordinate, respectively. Given $E\subseteq X\times X$ and $A,B\subseteq X$ , we say that $A$ and $B$ are $E$ -separated if $(A\times B)\cap E=\emptyset$ and $(B\times A)\cap E=\emptyset$ .

Definition 2.1.

A (connected) coarse structure on a set $X$ is a collection $\mathscr{E}\subseteq\mathcal{P}(X\times X)$ , called entourages, satisfying the following:

(1)

For any entourages $A$ and $B$ , then $A^{-1}$ , $A\circ B$ and $A\cup B$ are entourages;
(2)

Every finite subset of $X\times X$ is an entourage;
(3)

Any subset of an entourage is an entourage.

In this case, $(X,\mathscr{E})$ is called a coarse space. If additionally the diagonal

\Delta_{X}:=\{(x,x)~{}|~{}x\in X\}

is an entourage, then $\mathscr{E}$ (also the pair $(X,\mathscr{E})$ ) is called unital.

For $Y\subseteq X$ and $E\in\mathscr{E}$ , denote the $E$ -neighbourhood $\mathcal{N}_{E}(Y)$ of $Y$ by

\mathcal{N}_{E}(Y):=Y\cup\{x\in X~{}|~{}\exists~{}y\in Y\text{ such that }(x,y)\in E\}

and

n(E):=\sup_{x\in X}\mathrm{max}\{~{}|\mathcal{N}_{E}(\{x\})|,|\mathcal{N}_{E^{-1}}(\{x\})|~{}\}.

Definition 2.2.

A coarse structure $\mathscr{E}$ on a set $X$ is said to have bounded geometry (or to be uniformly locally finite) if $n(E)$ is finite for any entourage $E\in\mathscr{E}$ . In this case, we also say that the coarse space $(X,\mathscr{E})$ has bounded geometry.

For a set $X$ and a collection $\mathcal{S}\subseteq\mathcal{P}(X\times X)$ , the smallest coarse structure on $X$ containing $\mathcal{S}$ is called the coarse structure generated by $\mathcal{S}$ .

When $(X,d)$ is a discrete metric space, there is an associated unital coarse structure $\mathscr{E}_{d}$ (called the bounded coarse structure) generated by all the $R$ -entourages defined as $E_{R}:=\{(x,y)\in X\times X~{}|~{}d(x,y)\leq R\}$ for all $R\geq 0$ . In this case, we denote the closed ball by $B_{X}(x,r):=\mathcal{N}_{E_{r}}(\{x\})$ for $x\in X$ and $r\geq 0$ , and $\mathcal{N}_{r}(Y):=\mathcal{N}_{E_{r}}(Y)$ for $Y\subseteq X$ and $r\geq 0$ . We say that $(X,d)$ has bounded geometry if $\mathscr{E}_{d}$ has bounded geometry, i.e., the number $\sup_{x\in X}|B_{X}(x,r)|$ is finite for any $r\geq 0$ .

Definition 2.3.

Let $X,Y$ be sets, $f:X\to Y$ be a map and $\mathscr{E}_{Y}$ be a coarse structure on $Y$ .

(1)

Denote $f^{*}\mathscr{E}_{Y}:=\{E\subseteq X\times X~{}|~{}(f\times f)(E)\in\mathscr{E}_{Y}\}$ , which is a coarse structure on $X$ . Here $(f\times f)(E):=\{(f(x),f(y))~{}|~{}(x,y)\in E\}$ .
(2)

If $\mathscr{E}_{X}$ is a coarse structure on $X$ , we say that $f$ is coarse if $\mathscr{E}_{X}\subseteq f^{*}\mathscr{E}_{Y}$ . To specify the underlying coarse structures, we also write $f:(X,\mathscr{E}_{X})\to(Y,\mathscr{E}_{Y})$ .
(3)

Two coarse maps $f,g:(X,\mathscr{E}_{X})\to(Y,\mathscr{E}_{Y})$ are said to be close if for any $E\in\mathscr{E}_{X}$ , we have $(f\times g)(E):=\{(f(x),g(y))~{}|~{}(x,y)\in E\}\in\mathscr{E}_{Y}$ .

Definition 2.4.

Let $f:(X,\mathscr{E}_{X})\to(Y,\mathscr{E}_{Y})$ be a map between unital coarse spaces. We say that $f$ is a coarse equivalence if $f$ is coarse and there exists a coarse map $g:(Y,\mathscr{E}_{Y})\to(X,\mathscr{E}_{X})$ (called a coarse inverse to $f$ ) such that $f\circ g$ is close to $\mathrm{Id}_{Y}$ and $g\circ f$ is close to $\mathrm{Id}_{X}$ . In this case, we say that $(X,\mathscr{E}_{X})$ and $(Y,\mathscr{E}_{Y})$ are coarsely equivalent.

When $(X,\mathscr{E}_{X})$ and $(Y,\mathscr{E}_{Y})$ come from metric spaces, Definition 2.4 coincides with the one recalled in Section 1. Note that Definition 2.4 also makes sense for the general (non-unital) case, however, it does not work well for the rigidity problems (see Proposition D and Theorem E). It turns out that the suitable setting for the notion of coarse equivalence in the general case is Definition C. To obtain an appropriate picture, we will focus on Definition C in Section 3 based on an alternative version of coarse maps from [19].

2.3. Uniform Roe algebras and geometric ideals

Let $X$ be a set. Each operator $T\in\mathfrak{B}(\ell^{2}(X))$ can be written in the matrix form $T=(T(x,y))_{x,y\in X}$ , where $T(x,y)=\langle T\delta_{y},\delta_{x}\rangle\in\mathbb{C}$ . Denote by $\|T\|$ the operator norm of $T$ in $\mathfrak{B}(\ell^{2}(X))$ . Similarly for an operator $T\in\mathfrak{B}(\ell^{2}(X;\mathcal{H}))$ , we can also write $T=(T(x,y))_{x,y\in X}$ for $T(x,y)\in\mathfrak{B}(\mathcal{H})$ .

Given an operator $T\in\mathfrak{B}(\ell^{2}(X))$ , we define the support of $T$ to be

\mathrm{supp}(T):=\{(x,y)\in X\times X~{}|~{}T(x,y)\neq 0\}.

Given $\varepsilon>0$ , define the $\varepsilon$ -support of $T$ to be

\mathrm{supp}_{\varepsilon}(T):=\big{\{}(x,y)\in X\times X~{}\big{|}~{}|T(x,y)|\geq\varepsilon\big{\}}.

When $X$ is equipped with a metric $d$ , we define the propagation of $T$ to be

\mathrm{prop}(T):=\sup\{d(x,y)~{}|~{}(x,y)\in\mathrm{supp}(T)\}.

Definition 2.5.

Let $(X,\mathscr{E})$ be a coarse space of bounded geometry. The set of all operators in $\mathfrak{B}(\ell^{2}(X))$ whose supports belong to $\mathscr{E}$ forms a $\ast$ -algebra, called the algebraic uniform Roe algebra of $(X,\mathscr{E})$ and denoted by $\mathbb{C}_{u}[X,\mathscr{E}]$ . The uniform Roe algebra of $(X,\mathscr{E})$ is defined to be the operator norm closure of $\mathbb{C}_{u}[X,\mathscr{E}]$ in $\mathfrak{B}(\ell^{2}(X))$ , which forms a $C^{*}$ -algebra and is denoted by $C^{*}_{u}(X,\mathscr{E})$ .

When $(X,d)$ is a discrete metric space of bounded geometry, we simply write $\mathbb{C}_{u}[X]:=\mathbb{C}_{u}[X,\mathscr{E}_{d}]$ and $C^{*}_{u}(X):=C^{*}_{u}(X,\mathscr{E}_{d})$ .

For a coarse space $(X,\mathscr{E})$ of bounded geometry and $x,y\in X$ , denote the rank-one operator $\xi\mapsto\langle\xi,\delta_{y}\rangle\delta_{x}$ by $e_{xy}$ . It is clear that $e_{xy}\in\mathbb{C}_{u}[X,\mathscr{E}]$ . This kind of operators will play an important role when we study the rigidity problems later.

In [6, 23], Chen and Wang introduced the notion of geometric ideals (see Definition A) in uniform Roe algebras and provided a full description for these ideals. Here we only focus on the case of metric spaces, which are the main objects we are interested in for the rigidity problem. (Note that geometric ideals for general coarse spaces were also studied in [6].) In the following, we always assume that $(X,d)$ is a discrete metric space of bounded geometry and $\mathscr{E}_{d}$ is the associated bounded coarse structure.

Definition 2.6 ([6, Definition 4.1]).

A coarse structure $\mathscr{I}\subseteq\mathscr{E}_{d}$ is called an ideal in $\mathscr{E}_{d}$ if for any $E\in\mathscr{E}_{d}$ and $A\in\mathscr{I}$ , we have $E\circ A\in\mathscr{I}$ and $A\circ E\in\mathscr{I}$ .

Proposition 2.7 ([6, Proposition 4.2]).

Given an ideal $\mathscr{I}$ in $\mathscr{E}_{d}$ , the uniform Roe algebra $C^{*}_{u}(X,\mathscr{I})$ is a geometric ideal in $C^{*}_{u}(X)$ . Conversely, given a geometric ideal $I$ in $C^{*}_{u}(X)$ , the collection $\mathscr{I}(I):=\{\mathrm{supp}_{\varepsilon}(T)~{}|~{}T\in I,\varepsilon>0\}$ is an ideal in $\mathscr{E}_{d}$ . Moreover, we have $\mathscr{I}(C^{*}_{u}(X,\mathscr{I}))=\mathscr{I}$ and $C^{*}_{u}(X,\mathscr{I}(I))=I$ .

We also need the following notion of ideals in space:

Definition 2.8 ([6, Definition 6.1]).

An ideal in $(X,d)$ is a collection $\mathbf{L}\subseteq\mathcal{P}(X)$ satisfying the following:

(1)

If $Y\in\mathbf{L}$ and $Z\subseteq Y$ , then $Z\in\mathbf{L}$ ;
(2)

If $Y\in\mathbf{L}$ and $r>0$ , then $\mathcal{N}_{r}(Y)\in\mathbf{L}$ ;
(3)

If $Y,Z\in\mathbf{L}$ , then $Y\cup Z\in\mathbf{L}$ .

For $\mathcal{S}\subseteq\mathcal{P}(X)$ , we say that $\mathbf{L}$ is the ideal generated by $\mathcal{S}$ if $\mathbf{L}$ is the smallest ideal in $(X,d)$ containing $\mathcal{S}$ .

Proposition 2.9 ([6, Proposition 6.2]).

Given an ideal $\mathscr{I}$ in $\mathscr{E}_{d}$ , the collection $\mathbf{L}(\mathscr{I}):=\{\mathrm{r}(E)~{}|~{}E\in\mathscr{I}\}$ is an ideal in $(X,d)$ . Conversely, given an ideal $\mathbf{L}$ in $(X,d)$ , the collection $\mathscr{I}(\mathbf{L}):=\{E\in\mathscr{E}_{d}~{}|~{}\exists~{}L\in\mathbf{L}\text{ such that }E\subseteq L\times L\}$ is an ideal in $\mathscr{E}_{d}$ . Moreover, we have $\mathscr{I}(\mathbf{L}(\mathscr{I}))=\mathscr{I}$ and $\mathbf{L}(\mathscr{I}(\mathbf{L}))=\mathbf{L}$ .

Combining Proposition 2.7 and Proposition 2.9, we know that the lattice of geometric ideals in $C^{*}_{u}(X)$ is isomorphic to the lattice of ideals in $\mathscr{E}_{d}$ , which is also isomorphic to the lattice of ideals in $(X,d)$ . Hence we will drift among these three objects freely in the sequel.

Example 2.10.

As a special case, we consider the ideal $\mathfrak{K}(\ell^{2}(X))$ of compact operators in the uniform Roe algebra $C^{*}_{u}(X)$ for a discrete metric space $(X,d)$ . Note that $\mathfrak{K}(\ell^{2}(X))$ is a geometric ideal, and it is easy to see that $\mathscr{I}(\mathfrak{K}(\ell^{2}(X)))$ consists of all finite subsets of $X\times X$ , and $\mathbf{L}(\mathfrak{K}(\ell^{2}(X)))$ consists of all finite subsets of $X$ .

Remark 2.11.

We remark that the definition $\mathscr{I}(I)$ works for any (not necessarily geometric) ideal (see [6]) in the uniform Roe algebra $C^{*}_{u}(X)$ . However, the one-to-one correspondence in Proposition 2.7 only works for geometric ideals. As an easy example, one can consider the ideal $I_{G}$ consisting of all ghost operators in $C^{*}_{u}(X)$ . (Recall that an operator $T\in\mathfrak{B}(\ell^{2}(X))$ is a ghost operator if for any $\varepsilon>0$ , there exists a finite subset $F\subseteq X$ such that for any $(x,y)\notin F\times F$ , then $|T(x,y)|<\varepsilon$ .) One can calculate directly that $\mathscr{I}(I_{G})=\mathscr{I}(\mathfrak{K}(\ell^{2}(X)))$ , which consists of all finite subsets in $X\times X$ .

Finally, we record an elementary observation which will be used later.

Lemma 2.12.

Let $\mathbf{L}$ be an ideal in $(X,d)$ . Then the family $\{\chi_{L}~{}|~{}L\in\mathbf{L}\}$ is an approximate unit for the ideal $C^{*}_{u}(X,\mathscr{I}(\mathbf{L}))$ .

Proof.

Note that $C^{*}_{u}(X,\mathscr{I}(\mathbf{L}))$ is the direct limit of $\{C^{*}_{u}(L)~{}|~{}L\in\mathbf{L}\}$ where $C^{*}_{u}(L)$ is the uniform Roe algebra of the metric space $L$ with the induced metric of $d$ and $\mathbf{L}$ is a direct set under inclusion. Since $\chi_{L}$ is the unit of $C^{*}_{u}(L)$ , we conclude the proof. ∎

3. Coarse correspondence

In this section, we study the notion of coarse correspondence (Definition C) from [19] and provide a detailed picture for coarse equivalences between general coarse spaces (Proposition D).

Let $X$ be a set and $\mathscr{E}\subseteq\mathcal{P}(X\times X)$ . For $A,B\subseteq X$ , denote the restriction of $\mathscr{E}$ on $A\times B$ by

\mathscr{E}|_{A\times B}:=\{E\cap(A\times B)~{}|~{}E\in\mathscr{E}\}.

Recall from Definition C that for coarse spaces $(X,\mathscr{E}_{X})$ and $(Y,\mathscr{E}_{Y})$ , a coarse correspondence from $(X,\mathscr{E}_{X})$ to $(Y,\mathscr{E}_{Y})$ (or simply from $X$ to $Y$ ) is a coarse structure $\mathscr{E}$ on the disjoint union $X\sqcup Y$ satisfying the following:

(1)

$\mathscr{E}|_{Y\times Y}=\mathscr{E}_{Y}$ ;
(2)

$\mathscr{E}|_{X\times X}\supseteq\mathscr{E}_{X}$ ;
(3)

$\mathscr{E}$ is generated by $\mathscr{E}|_{Y\times(X\sqcup Y)}$ .

The following fact was implicitly stated in the proof of [19, Proposition 2.3]. It follows directly from condition (3) above, and hence we omit the proof.

Lemma 3.1.

Let $(X,\mathscr{E}_{X})$ and $(Y,\mathscr{E}_{Y})$ be coarse spaces, and $\mathscr{E}$ be a coarse correspondence from $(X,\mathscr{E}_{X})$ to $(Y,\mathscr{E}_{Y})$ . Then for any $E\in\mathscr{E}|_{X\times X}$ , there exists $F\in\mathscr{E}|_{Y\times X}$ such that $E\subseteq F^{-1}\circ F$ .

We also record the following elementary fact, whose proof is straightforward.

Lemma 3.2.

For a set $X$ and $E\subseteq X\times X$ , we have $E\subseteq(E^{\prime})^{-1}\circ E^{\prime}$ for $E^{\prime}:=E\cup(E\circ E^{-1})$ .

3.1. The unital case

It was shown in [19] that for unital coarse spaces, the notion of coarse correspondence can be identified with coarse maps in Definition 2.3(2). Here we recall the outline of the proof, and divide it into several pieces to clarify the dependence on the assumption of unitalness. Some of the results will also be used in the general case.

Let $X$ be a set, $(Y,\mathscr{E}_{Y})$ be a coarse space and $f:X\to Y$ be a map. Denote

\mathscr{E}_{YX}(f):=\{E\subseteq Y\times X~{}|~{}(\mathrm{Id}_{Y}\times f)(E)\in\mathscr{E}_{Y}\},

and

\mathscr{E}(f):=\{E_{X}\cup E_{Y}\cup E_{XY}\cup E_{YX}~{}|~{}E_{X}\in f^{*}\mathscr{E}_{Y},E_{Y}\in\mathscr{E}_{Y}\text{ and }E_{YX}\cup E_{XY}^{-1}\in\mathscr{E}_{YX}(f)\}.

The following result is contained in the proof of [19, Proposition 2.3]. For the reader’s convenience, here we recall the proof.

Lemma 3.3.

Let $X$ be a set, $(Y,\mathscr{E}_{Y})$ be a coarse space and $f:X\to Y$ be a map. Then $\mathscr{E}(f)$ is a coarse structure on $X\sqcup Y$ and generated by $\mathscr{E}(f)|_{Y\times(X\sqcup Y)}=\mathscr{E}_{Y}\cup\mathscr{E}_{YX}(f)$ .

Proof.

Consider the map $h:X\sqcup Y\to Y$ which restricts to $f$ on $X$ and $\mathrm{Id}_{Y}$ on $Y$ . It is easy to see that $\mathscr{E}(f)=h^{*}\mathscr{E}_{Y}$ , and hence $\mathscr{E}(f)$ is a coarse structure on $X\sqcup Y$ . For the second statement, it suffices to show that $\mathscr{E}(f)|_{X\times X}=f^{*}\mathscr{E}_{Y}$ can be generated by $\mathscr{E}(f)|_{Y\times X}=\mathscr{E}_{YX}(f)$ . Given $E\in f^{*}\mathscr{E}_{Y}$ , it follows from Lemma 3.2 that $E\subseteq(E^{\prime})^{-1}\circ E^{\prime}$ for $E^{\prime}:=E\cup(E\circ E^{-1})$ . It is clear that $E^{\prime}\in f^{*}\mathscr{E}_{Y}$ , and hence $F:=(f\times\mathrm{Id}_{X})(E^{\prime})\in\mathscr{E}_{YX}(f)$ . Finally, note that $E\subseteq(E^{\prime})^{-1}\circ E^{\prime}\subseteq F^{-1}\circ F$ , which concludes the proof. ∎

Consequently, we have the following:

Corollary 3.4.

Let $(X,\mathscr{E}_{X})$ and $(Y,\mathscr{E}_{Y})$ be coarse spaces, and $f:X\to Y$ be a map. Then $f$ is coarse if and only if $\mathscr{E}(f)$ is a coarse correspondence. In this case, we say that $\mathscr{E}(f)$ is the coarse correspondence associated to $f$ .

Recall that for a map $f:X\to Y$ , the graph of $f$ is defined to be

\mathrm{Gr}(f):=\{(f(x),x)\in Y\times X~{}|~{}x\in X\}.

We have the following:

Lemma 3.5.

Let $X$ be a set, $(Y,\mathscr{E}_{Y})$ be a coarse space and $f:X\to Y$ be a map. Assume that the set $\{(f(x),f(x))\in Y\times Y~{}|~{}x\in X\}$ belongs to $\mathscr{E}_{Y}$ . Then $\mathscr{E}(f)$ is generated by $\mathscr{E}_{Y}$ and $\mathrm{Gr}(f)$ .

Proof.

Since $\{(f(x),f(x))\in Y\times Y~{}|~{}x\in X\}\in\mathscr{E}_{Y}$ , we have $\mathrm{Gr}(f)\in\mathscr{E}_{YX}(f)$ . On the other hand, given $E\in\mathscr{E}_{YX}(f)$ , we have $E\subseteq(E\circ\mathrm{Gr}(f)^{-1})\circ\mathrm{Gr}(f)$ . Since $E\circ\mathrm{Gr}(f)^{-1}=(\mathrm{Id}_{Y}\times f)(E)\in\mathscr{E}_{Y}$ , we conclude the proof thanks to Lemma 3.3. ∎

Now we present the following result from [19] that for unital coarse spaces, each coarse correspondence is determined by a coarse map. For the reader’s convenience, here we recall the proof.

Proposition 3.6 ([19, Proposition 2.3]).

Let $(X,\mathscr{E}_{X})$ and $(Y,\mathscr{E}_{Y})$ be coarse spaces, and $\mathscr{E}$ be a coarse correspondence. Assume that $\mathscr{E}_{X}$ is unital. Then there exists a unique (up to closeness) coarse map $f:X\to Y$ such that $\mathscr{E}=\mathscr{E}(f)$ .

Proof.

By Lemma 3.1 and the assumption that $\mathscr{E}_{X}$ is unital, there exists $E\in\mathscr{E}|_{Y\times X}$ such that $\Delta_{X}\subseteq E^{-1}\circ E$ . It is easy to see that $E$ contains $\mathrm{Gr}(f)$ for some map $f:X\to Y$ . Note that $\{(f(x),f(x))\in Y\times Y~{}|~{}x\in X\}\subseteq\mathrm{Gr}(f)\circ\mathrm{Gr}(f)^{-1}\subseteq E\circ E^{-1}\in\mathscr{E}_{Y}$ . Hence it follows from Lemma 3.5 that $\mathscr{E}(f)\subseteq\mathscr{E}$ .

On the other hand, given $E\in\mathscr{E}|_{Y\times X}$ , we have $E\subseteq(E\circ\mathrm{Gr}(f)^{-1})\circ\mathrm{Gr}(f)$ . Since $E\circ\mathrm{Gr}(f)^{-1}\in\mathscr{E}|_{Y\times Y}=\mathscr{E}_{Y}$ , then we obtain $E\in\mathscr{E}(f)$ . Hence $\mathscr{E}=\mathscr{E}(f)$ , which implies that $f$ is coarse thanks to Corollary 3.4.

Finally, if $\mathscr{E}(f)=\mathscr{E}(g)$ for coarse maps $f,g:X\to Y$ , then $\mathrm{Gr}(f)\circ\mathrm{Gr}(g)^{-1}\in\mathscr{E}_{Y}$ . Hence $f$ and $g$ are close. ∎

The following result shows that the notion of coarse equivalence from Definition C coincides with the one from Definition 2.4 for unital coarse spaces.

Corollary 3.7.

Let $(X,\mathscr{E}_{X})$ and $(Y,\mathscr{E}_{Y})$ be unital coarse spaces.

(1)

Let $f:X\to Y$ be a coarse equivalence with a coarse inverse $g:Y\to X$ . Then we have $\mathscr{E}(f)=\mathscr{E}(g)$ , which is a coarse correspondence both from $X$ to $Y$ and from $Y$ to $X$ . In this case, we have $\mathscr{E}(f)|_{X\times X}=\mathscr{E}_{X}$ .
(2)

Let $\mathscr{E}$ be a coarse correspondence from $X$ to $Y$ and $Y$ to $X$ . Let $f:X\to Y$ be a coarse map from Proposition 3.6 such that $\mathscr{E}=\mathscr{E}(f)$ , and similarly $g:Y\to X$ be a coarse map such that $\mathscr{E}=\mathscr{E}(g)$ . Then $f$ is a coarse equivalence with a coarse inverse $g$ .

Proof.

(1) is straightforward, and hence we omit the proof. For (2), it suffices to note that $\{(f\circ g(y),y)~{}|~{}y\in Y\}\subseteq\mathrm{Gr}(f)\circ\mathrm{Gr}(g)\in\mathscr{E}|_{Y\times Y}=\mathscr{E}_{Y}$ and similarly, $\{(g\circ f(x),x)~{}|~{}x\in X\}=\mathrm{Gr}(g)\circ\mathrm{Gr}(f)\in\mathscr{E}|_{X\times X}=\mathscr{E}_{X}$ . Hence we obtain the result. ∎

3.2. The general case

Now we move to the general case. Firstly, let us introduce the following notion:

Definition 3.8.

Let $X$ be a set and $\mathbf{L}\subseteq\mathcal{P}(X)$ . We say that $\mathbf{L}$ is admissible if the following holds:

(1)

Every finite subset of $X$ belongs to $\mathbf{L}$ ;
(2)

For any $Y\in\mathbf{L}$ and $Z\subseteq Y$ , then $Z\in\mathbf{L}$ ;
(3)

For any $Y,Z\in\mathbf{L}$ , then $Y\cup Z\in\mathbf{L}$ .

The example of admissible collection we are interested in comes from coarse structures:

Definition 3.9.

Let $(X,\mathscr{E}_{X})$ be a coarse space. The collection

\mathbf{L}(\mathscr{E}_{X}):=\{\mathrm{r}(E)~{}|~{}E\in\mathscr{E}_{X}\}=\{\mathrm{s}(E)~{}|~{}E\in\mathscr{E}_{X}\}

is called associated to $(X,\mathscr{E}_{X})$ , also denoted by $\mathbf{L}_{X}$ , if the coarse structure is clear from the context.

Clearly $\mathbf{L}(\mathscr{E}_{X})$ associated to a coarse space $(X,\mathscr{E}_{X})$ is always admissible. When $(X,d)$ is a discrete metric space and $\mathscr{I}\subseteq\mathscr{E}_{d}$ is an ideal (in the sense of Definition 2.6), the definition for $\mathbf{L}(\mathscr{I})$ above coincides with the one in Proposition 2.9. In particular, any ideal $\mathbf{L}$ in $(X,d)$ (in the sense of Definition 2.8) is admissible.

The following is straightforward, and hence we omit the proof.

Lemma 3.10.

Let $(X,\mathscr{E}_{X})$ be a coarse space, and $\mathbf{L}_{X}$ be the associated collection. For $L\subseteq X$ , we have that $L\in\mathbf{L}_{X}$ if and only if the set $\Delta_{L}=\{(x,x)~{}|~{}x\in L\}$ belongs to $\mathscr{E}_{X}$ .

To study the non-unital case, we need to consider a family of maps instead of a single map:

Definition 3.11.

Let $X$ be a set, $\mathbf{L}$ be an admissible collection in $\mathcal{P}(X)$ and $(Y,\mathscr{E}_{Y})$ be a coarse space.

(1)

We say that a family of maps $\mathcal{F}=\{f_{L}:L\to Y~{}|~{}L\in\mathbf{L}\}$ is admissible if for any $L_{1},L_{2}\in\mathbf{L}$ , the set $\{(f_{L_{1}}(x),f_{L_{2}}(x))~{}|~{}x\in L_{1}\cap L_{2}\}$ belongs to $\mathscr{E}_{Y}$ .

(2)

For an admissible family $\mathcal{F}=\{f_{L}:L\to Y~{}|~{}L\in\mathbf{L}\}$ , denote

\mathcal{F}^{*}\mathscr{E}_{Y}:=\{E\in X\times X~{}|~{}\exists~{}L\in\mathbf{L}\text{ such that }E\subseteq L\times L\text{ and }(f_{L}\times f_{L})(E)\in\mathscr{E}_{Y}\}.

We also need to consider the relation of closeness between families:

Definition 3.12.

Let $X$ be a set, $\mathbf{L}$ be an admissible collection in $\mathcal{P}(X)$ and $(Y,\mathscr{E}_{Y})$ be a coarse space. Two admissible families $\mathcal{F}=\{f_{L}:L\to Y~{}|~{}L\in\mathbf{L}\}$ and $\mathcal{G}=\{g_{L}:L\to Y~{}|~{}L\in\mathbf{L}\}$ are called close if for any $L\in\mathbf{L}$ , the set $\{(f_{L}(x),g_{L}(x))~{}|~{}x\in L\}$ belongs to $\mathscr{E}_{Y}$ .

Lemma 3.13.

Let $X$ be a set and $\mathbf{L}$ be an admissible collection in $\mathcal{P}(X)$ . Let $(Y,\mathscr{E}_{Y})$ be a coarse space, and $\mathcal{F}=\{f_{L}:L\to Y~{}|~{}L\in\mathbf{L}\}$ be an admissible family. Then

(1)

For any $L\in\mathbf{L}$ , we have $f_{L}(L)\in\mathbf{L}_{Y}$ .
(2)

$\mathcal{F}^{*}\mathscr{E}_{Y}$ is a coarse structure on $X$ .

Proof.

(1). Since $\mathbf{L}$ is admissible, the set $\{(f_{L}(x),f_{L}(x))~{}|~{}x\in L\}\in\mathscr{E}_{Y}$ for any $L\in\mathbf{L}$ . Hence we obtain $f_{L}(L)\in\mathbf{L}_{Y}$ .

(2). Given $E_{1},E_{2}\in\mathcal{F}^{*}\mathscr{E}_{Y}$ , assume that there exist $L_{1},L_{2}\in\mathbf{L}$ such that $E_{i}\subseteq L_{i}\times L_{i}$ and $(f_{L_{i}}\times f_{L_{i}})(E_{i})\in\mathscr{E}_{Y}$ for $i=1,2$ . Take $L=L_{1}\cup L_{2}\in\mathbf{L}$ , then we have $E_{1}\cup E_{2}\subseteq L\times L$ and $E_{1}\circ E_{2}\subseteq L\times L$ . Note that $F_{i}:=\{(f_{L}(x),f_{L_{i}}(x))~{}|~{}x\in L_{i}\}\in\mathscr{E}_{Y}$ for $i=1,2$ . Hence we have

(f_{L}\times f_{L})(E_{i})\subseteq F_{i}\circ(f_{L_{i}}\times f_{L_{i}})(E_{i})\circ F_{i}^{-1}\in\mathscr{E}_{Y}

for $i=1,2$ , which implies that $(f_{L}\times f_{L})(E_{1}\cup E_{2})\in\mathscr{E}_{Y}$ . Also note that

(f_{L}\times f_{L})(E_{1}\circ E_{2})\subseteq(f_{L}\times f_{L})(E_{1})\circ(f_{L}\times f_{L})(E_{2})\in\mathscr{E}_{Y}.

The rest is trivial, and hence we omit the details. ∎

Now we introduce the notion of coarse family, which is the replacement of coarse maps in the general case.

Definition 3.14.

Let $(X,\mathscr{E}_{X}),(Y,\mathscr{E}_{Y})$ be coarse spaces and $\mathbf{L}$ be an admissible collection in $\mathcal{P}(X)$ . A family of admissible maps $\mathcal{F}=\{f_{L}:L\to Y~{}|~{}L\in\mathbf{L}\}$ is called coarse if $\mathscr{E}_{X}\subseteq\mathcal{F}^{*}\mathscr{E}_{Y}$ .

It is clear from the definition that for a coarse family $\mathcal{F}=\{f_{L}:L\to Y~{}|~{}L\in\mathbf{L}\}$ , we have $\mathbf{L}_{X}\subseteq\mathbf{L}$ . Note that we do not require $\mathbf{L}_{X}=\mathbf{L}$ in general.

Definition 3.15.

Let $X$ be a set and $\mathbf{L}$ be an admissible collection in $\mathcal{P}(X)$ . Let $(Y,\mathscr{E}_{Y})$ be a coarse space and $\mathcal{F}=\{f_{L}:L\to Y~{}|~{}L\in\mathbf{L}\}$ be an admissible family. Denote

\mathscr{E}_{YX}(\mathcal{F}):=\{E\subseteq Y\times X~{}|~{}\exists~{}L\in\mathbf{L}\text{ such that }E\subseteq Y\times L\text{ and }(\mathrm{Id}_{Y}\times f_{L})(E)\in\mathscr{E}_{Y}\},

and

\mathscr{E}(\mathcal{F}):=\{E_{X}\cup E_{Y}\cup E_{XY}\cup E_{YX}~{}|~{}E_{X}\in\mathcal{F}^{*}\mathscr{E}_{Y},E_{Y}\in\mathscr{E}_{Y}\text{ and }E_{YX}\cup E_{XY}^{-1}\in\mathscr{E}_{YX}(\mathcal{F})\}.

We record the following elementary observation, whose proof is straightforward and hence omitted.

Lemma 3.16.

Given $X,\mathbf{L},(Y,\mathscr{E}_{Y})$ and $\mathcal{F}$ as in Definition 3.15, we have the following:

(1)

For any $E\in\mathscr{E}_{YX}(\mathcal{F})$ , there exist $L\in\mathbf{L}$ and $B\in\mathbf{L}_{Y}$ such that $E\subseteq B\times L$ .
(2)

For $L_{1},L_{2}\in\mathbf{L}$ and $E\subseteq Y\times X$ , if $E\subseteq Y\times L_{i}$ for $i=1,2$ and $(\mathrm{Id}_{Y}\times f_{L_{1}})(E)\in\mathscr{E}_{Y}$ , then $(\mathrm{Id}_{Y}\times f_{L_{2}})(E)\in\mathscr{E}_{Y}$ .

The following result is analogous to Lemma 3.3.

Lemma 3.17.

Given $X,\mathbf{L},(Y,\mathscr{E}_{Y})$ and $\mathcal{F}$ as in Definition 3.15, then $\mathscr{E}(\mathcal{F})$ is a coarse structure on $X\sqcup Y$ generated by $\mathscr{E}(\mathcal{F})|_{Y\times(X\sqcup Y)}=\mathscr{E}_{Y}\cup\mathscr{E}_{YX}(\mathcal{F})$ .

Proof.

Firstly, we show that $\mathscr{E}(\mathcal{F})$ is a coarse structure on $X\sqcup Y$ . Consider the admissible collection $\mathbf{L}^{\prime}:=\{A\sqcup B~{}|~{}A\in\mathbf{L}\text{ and }B\in\mathbf{L}_{Y}\}$ in $\mathcal{P}(X\sqcup Y)$ . For any $L=A\sqcup B\in\mathbf{L}^{\prime}$ with $A\in\mathbf{L}$ and $B\in\mathbf{L}_{Y}$ , define a map $h_{L}:L\to Y$ which restricts to $f_{A}$ on $A$ and $\mathrm{Id}_{B}$ on $B$ . Given $L_{i}=A_{i}\sqcup B_{i}$ with $A_{i}\in\mathbf{L}$ and $B_{i}\in\mathbf{L}_{Y}$ for $i=1,2$ , we have

\{(h_{L_{1}}(z),h_{L_{2}}(z))~{}|~{}z\in L_{1}\cap L_{2}\}=\{(f_{A_{1}}(x),f_{A_{2}}(x))~{}|~{}x\in A_{1}\cap A_{2}\}\cup\Delta_{B_{1}\cap B_{2}},

which belongs to $\mathscr{E}_{Y}$ due to Lemma 3.10. Hence $\mathcal{H}:=\{h_{L}~{}|~{}L\in\mathbf{L}^{\prime}\}$ is admissible.

Moreover, $E=E_{X}\cup E_{Y}\cup E_{XY}\cup E_{YX}$ belongs to $\mathcal{H}^{*}\mathscr{E}_{Y}$ if and only if there exists $L=A\sqcup B\in\mathbf{L}^{\prime}$ with $A\in\mathbf{L}$ and $B\in\mathbf{L}_{Y}$ such that $E\subseteq L\times L$ and $(h_{L}\times h_{L})(E)\in\mathscr{E}_{Y}$ , which is equivalent to that $E\in\mathscr{E}(\mathcal{F})$ thanks to Lemma 3.16. Hence we conclude that $\mathscr{E}(\mathcal{F})=\mathcal{H}^{*}\mathscr{E}_{Y}$ , which is a coarse structure on $X\sqcup Y$ due to Lemma 3.13(2).

The rest of the proof is similar to that for Lemma 3.3, and hence we omit the details. ∎

Consequently, we obtain the following:

Corollary 3.18.

Let $(X,\mathscr{E}_{X}),(Y,\mathscr{E}_{Y})$ be coarse spaces, $\mathbf{L}$ be an admissible collection in $\mathcal{P}(X)$ and $\mathcal{F}=\{f_{L}:L\to Y~{}|~{}L\in\mathbf{L}\}$ be a family of admissible maps. Then $\mathcal{F}$ is coarse if and only if $\mathscr{E}(\mathcal{F})$ is a coarse correspondence. In this case, we say that $\mathscr{E}(\mathcal{F})$ is the coarse correspondence associated to $\mathcal{F}$ .

Similar to Lemma 3.5, we have the following:

Lemma 3.19.

Given $X,\mathbf{L},(Y,\mathscr{E}_{Y})$ and $\mathcal{F}$ as in Definition 3.15, then $\mathscr{E}(\mathcal{F})$ is generated by $\mathscr{E}_{Y}$ and $\{\mathrm{Gr}(f_{L})~{}|~{}L\in\mathbf{L}\}$ .

Proof.

Note that for any $L\in\mathbf{L}$ , we have $\{(f_{L}(x),f_{L}(x))~{}|~{}x\in L\}\in\mathscr{E}_{Y}$ and hence $\mathrm{Gr}(f_{L})\in\mathscr{E}_{YX}(\mathcal{F})$ . Now given $E\in\mathscr{E}_{YX}(\mathcal{F})$ , there exists $L\in\mathbf{L}$ such that $E\subseteq Y\times L$ and $(\mathrm{Id}_{Y}\times f_{L})(E)\in\mathscr{E}_{Y}$ . Hence we have $E\subseteq(E\circ\mathrm{Gr}(f_{L})^{-1})\circ\mathrm{Gr}(f_{L})$ , where $E\circ\mathrm{Gr}(f_{L})^{-1}\in\mathscr{E}_{Y}$ . Therefore we conclude the proof thanks to Lemma 3.17. ∎

Now we are in the position to provide a detailed picture for coarse correspondence in the general case. This is a crucial step to achieve Proposition D.

Proposition 3.20.

Let $(X,\mathscr{E}_{X}),(Y,\mathscr{E}_{Y})$ be coarse spaces, and $\mathscr{E}$ be a coarse correspondence from $X$ to $Y$ . Then there exist an admissible collection $\mathbf{L}\subseteq\mathcal{P}(X)$ and an admissible family of maps $\mathcal{F}=\{f_{L}:L\to Y~{}|~{}L\in\mathbf{L}\}$ such that $\mathscr{E}=\mathscr{E}(\mathcal{F})$ . Moreover, such a family $\mathcal{F}$ is unique up to closeness.

Proof.

Set $\mathbf{L}:=\{\mathrm{r}(E)~{}|~{}E\in\mathscr{E}|_{X\times X}\}$ , which is clearly admissible. For any $L\in\mathbf{L}$ , Lemma 3.10 shows that $\Delta_{L}\in\mathscr{E}_{Y}$ . By Lemma 3.1, there exists $E_{L}\in\mathscr{E}|_{Y\times X}$ such that $\Delta_{L}\subseteq E_{L}^{-1}\circ E_{L}$ . Hence there exists a map $f_{L}:L\to Y$ such that $E_{L}\supseteq\mathrm{Gr}(f_{L})$ . For $L_{1},L_{2}\in\mathbf{L}$ , we have

\{(f_{L_{1}}(x),f_{L_{2}}(x))~{}|~{}x\in L_{1}\cap L_{2}\}\subseteq\mathrm{Gr}(f_{L_{1}})\circ\mathrm{Gr}(f_{L_{2}})^{-1}\subseteq E_{L_{1}}\circ E_{L_{2}}^{-1}\in\mathscr{E}_{Y},

which implies that the family $\mathcal{F}=\{f_{L}:L\to Y~{}|~{}L\in\mathbf{L}\}$ is admissible. Applying Lemma 3.19, we obtain that $\mathscr{E}(\mathcal{F})\subseteq\mathscr{E}$ .

Conversely, we claim that $\mathbf{L}=\{\mathrm{s}(F)~{}|~{}F\in\mathscr{E}|_{Y\times X}\}$ . In fact, for any $F\in\mathscr{E}|_{Y\times X}$ , note that $F^{-1}\circ F\in\mathscr{E}|_{X\times X}$ and $\mathrm{r}(F^{-1}\circ F)=\mathrm{s}(F)\in\mathbf{L}$ . For any $E\in\mathscr{E}|_{X\times X}$ , Lemma 3.1 implies that there exists $F\in\mathscr{E}|_{Y\times X}$ such that $E\subseteq F^{-1}\circ F$ , and hence $\mathrm{r}(E)\subseteq\mathrm{r}(F^{-1})=\mathrm{s}(F)$ . This concludes the claim. Now given $E\in\mathscr{E}|_{Y\times X}$ , the claim implies that there exists $L\in\mathbf{L}$ such that $E\subseteq Y\times L$ . Then $E\subseteq(E\circ\mathrm{Gr}(f_{L})^{-1})\circ\mathrm{Gr}(f_{L})\in\mathscr{E}(\mathcal{F})$ . Hence we obtain that $\mathscr{E}=\mathscr{E}(\mathcal{F})$ .

The last statement is easy to prove, and hence we omit the details. ∎

Finally, we consider the notion of coarse equivalence from Definition C. Recall that a coarse equivalence between two coarse spaces $(X,\mathscr{E}_{X})$ and $(Y,\mathscr{E}_{Y})$ is a coarse correspondence $\mathscr{E}$ from both $X$ to $Y$ and $Y$ to $X$ . We say that $(X,\mathscr{E}_{X})$ and $(Y,\mathscr{E}_{Y})$ are coarsely equivalent if there exists a coarse equivalence between $X$ and $Y$ .

Now we would like to apply Proposition 3.20 to unpack Definition C using families of maps, and our main target is to prove Proposition D. To make it more clear, we divide into two parts.

Corollary 3.21.

Let $(X,\mathscr{E}_{X}),(Y,\mathscr{E}_{Y})$ be coarse spaces, and $\mathscr{E}$ be a coarse equivalence between them. Then there exist coarse families $\mathcal{F}=\{f_{L}:L\to Y~{}|~{}L\in\mathbf{L}_{X}\}$ and $\mathcal{G}=\{g_{L^{\prime}}:L^{\prime}\to X~{}|~{}L^{\prime}\in\mathbf{L}_{Y}\}$ such that $\mathscr{E}=\mathscr{E}(\mathcal{F})=\mathscr{E}(\mathcal{G})$ and satisfy the following:

(3.1)

\{(g_{f_{L}(L)}\circ f_{L}(x),x)~{}|~{}x\in L\}\in\mathscr{E}_{X}\quad\text{for any}\quad L\in\mathbf{L}_{X},

and

(3.2)

\{(f_{g_{L^{\prime}}(L^{\prime})}\circ g_{L^{\prime}}(y),y)~{}|~{}y\in L^{\prime}\}\in\mathscr{E}_{Y}\quad\text{for any}\quad L^{\prime}\in\mathbf{L}_{Y}.

In this case, we have $\mathcal{F}^{*}\mathscr{E}_{Y}=\mathscr{E}_{X}$ and $\mathcal{G}^{*}\mathscr{E}_{X}=\mathscr{E}_{Y}$ .

The proof is similar to that for Corollary 3.7 using Proposition 3.20 instead, and hence we omit the proof.

Corollary 3.22.

Let $(X,\mathscr{E}_{X}),(Y,\mathscr{E}_{Y})$ be coarse spaces, and $\mathcal{F}=\{f_{L}:L\to Y~{}|~{}L\in\mathbf{L}_{X}\}$ , $\mathcal{G}=\{g_{L^{\prime}}:L^{\prime}\to X~{}|~{}L^{\prime}\in\mathbf{L}_{Y}\}$ be coarse families satisfying (3.1) and (3.2). Then $\mathcal{F}^{*}\mathscr{E}_{Y}=\mathscr{E}_{X},\mathcal{G}^{*}\mathscr{E}_{X}=\mathscr{E}_{Y}$ and $\mathscr{E}(\mathcal{F})=\mathscr{E}(\mathcal{G})$ is a coarse equivalence between $X$ and $Y$ .

Proof.

Firstly, we show that $\mathcal{F}^{*}\mathscr{E}_{Y}=\mathscr{E}_{X}$ . Since $\mathcal{F}$ is a coarse family, we have $\mathscr{E}_{X}\subseteq\mathcal{F}^{*}\mathscr{E}_{Y}$ . Conversely given $E\in\mathcal{F}^{*}\mathscr{E}_{Y}$ , there exists $L\in\mathbf{L}_{X}$ such that $E\subseteq L\times L$ and $(f_{L}\times f_{L})(E)\in\mathscr{E}_{Y}$ . Take $L^{\prime}:=f_{L}(L)$ , which belongs to $\mathbf{L}_{Y}$ by Lemma 3.13. Then $\mathcal{G}$ being coarse implies:

F:=\{(g_{L^{\prime}}\circ f_{L}(x),g_{L^{\prime}}\circ f_{L}(y))~{}|~{}(x,y)\in E\}=(g_{L^{\prime}}\times g_{L^{\prime}})\circ(f_{L}\times f_{L})(E)\in\mathscr{E}_{X}.

Hence

E\subseteq\{(g_{L^{\prime}}\circ f_{L}(x),x)~{}|~{}x\in L\}^{-1}\circ F\circ\{(g_{L^{\prime}}\circ f_{L}(y),y)~{}|~{}y\in L\}\in\mathscr{E}_{X},

which concludes that $\mathcal{F}^{*}\mathscr{E}_{Y}=\mathscr{E}_{X}$ . Similarly, we have $\mathcal{G}^{*}\mathscr{E}_{X}=\mathscr{E}_{Y}$ .

To see $\mathscr{E}(\mathcal{F})=\mathscr{E}(\mathcal{G})$ , it suffices to show that $\mathscr{E}_{YX}(\mathcal{F})=\mathscr{E}_{XY}(\mathcal{G})^{-1}$ . For $E\in\mathscr{E}_{YX}(\mathcal{F})$ , Lemma 3.16(1) shows that there exist $B\in L_{Y}$ and $L\in\mathbf{L}_{X}$ such that $E\subseteq L\times B$ and $(\mathrm{Id}_{Y}\times f_{L})(E)\in\mathscr{E}_{Y}$ . Taking $L^{\prime}:=B\cup f_{L}(L)\in\mathbf{L}_{Y}$ , we have

F^{\prime}:=\{(g_{L^{\prime}}(y),g_{L^{\prime}}f_{L}(x))~{}|~{}(y,x)\in E\}=(g_{L^{\prime}}\times g_{L^{\prime}})\circ(\mathrm{Id}_{Y}\times f_{L})(E)\in\mathscr{E}_{X}.

Note that

(g_{B}\times\mathrm{Id}_{X})(E)\subseteq F_{1}\circ F^{\prime}\circ F_{2},

where $F_{1}:=\{(g_{B}(y),g_{L^{\prime}}(y))~{}|~{}y\in B\}$ and $F_{2}:=\{(g_{L^{\prime}}f_{L}(x),x)~{}|~{}x\in L\}$ . Since $\mathcal{G}$ is admissible, then $F_{1}\in\mathscr{E}_{X}$ . Also (3.2) implies that $\{(g_{f(L)}f_{L}(x),x)~{}|~{}x\in L\}\in\mathscr{E}_{X}$ . By Lemma 3.16, we obtain $F_{2}\in\mathscr{E}_{X}$ . Hence $(g_{B}\times\mathrm{Id}_{X})(E)\in\mathscr{E}_{X}$ , which means that $E\in\mathscr{E}_{XY}(\mathcal{G})^{-1}$ . Similarly, we have $\mathscr{E}_{XY}(\mathcal{G})\subseteq\mathscr{E}_{YX}(\mathcal{F})^{-1}$ , which concludes the proof. ∎

Motivated by Corollary 3.21 and Corollary 3.22, we introduce the following:

Definition 3.23.

Let $(X,\mathscr{E}_{X}),(Y,\mathscr{E}_{Y})$ be coarse spaces, and $\mathbf{L}_{X},\mathbf{L}_{Y}$ be the associated admissible families. A coarse family $\mathcal{F}=\{f_{L}:L\to Y~{}|~{}L\in\mathbf{L}_{X}\}$ is called a coarse equivalence if there exists a coarse family $\mathcal{G}=\{g_{L^{\prime}}:L^{\prime}\to X~{}|~{}L^{\prime}\in\mathbf{L}_{Y}\}$ (which is called a coarse inverse to $\mathcal{F}$ ) satisfying (3.1) and (3.2).

Consequently, Corollary 3.21 and Corollary 3.22 can be rewritten in the following form, which concludes Proposition D.

Corollary 3.24.

Let $(X,\mathscr{E}_{X})$ and $(Y,\mathscr{E}_{Y})$ be coarse spaces. Then there exists a coarse equivalence $\mathscr{E}$ between $X$ and $Y$ in the sense of Definition C if and only if there exists a coarse equivalence $\mathcal{F}$ in the sense of Definition 3.23.

Finally, we discuss the condition of bounded geometry, which will be used in the next section.

Lemma 3.25.

Let $(X,\mathscr{E}_{X})$ and $(Y,\mathscr{E}_{Y})$ be coarse spaces of bounded geometry, and $\mathscr{E}$ be a coarse equivalence between them. Then $\mathscr{E}$ also has bounded geometry as a coarse structure on $X\sqcup Y$ .

Proof.

For $F\subseteq X\times Y$ , $x\in X$ and $y\in Y$ , denote

F_{x}:=\{y\in Y~{}|~{}(x,y)\in F\}\quad\text{and}\quad F^{y}:=\{x\in X~{}|~{}(x,y)\in F\}.

It suffices to show that for $F\in\mathscr{E}|_{X\times Y}$ , the number $\sup_{x\in X}|F_{x}|$ and $\sup_{y\in Y}|F^{y}|$ are finite. Note that

F^{-1}\circ F=\bigcup_{x\in X}(F_{x}\times F_{x})\quad\text{and}\quad F\circ F^{-1}=\bigcup_{y\in Y}(F^{y}\times F^{y}).

Hence we have

\sup_{x\in X}|F_{x}|\leq n(F^{-1}\circ F)\quad\text{and}\quad\sup_{y\in Y}|F^{y}|\leq n(F\circ F^{-1}).

Note that $F^{-1}\circ F\in\mathscr{E}_{X}$ and $F\circ F^{-1}\in\mathscr{E}_{Y}$ . Hence we conclude the proof since $\mathscr{E}_{X}$ and $\mathscr{E}_{Y}$ have bounded geometry. ∎

4. Coarse equivalences induce Morita equivalences

In this section, we recall the known result that a coarse equivalence between coarse spaces induces a Morita equivalence between the associated uniform Roe algebras. As a special case, we obtain the proof for Theorem E “(2) $\Rightarrow$ (3)”. Here we provide a detailed proof since similar idea will be used later to treat ghostly ideals (see Section 7).

Firstly, let us recall the following notion of Morita equivalence for $C^{*}$ -algebras due to Rieffel ([15], see also [8, Definition 2.5.2]).

Definition 4.1.

Let $A$ and $B$ be $C^{*}$ -algebras. An $A$ - $B$ imprimitivity bimodule $\mathcal{M}$ is a Banach space that carries the structure of both a right Hilbert $B$ -module with $B$ -inner product $\langle\cdot,\cdot\rangle_{B}$ and a left Hilbert $A$ -module with $A$ -inner product ${}_{A}\langle\cdot,\cdot\rangle$ satisfying the following:

(1)

Both inner products on $\mathcal{M}$ are full, i.e., ${}_{A}{\langle\mathcal{M},\mathcal{M}\rangle}=A$ and $\langle\mathcal{M},\mathcal{M}\rangle_{B}=B$ ;
(2)

${}_{A}{\langle\xi,\eta\rangle}\cdot\zeta=\xi\cdot\langle\eta,\zeta\rangle_{B}$ for all $\xi,\eta,\zeta\in\mathcal{M}$ ;
(3)

The actions of $A$ and $B$ on $\mathcal{M}$ commutes.

Say that $A$ and $B$ are Morita equivalent if such an $A$ - $B$ imprimitivity bimodule $\mathcal{M}$ exists.

Also recall from Section 1 that two $C^{*}$ -algebras $A$ and $B$ are stably isomorphic if $A\otimes\mathfrak{K}(\mathcal{H})$ is $\ast$ -isomorphic to $B\otimes\mathfrak{K}(\mathcal{H})$ , where $\mathcal{H}$ is a separable infinite dimensional Hilbert space. It was proved in [5] (see also [10, Chapter 7]) that if $A$ and $B$ are $\sigma$ -unital (i.e., they admit countable approximate units), then $A$ and $B$ are Morita equivalent if and only if they are stably isomorphic.

Let us restate Theorem E “(2) $\Rightarrow$ (3)” in a more general form as follows:

Proposition 4.2.

Let $(X,\mathscr{E}_{X})$ and $(Y,\mathscr{E}_{Y})$ be coarse spaces of bounded geometry which are coarsely equivalent. Then the uniform Roe algebras $C^{*}_{u}(X,\mathscr{E}_{X})$ and $C^{*}_{u}(Y,\mathscr{E}_{Y})$ are Morita equivalent. Additionally if $\mathscr{E}_{X}$ and $\mathscr{E}_{Y}$ are unital, then $C^{*}_{u}(X,\mathscr{E}_{X})$ and $C^{*}_{u}(Y,\mathscr{E}_{Y})$ are stably isomorphic.

This can be proved essentially by combining [19, Corollary 3.6] and [12, Theorem 2.8] (where second countability is assumed) using the notion of Morita equivalence for groupoids. Also note that the metric space case was proved directly in [4, Theorem 4]. For the reader’s convenience, here we provide a direct proof for the general case.

Proof of Proposition 4.2.

Let $\mathscr{E}$ be a coarse equivalence between $(X,\mathscr{E}_{X})$ and $(Y,\mathscr{E}_{Y})$ , which is a coarse structure on $X\sqcup Y$ . For simplicity, denote $A:=C^{*}_{u}(X,\mathscr{E}_{X})$ and $B:=C^{*}_{u}(Y,\mathscr{E}_{Y})$ . Define $\mathcal{M}$ to be the norm closure of

\{T\in\mathbb{C}_{u}[X\sqcup Y,\mathscr{E}]~{}|~{}\mathrm{supp}(T)\subseteq X\times Y\}

in $\mathfrak{B}(\ell^{2}(X\sqcup Y))$ , which is a left $A$ -module as well as a right $B$ -module under composition of operators. Also define the inner products

{}_{A}\langle\cdot,\cdot\rangle:\mathcal{M}\times\mathcal{M}\longrightarrow A\quad\text{by}\quad_{A}\langle T,S\rangle:=TS^{*}\quad\text{for}\quad T,S\in\mathcal{M},

and

\langle\cdot,\cdot\rangle_{B}:\mathcal{M}\times\mathcal{M}\longrightarrow B\quad\text{by}\quad\langle T,S\rangle_{B}:=T^{*}S\quad\text{for}\quad T,S\in\mathcal{M}.

It is straightforward to check that $\mathcal{M}$ is a right Hilbert $B$ -module as well as a left Hilbert $A$ -module with commuting actions of $A$ and $B$ on $\mathcal{M}$ . Moreover, we have

{}_{A}{\langle T,S\rangle}\cdot R=(TS^{*})R=T(S^{*}R)=T\cdot\langle S,R\rangle_{B}

for all $T,S,R\in\mathcal{M}$ .

It remains to show that both inner products are full. Given $a\in\mathbb{C}_{u}[X,\mathscr{E}_{X}]$ , denote $E:=\mathrm{supp}(a)$ . It follows from Lemma 3.1 that there exists $F\in\mathscr{E}|_{X\times Y}$ such that $\Delta_{\mathrm{s}(E)}\subseteq F\circ F^{-1}$ . Hence $F^{-1}$ contains a graph of a map $\phi:\mathrm{s}(E)\to Y$ . Thanks to Lemma 3.25, we can apply [18, Lemma 4.10] to write $\mathrm{Gr}(\phi)$ as a finite union of $\mathrm{Gr}(\phi_{i})$ for $i=1,2,\cdots,n$ where each $\phi_{i}:D_{i}\rightarrow Y$ is an injective map. Moreover, we can further assume that $\{D_{i}\}_{i=1}^{n}$ are mutually disjoint.

For each $i=1,2,\cdots,n$ , we set $T_{i}=\chi_{\mathrm{Gr}(\phi_{i})}$ , which is regarded as a bounded operator from $\ell^{2}(X)$ to $\ell^{2}(Y)$ . Note that $\mathrm{supp}(T_{i})\subseteq F^{-1}$ , and hence $T^{*}_{i}\in\mathcal{M}$ . Moreover, for each $i=1,2,\cdots,n$ , we have ${}_{A}{\langle aT^{*}_{i},T^{*}_{i}\rangle}=aT^{*}_{i}T_{i}=a\chi_{\Delta_{D_{i}}}$ . Hence

{\sum_{i=1}^{n}}~{}{{}_{A}{\langle aT^{*}_{i},T^{*}_{i}\rangle}}=\sum_{i}a\chi_{\Delta_{D_{i}}}=a\chi_{\Delta_{\mathrm{s}(E)}}=a,

which concludes that the inner $A$ -product is full. Similarly, we obtain that the inner $B$ -product is also full. Hence $\mathcal{M}$ is an $A$ - $B$ imprimitivity bimodule, which concludes that $A$ and $B$ are Morita equivalent. ∎

5. Rigidity for geometric ideals

In this section and the next, we focus on the rigidity problem for geometric ideals and prove Theorem E “(1) $\Rightarrow$ (2)”, which is the main task of this paper. To make the proof more transparent, we prove a weak version in this section. The proof is relatively easier, while contains almost all the necessary ingredients to treat the general case.

Recall that for an ideal $I$ in the uniform Roe algebra of a discrete metric space $(X,d)$ of bounded geometry, the associated coarse structure is $\mathscr{I}(I):=\{\mathrm{supp}_{\varepsilon}(T)~{}|~{}T\in I,\varepsilon>0\}$ from Proposition 2.7. We aim to prove the following:

Theorem 5.1.

Let $(X,d_{X}),(Y,d_{Y})$ be discrete metric spaces of bounded geometry, and $I_{X},I_{Y}$ be geometric ideals in the uniform Roe algebras $C^{*}_{u}(X)$ and $C^{*}_{u}(Y)$ , respectively. Assume that $I_{X}$ and $I_{Y}$ are isomorphic. Then the coarse spaces $(X,\mathscr{I}(I_{X}))$ and $(Y,\mathscr{I}(I_{Y}))$ are coarsely equivalent.

The proof of Theorem 5.1 is divided into two parts. We follow the outline of [1], while requiring more techniques to overcome the issue of lacking units.

Throughout the rest of this section, let $(X,d_{X}),(Y,d_{Y})$ be discrete metric spaces of bounded geometry, and $I_{X},I_{Y}$ be geometric ideals in the uniform Roe algebras $C^{*}_{u}(X)$ and $C^{*}_{u}(Y)$ , respectively. Denote the associated ideal in $\mathscr{E}_{d_{X}}$ by $\mathscr{I}_{X}:=\mathscr{I}(I_{X})$ , and the associated ideal in $(X,d_{X})$ by $\mathbf{L}_{X}:=\mathbf{L}(\mathscr{I}_{X})$ (from Proposition 2.7 and 2.9). Similarly, denote $\mathscr{I}_{Y}:=\mathscr{I}(I_{Y})$ and $\mathbf{L}_{Y}:=\mathbf{L}(\mathscr{I}_{Y})$ . Also let $\Phi\colon I_{X}\to I_{Y}$ be an isomorphism.

5.1. Rigid isomorphisms for geometric ideals

Firstly, recall that the isomorphism $\Phi\colon I_{X}\to I_{Y}$ can always be spatially implemented. The proof is similar to that for [21, Lemma 3.1] based on the representation theory for compact operators together with the fact that finite subsets are entourages, and hence we omit the details.

Lemma 5.2.

There exists a unitary $U:\ell^{2}(X)\to\ell^{2}(Y)$ such that $\Phi(a)=UaU^{*}$ for any $a\in I_{X}$ .

In [3], the authors introduced a notion of rigid isomorphism which plays a key role to attack the rigidity problem. The idea can be traced back to the work [21]. We introduce the following to deal with the non-unital case.

Definition 5.3.

With the same notation as above, the isomorphism $\Phi\colon I_{X}\rightarrow I_{Y}$ is called a rigid isomorphism if $\Phi$ can be spatially implemented by a unitary $U:\ell^{2}(X)\to\ell^{2}(Y)$ satisfying:

\inf_{x\in L}\sup_{y\in Y}|\langle U\delta_{x},\delta_{y}\rangle|>0\quad\text{for all }\quad L\in\mathbf{L}_{X},

and

\inf_{y\in L^{\prime}}\sup_{x\in X}|\langle U^{*}\delta_{y},\delta_{x}\rangle|>0\quad\text{for all }\quad L^{\prime}\in\mathbf{L}_{Y}.

In this case, we say that $I_{X}$ and $I_{Y}$ are rigidly isomorphic.

The first step to attack the rigidity problem is the following (comparing [3, Theorem 4.12]):

Proposition 5.4.

With the same notation as above, assume that $I_{X}$ and $I_{Y}$ is rigidly isomorphic. Then the coarse spaces $(X,\mathscr{I}_{X})$ and $(Y,\mathscr{I}_{Y})$ are coarsely equivalent.

To prove Proposition 5.4, we need some preparations.

Definition 5.5 ([3, Definition 4.3]).

Let $(X,d)$ be a discrete metric of bounded geometry, $\varepsilon>0$ and $r\geq 0$ . An operator $a\in\mathfrak{B}(\ell^{2}(X))$ is called $\varepsilon$ - $r$ -approximable if there exists $b\in\mathfrak{B}(\ell^{2}(X))$ with propagation at most $r$ such that $\|a-b\|\leq\varepsilon$ .

The following key “equi-approximability lemma” was originally obtained in [3, Section 4] (see also [1, Lemma 1.9]), which will be used frequently in the sequel.

Lemma 5.6.

Let $(X,d)$ be a discrete metric of bounded geometry, and $\{a_{n}\}_{n}$ be a sequence of operators such that $\mathrm{SOT}$ - $\sum_{n\in M}a_{n}$ converges to an element in $C^{*}_{u}(X)$ for all $M\subseteq\mathbb{N}$ . Then for all $\varepsilon>0$ , there exists $r>0$ such that $\mathrm{SOT}$ - $\sum_{n\in M}a_{n}$ is $\varepsilon$ - $r$ -approximable for all $M\subseteq\mathbb{N}$ .

We also need the following lemma, which is analogous to [3, Lemma 4.11]. As explained in Section 1, it is unclear whether the coarse structures $\mathscr{I}_{X}$ and $\mathscr{I}_{Y}$ are small in the sense of [3, Definition 4.2]. This obstructs us from using the results in [3, Section 4] directly, and hence more techniques are required.

Lemma 5.7.

With the notation as above, for any $E\in\mathscr{I}_{X}$ and $\delta>0$ , the following set

F(E,\delta):=\left\{(y_{1},y_{2})\in Y\times Y\;\middle|\;\begin{array}[]{l}\text{ there exists }(x_{1},x_{2})\in E\text{ such that}\\ |\langle U\delta_{x_{1}},\delta_{y_{1}}\rangle|\geq\delta\text{ and }|\langle U\delta_{x_{2}},\delta_{y_{2}}\rangle|\geq\delta\end{array}\right\}

belongs to $\mathscr{I}_{Y}$ .

Proof.

Suppose otherwise. Then there exist $E\in\mathscr{I}_{X}$ and $\delta>0$ such that $F(E,\delta)\notin\mathscr{I}_{Y}$ . Hence for any $F\in\mathscr{I}_{Y}$ , we have $F(E,\delta)\neq F$ . Note that subsets of any entourage are entourages (see Definition 2.1(3)), so we have $F(E,\delta)\setminus F\neq\emptyset$ . Therefore, we can find $(x_{1}^{F},x_{2}^{F})\in E$ and $(y_{1}^{F},y_{2}^{F})\in Y\times Y$ such that $|\langle U\delta_{x_{1}^{F}},\delta_{y_{1}^{F}}\rangle|\geq\delta$ and $|\langle U\delta_{x_{2}^{F}},\delta_{y_{2}^{F}}\rangle|\geq\delta$ while $(y_{1}^{F},y_{2}^{F})\notin F$ . Ordering $\mathscr{I}_{Y}$ by inclusion, it follows from [3, Lemma 4.10] that there exist cofinal $I\subset\mathscr{I}_{Y}$ , $J\subset\mathscr{I}_{Y}$ and a map $\varphi:I\rightarrow J$ such that

(1)

$x_{1}^{F}\neq x_{1}^{F^{\prime}}$ and $x_{2}^{F}\neq x_{2}^{F^{\prime}}$ for all $F\neq F^{\prime}\in J$ ;
(2)

$x_{1}^{F}=x_{1}^{\varphi(F)}$ and $x_{2}^{F}=x_{2}^{\varphi(F)}$ for all $F\in I$ .

Since elements in $\{x_{1}^{F}\}_{F\in J}\subseteq X$ are distinct, then $J$ is countable. For $A\subseteq J$ , condition (1) and $\mathscr{I}_{X}$ having bounded geometry imply that $\mathrm{SOT}$ - $\sum_{F\in A}e_{x_{1}^{F}x_{2}^{F}}\in\mathbb{C}_{u}[X,\mathscr{I}_{X}]$ with support in $E$ . Hence $\mathrm{SOT}$ - $\sum_{F\in A}\Phi(e_{x_{1}^{F}x_{2}^{F}})=\mathrm{SOT}$ - $\Phi(\sum_{F\in A}e_{x_{1}^{F}x_{2}^{F}})$ belongs to $C^{*}_{u}(Y)$ since $\Phi$ is $\mathrm{SOT}$ -continuous. Applying Lemma 5.6, for $\epsilon=\delta^{2}/3$ there exists $r>0$ such that $\mathrm{SOT}$ - $\sum_{F\in A}\Phi(e_{x_{1}^{F}x_{2}^{F}})$ is $\epsilon$ - $r$ -approximable for all $A\subseteq J$ . In particular, we have

(5.1)

\|\chi_{A}\Phi(e_{x_{1}^{F}x_{2}^{F}})\chi_{B}\|<\epsilon

for all $F\in J$ and $A,B\subseteq Y$ with $d_{Y}(A,B)\geq r$ .

By Proposition 2.9, we can assume that $E\subseteq L\times L$ for some $L\in\mathbf{L}_{X}$ . Since $\{\chi_{L^{\prime}}\}_{{L^{\prime}\in\mathbf{L}_{Y}}}$ is an approximate unit for $I_{Y}$ due to Lemma 2.12, there exist $L^{\prime}\in\mathbf{L}_{Y}$ such that

\|\Phi(\chi_{L})\chi_{L^{\prime}}-\Phi(\chi_{L})\|<\epsilon\quad\text{and}\quad\|\chi_{L^{\prime}}\Phi(\chi_{L})-\Phi(\chi_{L})\|<\epsilon.

Thus, we have

	$\displaystyle\\|\chi_{L^{\prime}}\Phi(e_{x_{1}^{F}x_{2}^{F}})\chi_{L^{\prime}}-\Phi(e_{x_{1}^{F}x_{2}^{F}})\\|\leq\\|\chi_{L^{\prime}}\Phi(e_{x_{1}^{F}x_{2}^{F}})\chi_{L^{\prime}}-\Phi(e_{x_{1}^{F}x_{2}^{F}})\chi_{L^{\prime}}\\|+\\|\Phi(e_{x_{1}^{F}x_{2}^{F}})\chi_{L^{\prime}}-\Phi(e_{x_{1}^{F}x_{2}^{F}})\\|$
	$\displaystyle\leq\\|\chi_{L^{\prime}}\Phi(\chi_{L})\Phi(e_{x_{1}^{F}x_{2}^{F}})-\Phi(\chi_{L})\Phi(e_{x_{1}^{F}x_{2}^{F}})\\|+\\|\Phi(e_{x_{1}^{F}x_{2}^{F}})\Phi(\chi_{L})\chi_{L^{\prime}}-\Phi(e_{x_{1}^{F}x_{2}^{F}})\Phi(\chi_{L})\\|$
	$\displaystyle\leq 2\varepsilon.$

Setting $F_{1}:=\{(y_{1},y_{2})\in L^{\prime}\times L^{\prime}~{}|~{}d_{Y}(y_{1},y_{2})<r\}$ , it is clear that $F_{1}\in\mathscr{I}_{Y}$ . For all $A,B\subseteq Y$ which are $F_{1}$ -separated, then $d_{Y}(L^{\prime}\cap A,L^{\prime}\cap B)\geq r$ . Hence

\|\chi_{A}\Phi(e_{x_{1}^{F}x_{2}^{F}})\chi_{B}\|\leq\|\chi_{A}\chi_{L^{\prime}}\Phi(e_{x_{1}^{F}x_{2}^{F}})\chi_{L^{\prime}}\chi_{B}\|+2\varepsilon=\|\chi_{A\cap L^{\prime}}\Phi(e_{x_{1}^{F}x_{2}^{F}})\chi_{L^{\prime}\cap B}\|+2\varepsilon<3\varepsilon=\delta^{2},

where the penultimate inequality comes from (5.1). Since $I$ is cofinal in $\mathscr{I}_{Y}$ , we can assume that $F_{1}\in I$ .

Then we have

\left\|\chi_{\left\{y_{1}^{F_{1}}\right\}}\Phi(e_{x_{1}^{F_{1}}x_{2}^{F_{1}}})\chi_{\left\{y_{2}^{F_{1}}\right\}}\right\|=\left\|\chi_{\left\{y_{1}^{F_{1}}\right\}}\Phi(e_{x_{1}^{\varphi(F_{1})}x_{2}^{\varphi(F_{1})}})\chi_{\left\{y_{2}^{F_{1}}\right\}}\right\|<\delta^{2}.

On the other hand, a direct calculation shows that

\left\|\chi_{\left\{y_{1}^{F_{1}}\right\}}\Phi(e_{x_{1}^{F_{1}}x_{2}^{F_{1}}})\chi_{\left\{y_{2}^{F_{1}}\right\}}\right\|=\left|\left\langle\delta_{y_{1}^{F_{1}}},U\left(\delta_{x_{1}^{F_{1}}}\right)\right\rangle\right|\cdot\left|\left\langle U\left(\delta_{x_{2}^{F_{1}}}\right),\delta_{y_{2}^{F_{1}}}\right\rangle\right|\geq\delta^{2},

which leads to a contradiction. ∎

Now we are in the position to prove Proposition 5.4.

Proof of Proposition 5.4.

Let $U\colon\ell^{2}(X)\rightarrow\ell^{2}(Y)$ be a unitary operator which spatially implements the rigid isomorphism $\Phi:I_{X}\to I_{Y}$ . Hence:

•

for any $L\in\mathbf{L}_{X}$ , there exist $\delta_{L}>0$ and $f_{L}:L\rightarrow Y$ such that

$|\langle U\delta_{x},\delta_{f_{L}(x)}\rangle|\geq\delta_{L}\quad\text{for any}\quad x\in L;$
•

for any $L^{\prime}\in\mathbf{L}_{Y}$ , there exist $\delta_{L^{\prime}}>0$ and $g_{L^{\prime}}:L^{\prime}\rightarrow X$ such that

$|\langle U^{*}\delta_{y},\delta_{g_{L^{\prime}}(y)}\rangle|\geq\delta_{L^{\prime}}\quad\text{for any}\quad y\in L^{\prime}.$

We aim to show that the families $\mathcal{F}:=\{f_{L}~{}|~{}L\in\mathbf{L}_{X}\}$ and $\mathcal{G}:=\{g_{L^{\prime}}~{}|~{}L^{\prime}\in\mathbf{L}_{Y}\}$ provide a coarse equivalence between $(X,\mathscr{I}_{X})$ and $(Y,\mathscr{I}_{Y})$ in the sense of Definition 3.23, and hence conclude the proof thanks to Corollary 3.24.

Firstly, we show that the families $\mathcal{F}$ and $\mathcal{G}$ are coarse. For $L_{1},L_{2}\in\mathbf{L}_{X}$ , consider the set $\{(f_{L_{1}}(x),f_{L_{2}}(x))\;|\;x\in L_{1}\cap L_{2}\}$ , which is contained in $F(\Delta_{L_{1}\cap L_{2}},\delta_{L_{1}}\wedge\delta_{L_{2}})$ . By Lemma 5.7, the set $F(\Delta_{L_{1}\cap L_{2}},\delta_{L_{1}}\wedge\delta_{L_{2}})$ belongs to $\mathscr{I}_{Y}$ . Hence $\mathcal{F}$ is admissible.

Given $E\in\mathscr{I}_{X}$ , we can assume that $E\subseteq L\times L$ for some $L\in\mathbf{L}_{X}$ thanks to Proposition 2.9. Then for any $(x_{1},x_{2})\in E$ , it follows from the definition that $(f_{L}(x_{1}),f_{L}(x_{2}))\in F(E,\delta_{L})$ . In other words, we obtain $(f_{L}\times f_{L})(E)\subseteq F(E,\delta_{L})$ , where the latter belongs to $\mathscr{I}_{Y}$ due to Lemma 5.7. Hence we have $\mathscr{I}_{X}\subseteq\mathcal{F}^{*}(\mathscr{I}_{Y})$ , which concludes that the family $\mathcal{F}$ is coarse. Similarly, we obtain that the family $\mathcal{G}$ is also coarse. Moreover by Lemma 3.13, we have $f_{L}(L)\in\mathbf{L}_{Y}$ for any $L\in\mathbf{L}_{X}$ , and $g_{L^{\prime}}(L^{\prime})\in\mathbf{L}_{Y}$ for any $L^{\prime}\in\mathbf{L}_{Y}$ .

It remains to show that (3.1) and (3.2) holds for $\mathcal{F}$ and $\mathcal{G}$ . Fix $L^{\prime}\in\mathbf{L}_{Y}$ and denote $L^{\prime\prime}:=g_{L^{\prime}}(L^{\prime})$ . By the choice of $g_{L^{\prime}}$ and $f_{L^{\prime\prime}}$ , we have

|\langle U\delta_{g_{L^{\prime}}(y)},\delta_{y}\rangle|=|\langle\delta_{g_{L^{\prime}}(y)},U^{*}\delta_{y}\rangle|\geq\delta_{L^{\prime}}\text{ for any }y\in L^{\prime}

and

|\langle U\delta_{g_{L^{\prime}}(y)},\delta_{f_{L^{\prime\prime}}(g_{L^{\prime}}(y))}\rangle|\geq\delta_{L^{\prime\prime}}\text{ for any }y\in L^{\prime}.

This implies that

\{(y,f_{L^{\prime\prime}}(g_{L^{\prime}}(y)))\;|\;y\in L^{\prime}\}\subseteq F(\Delta_{L^{\prime\prime}},\delta_{L^{\prime}}\wedge\delta_{L^{\prime\prime}}),

which concludes (3.2). Similarly, we obtain (3.1) and finish the proof. ∎

5.2. From isomorphisms to rigid isomorphisms

Now we prove that any isomorphism between geometric ideals is always a rigid isomorphism, and hence conclude the proof for Theorem 5.1. We will follow the outline of the proof for [1, Theorem 1.2] with the following extra piece:

Lemma 5.8.

With that notation as above, for any $L\in\mathbf{L}_{X}$ and $\epsilon>0$ , there exist $L^{\prime}\in\mathbf{L}_{Y}$ such that $\|\chi_{L}-\Phi^{-1}(\chi_{L^{\prime}})\chi_{L}\|<\epsilon$ .

Proof.

By Lemma 2.12, $\{\chi_{L^{\prime}}~{}|~{}L^{\prime}\in\mathbf{L}_{Y}\}$ is an approximate unit for $I_{Y}$ . Hence, $\{\Phi^{-1}(\chi_{L^{\prime}})~{}|~{}L^{\prime}\in\mathbf{L}_{Y}\}$ is an approximate unit for $I_{X}$ , which concludes the proof. ∎

Proposition 5.9.

With the same notation as above, assume that $I_{X}$ and $I_{Y}$ are isomorphic. Then they are rigidly isomorphic.

Proof.

Fixing $L\in\mathbf{L}_{X}$ and $\epsilon>0$ , take $L^{\prime}\in\mathbf{L}_{Y}$ as in Lemma 5.8. Consider $\{\Phi^{-1}(e_{yy})\}_{y\in L^{\prime}}\subseteq I_{X}$ , which is a sequence of orthogonal projections in $C^{*}_{u}(X)$ and satisfies the condition in Lemma 5.6. Hence, there is $r>0$ such that

\Phi^{-1}(\chi_{A})=\Phi^{-1}\big{(}\mathrm{SOT}\text{-}\sum_{y\in A}e_{yy}\big{)}=\mathrm{SOT}\text{-}\sum_{y\in A}\Phi^{-1}(e_{yy})

is $\epsilon$ - $r$ -approximable for any $A\subseteq L^{\prime}$ .

Set $N_{r}:=\sup_{x\in X}|B_{X}(x,r)|$ . It follows from Lemma 5.8 that for any $x\in L$ , we have

(5.2)

\|\chi_{\{x\}}-\Phi^{-1}(\chi_{L^{\prime}})\chi_{\{x\}}\|=\|\chi_{\{x\}}\chi_{L}-\Phi^{-1}(\chi_{L^{\prime}})\chi_{\{x\}}\chi_{L}\|\leq\epsilon.

Taking $\delta\leq\frac{\epsilon}{2N_{r}}$ , we denote

M(x,\delta):=\{y\in L^{\prime}\;|\;\|\Phi^{-1}(e_{yy})\delta_{x}\|\geq\delta\}\quad\text{and}\quad M^{\prime}(x,\delta):=L^{\prime}\backslash M(x,\delta).

Let $\pi:\ell^{2}(X)\rightarrow\ell^{2}(B_{X}(x,r))$ be the canonical orthogonal projection. Define $\mu:\mathcal{P}(M^{\prime}(x,\delta))\rightarrow\ell^{2}(B_{X}(x,r))$ by

\mu(A)=\pi(\Phi^{-1}(\chi_{A})\delta_{x}),\text{ for all }A\subseteq M^{\prime}(x,\delta).

Then we have $\|\mu(\{y\})\|=\|\pi(\Phi^{-1}(e_{yy})\delta_{x})\|\leq\|\Phi^{-1}(e_{yy})\delta_{x}\|<\delta$ for all $y\in M^{\prime}(x,\delta)$ .

Since $\frac{\mu(M^{\prime}(x,\delta))}{2}$ belongs to the convex hull of the range of $\mu$ and $\ell^{2}(B_{X}(x,r))$ has real dimension at most $2N_{r}$ , [1, Lemma 2.1] implies that there exists $A\subseteq M^{\prime}(x,\delta)$ such that

(5.3)

\left\|\mu(A)-\frac{\mu(M^{\prime}(x,\delta))}{2}\right\|<2N_{r}\cdot\delta\leq\epsilon.

Moreover,

(5.4)		$\displaystyle\left\\|\mu(A)-\frac{\mu(M^{\prime}(x,\delta))}{2}\right\\|$	$\displaystyle=\left\\|\pi\left(\left(\Phi^{-1}(\chi_{A})-\frac{1}{2}\Phi^{-1}(\chi_{M^{\prime}(x,\delta)})\right)\delta_{x}\right)\right\\|$
		$\displaystyle=\left\\|\pi\left(\left(\frac{1}{2}\Phi^{-1}(\chi_{A})-\frac{1}{2}\Phi^{-1}(\chi_{M^{\prime}(x,\delta)\backslash A})\right)\delta_{x}\right)\right\\|.$

Since $\Phi^{-1}(\chi_{A})$ and $\Phi^{-1}(\chi_{M^{\prime}(x,\delta)\backslash A})$ are $\epsilon$ - $r$ -approximable, the convex combination $\frac{1}{2}\Phi^{-1}(\chi_{A})-\frac{1}{2}\Phi^{-1}(\chi_{M^{\prime}(x,\delta)\backslash A})$ is also $\epsilon$ - $r$ -approximable. Hence we have

(5.5)

\left\|(1-\pi)\left(\left(\frac{1}{2}\Phi^{-1}(\chi_{A})-\frac{1}{2}\Phi^{-1}(\chi_{M^{\prime}(x,\delta)\backslash A})\right)\delta_{x}\right)\right\|<\epsilon.

Using the fact that $\Phi^{-1}(\chi_{A})\Phi^{-1}(\chi_{M^{\prime}(x,\delta)})=\Phi^{-1}(\chi_{A})$ together with (5.3)-(5.5), we obtain

	$\displaystyle\left\\|\Phi^{-1}(\chi_{A})\Phi^{-1}(\chi_{M^{\prime}(x,\delta)})\delta_{x}-\frac{1}{2}\Phi^{-1}(\chi_{M^{\prime}(x,\delta)})\delta_{x}\right\\|$
	$\displaystyle\leq\left\\|(1-\pi)\left(\left(\frac{1}{2}\Phi^{-1}(\chi_{A})-\frac{1}{2}\Phi^{-1}(\chi_{M^{\prime}(x,\delta)\backslash A})\right)\delta_{x}\right)\right\\|+\left\\|\mu(A)-\frac{\mu(M^{\prime}(x,\delta))}{2}\right\\|<2\epsilon.$

Hence [1, Lemma 3.1] implies that $\|\Phi^{-1}(\chi_{M^{\prime}(x,\delta)})\delta_{x}\|<4\epsilon$ . Moreover, note that

\Phi^{-1}(\chi_{M^{\prime}(x,\delta)})\delta_{x}+\Phi^{-1}(\chi_{M(x,\delta)})\delta_{x}=\Phi^{-1}(\chi_{L^{\prime}})\delta_{x}.

Hence combining (5.2), we obtain $\|\Phi^{-1}(\chi_{M(x,\delta)})\delta_{x}\|\geq 1-5\epsilon$ .

Taking $\epsilon=1/10$ , we have $\|\Phi^{-1}(\chi_{M(x,\delta)})\delta_{x}\|\geq\frac{1}{2}$ . Set $\delta=\frac{1}{20N_{r}}$ , then $M(x,\delta)$ is non-empty for any $x\in L$ . Hence we obtain

\inf_{x\in L}\sup_{y\in L^{\prime}}\|\Phi^{-1}(e_{yy})\delta_{x}\|\geq\frac{1}{20N_{r}}.

Finally, note that

\|\Phi^{-1}(e_{yy})\delta_{x}\|^{2}=|\langle U^{*}\delta_{y},\delta_{x}\rangle|^{2}=|\langle U\delta_{x},\delta_{y}\rangle|^{2}.

Hence $\Phi$ is a rigid isomorphism. ∎

Proof of Theorem 5.1.

By Proposition 5.9, we know that $\Phi$ is a rigid isomorphism. Hence applying Proposition 5.4, we conclude the proof. ∎

6. Rigidity for stable geometric ideals

Now we move to the case of stable isomorphism and finish the proof of Theorem E “(1) $\Rightarrow$ (2)”. Recall that given a coarse space $(X,\mathscr{E})$ of bounded geometry, the stable uniform Roe algebra of $(X,\mathscr{E})$ is defined to be

C^{*}_{s}(X,\mathscr{E}):=C^{*}_{u}(X,\mathscr{E})\otimes\mathfrak{K}(\mathcal{H}),

where $\mathcal{H}$ is a separable infinite dimensional Hilbert space. Note that $C^{*}_{s}(X,\mathscr{E})$ can be regarded as a $C^{*}$ -subalgebra of $\mathfrak{B}(\ell^{2}(X;\mathcal{H}))$ . Similar to the case of the uniform Roe algebra, there is a dense $\ast$ -subalgebra $\mathbb{C}_{s}[X,\mathscr{E}]$ in $C^{*}_{s}(X,\mathscr{E})$ under this viewpoint. More precisely, an operator $T\in\mathfrak{B}(\ell^{2}(X;\mathcal{H}))$ belongs to $\mathbb{C}_{s}[X,\mathscr{E}]$ if and only if its support (similar to the definition in Section 2.3) belongs to $\mathscr{E}$ and there exists a finite-dimensional subspace $\mathcal{H}^{\prime}\subseteq\mathcal{H}$ such that each matrix entry $T(x,y)$ belongs to $\mathfrak{B}(\mathcal{H}^{\prime})$ .

Hence Theorem E “(1) $\Rightarrow$ (2)” can be rewritten as follows:

Theorem 6.1.

Let $(X,d_{X}),(Y,d_{Y})$ be discrete metric spaces of bounded geometry, and $I_{X},I_{Y}$ be geometric ideals in the uniform Roe algebras $C^{*}_{u}(X)$ and $C^{*}_{u}(Y)$ , respectively. Assume that the stable geometric ideals $C^{*}_{s}(X,\mathscr{I}(I_{X}))$ and $C^{*}_{s}(Y,\mathscr{I}(I_{Y}))$ are isomorphic. Then the coarse spaces $(X,\mathscr{I}(I_{X}))$ and $(Y,\mathscr{I}(I_{Y}))$ are coarsely equivalent.

The proof is similar to that for Theorem 5.1 but more technical, which follows the outline of that for [1, Theorem 4.1]. Again throughout the rest of this section, let $(X,d_{X}),(Y,d_{Y})$ be discrete metric spaces of bounded geometry, and $I_{X},I_{Y}$ be geometric ideals in the uniform Roe algebras $C^{*}_{u}(X)$ and $C^{*}_{u}(Y)$ , respectively. Denote $\mathscr{I}_{X}:=\mathscr{I}(I_{X}),\mathscr{I}_{Y}:=\mathscr{I}(I_{Y})$ and $\mathbf{L}_{X}:=\mathbf{L}(\mathscr{I}_{X}),\mathbf{L}_{Y}:=\mathbf{L}(\mathscr{I}_{Y})$ . Also let $\Phi\colon C^{*}_{s}(X,\mathscr{I}(I_{X}))\to C^{*}_{s}(Y,\mathscr{I}(I_{Y}))$ be an isomorphism with the inverse $\Psi:=\Phi^{-1}$ .

Firstly, we have the following analogue of Proposition 5.9.

Proposition 6.2.

With the same notation as above, for any unit vector $\xi\in\mathcal{H}$ the following holds:

(1)

for any $L\in\mathbf{L}_{X}$ , there exist $f_{L}:L\to Y$ and a finite-rank projection $p_{L}$ on $\mathcal{H}$ such that

$\inf_{x\in L}\|\Phi(\chi_{\{x\}}\otimes p_{\xi})(\chi_{\{f_{L}(x)\}}\otimes p_{L})\|>0;$
(2)

for any $L^{\prime}\in\mathbf{L}_{Y}$ , there exist $g_{L^{\prime}}:L^{\prime}\to X$ and a finite-rank projection $p_{L^{\prime}}$ on $\mathcal{H}$ such that

$\inf_{y\in L^{\prime}}\|\Phi(\chi_{\{y\}}\otimes p_{\xi})(\chi_{\{g_{L^{\prime}}(y)\}}\otimes p_{L^{\prime}})\|>0.$

Here $p_{\xi}$ is the orthogonal projection from $\mathcal{H}$ onto $\mathbb{C}\xi$ .

The proof is similar to that for [1, Theorem 4.1] (with the same idea presented in the proof of Proposition 5.9), and hence we only provide a sketch here.

Sketch of proof for Proposition 6.2.

Note that the set

\{\chi_{L^{\prime}}\otimes q~{}|~{}L^{\prime}\in\mathbf{L}_{Y},q\text{ is a finite dimensional projection on }\mathcal{H}\}

is an approximate unit for $I_{Y}\otimes\mathfrak{K}(\mathcal{H})$ . Hence given $\varepsilon>0$ and $L\in\mathbf{L}_{X}$ , there exists $L^{\prime}\in\mathbf{L}_{Y}$ and a finite dimensional projection $p_{L}$ on $\mathcal{H}$ such that

\|\Psi(\chi_{L^{\prime}}\otimes p_{L})(\chi_{L}\otimes p_{\xi})-\chi_{L}\otimes p_{\xi}\|<\varepsilon.

Therefore, we have

\|(\mathrm{Id}_{\ell^{2}(L^{\prime};\mathcal{H})}-\chi_{L^{\prime}}\otimes p_{L})\Phi(\chi_{\{x\}}\otimes\xi)\|<\varepsilon,\quad\forall x\in L.

This provides a similar condition as in the hypothesis of [1, Lemma 4.3], which allows us to apply the same argument therein to obtain a constant $\delta=\delta(\varepsilon,L)>0$ and a map $f_{L}:L\to L^{\prime}$ satisfying

\|\Phi(\chi_{\{x\}}\otimes p_{\xi})(\chi_{\{f_{L}(x)\}}\otimes p_{L})\|=\|\Psi(\chi_{\{f_{L}(x)\}}\otimes p_{L})(\delta_{x}\otimes\xi)\|>\delta,\quad\forall x\in L.

Hence we obtain (1), and (2) can be treated similarly. ∎

Now we show that the conditions in Proposition 6.2 imply the required coarse equivalence, analogous to Proposition 5.4.

Proposition 6.3.

With the same notation as above, assume that for any unit vector $\xi\in\mathcal{H}$ , condition (1) and (2) in Proposition 6.2 hold with functions $f_{L}$ and $g_{L^{\prime}}$ for $L\in\mathbf{L}_{X}$ and $L^{\prime}\in\mathbf{L}_{Y}$ . Then the family $\mathcal{F}:=\{f_{L}~{}|~{}L\in\mathbf{L}_{X}\}$ is a coarse equivalence (in the sense of Definition 3.23) from $(X,\mathscr{I}_{X})$ to $(Y,\mathscr{I}_{Y})$ with a coarse inverse $\mathcal{G}:=\{g_{L^{\prime}}~{}|~{}L^{\prime}\in\mathbf{L}_{Y}\}$ .

To prove Proposition 6.3, first note that there exists a unitary $U:\ell^{2}(X)\otimes\mathcal{H}\to\ell^{2}(Y)\otimes\mathcal{H}$ such that $\Phi(a)=UaU^{*}$ for any $a\in I_{X}\otimes\mathfrak{K}(\mathcal{H})$ , which is similar to Lemma 5.2. Analogous to Lemma 5.7, we also need the following lemma:

Lemma 6.4.

With the notation as above, for any $E\in\mathscr{I}_{X},\delta>0$ and a finite dimensional subspace $V\subseteq\mathcal{H}$ , the following set

F(E,\delta,V):=\left\{(y_{1},y_{2})\in Y\times Y\;\middle|\;\begin{array}[]{l}\text{ there exist }(x_{1},x_{2})\in E,\text{unit vectors }v_{1},v_{2}\in V\\ \text{ and unit vectors }w_{1},w_{2}\in\mathcal{H}\text{ such that}\\ |\langle U(\delta_{x_{1}}\otimes v_{1}),\delta_{y_{1}}\otimes w_{1}\rangle|\geq\delta\text{ and }|\langle U(\delta_{x_{2}}\otimes v_{2}),\delta_{y_{2}}\otimes w_{2}\rangle|\geq\delta\end{array}\right\}

belongs to $\mathscr{I}_{Y}$ .

The proof for Lemma 6.4 is similar to that for Lemma 5.7 with minor modifications, and hence we only provide a sketch to explain the difference.

Sketch of proof for Lemma 6.4.

Suppose otherwise. Then there exist $E\in\mathscr{I}_{X},\delta>0$ and a finite dimensional subspace $V\subseteq\mathcal{H}$ such that $F(E,\delta,V)\notin\mathscr{I}_{Y}$ . Therefore for any $F\in\mathscr{I}_{Y}$ , we can find $(x_{1}^{F},x_{2}^{F})\in E,(y_{1}^{F},y_{2}^{F})\in Y\times Y$ , unit vectors $v_{1}^{F},v_{2}^{F}\in V$ and unit vectors $w_{1}^{F},w_{2}^{F}\in\mathcal{H}$ such that

\left|\left\langle U\left(\delta_{x^{F}_{i}}\otimes v^{F}_{i}\right),\delta_{y^{F}_{i}}\otimes w^{F}_{i}\right\rangle\right|\geq\delta

for $i=1,2$ , while $(y_{1}^{F},y_{2}^{F})\notin F$ . Note that $v_{1}^{F},v_{2}^{F}$ are unit vectors in $V$ and $V$ is finite dimensional. Hence after a small perturbation, we can assume that the set $\left\{v_{1}^{F},v_{2}^{F}\right\}_{F\in\mathscr{I}_{Y}}$ is finite. Ordering $\mathscr{I}_{Y}$ by inclusion, it follows from [3, Lemma 4.10] that there exist cofinal $I\subset\mathscr{I}_{Y}$ , $J\subset\mathscr{I}_{Y}$ and a map $\varphi:I\rightarrow J$ such that

(1)

$x_{1}^{F}\neq x_{1}^{F^{\prime}}$ and $x_{2}^{F}\neq x_{2}^{F^{\prime}}$ for all $F\neq F^{\prime}\in J$ ;
(2)

$x_{1}^{F}=x_{1}^{\varphi(F)}$ , $x_{2}^{F}=x_{2}^{\varphi(F)}$ , $v_{1}^{F}=v_{1}^{\varphi(F)}$ and $v_{2}^{F}=v_{2}^{\varphi(F)}$ for all $F\in I$ .

The rest is similar to that of Lemma 5.7, and hence we omit the details. ∎

Now we use Lemma 6.4 to prove Proposition 6.3.

Proof of Proposition 6.3.

As remarked above, there exists a unitary $U:\ell^{2}(X)\otimes\mathcal{H}\to\ell^{2}(Y)\otimes\mathcal{H}$ such that $\Phi(a)=UaU^{*}$ for any $a\in I_{X}\otimes\mathfrak{K}(\mathcal{H})$ . Fix a unit vector $\xi\in\mathcal{H}$ . Direct calculations show that conditions in Proposition 6.2 can be translated as follows:

•

for any $L\in\mathbf{L}_{X}$ , there exist $\delta_{L}>0$ , $f_{L}:L\to Y$ and a finite-rank projection $p_{L}$ on $\mathcal{H}$ such that for any $x\in L$ , there exists a unit vector $w_{x}\in p_{L}\mathcal{H}$ satisfying:

$|\langle U(\delta_{x}\otimes\xi),\delta_{f_{L}(x)}\otimes w_{x}\rangle|\geq\delta_{L};$
•

for any $L^{\prime}\in\mathbf{L}_{Y}$ , there exist $\delta_{L^{\prime}}>0$ , $g_{L^{\prime}}:L^{\prime}\to X$ and a finite-rank projection $p_{L^{\prime}}$ on $\mathcal{H}$ such that for any $y\in L^{\prime}$ , there exists a unit vector $v_{y}\in p_{L^{\prime}}\mathcal{H}$ satisfying:

$|\langle U^{*}(\delta_{y}\otimes\xi),\delta_{g_{L^{\prime}}(y)}\otimes v_{y}\rangle|\geq\delta_{L^{\prime}}.$

We aim to show that the families $\mathcal{F}:=\{f_{L}~{}|~{}L\in\mathbf{L}_{X}\}$ and $\mathcal{G}:=\{g_{L^{\prime}}~{}|~{}L^{\prime}\in\mathbf{L}_{Y}\}$ provide a coarse equivalence between $(X,\mathscr{I}_{X})$ and $(Y,\mathscr{I}_{Y})$ , and hence conclude the proof thanks to Corollary 3.24.

Firstly, we show that the families $\mathcal{F}$ and $\mathcal{G}$ are coarse. For $L_{1},L_{2}\in\mathbf{L}_{X}$ , consider the set $\{(f_{L_{1}}(x),f_{L_{2}}(x))\;|\;x\in L_{1}\cap L_{2}\}$ , which is contained in $F(\Delta_{L_{1}\cap L_{2}},\delta_{L_{1}}\wedge\delta_{L_{2}},\mathbb{C}\xi)$ . By Lemma 6.4, the set $F(\Delta_{L_{1}\cap L_{2}},\delta_{L_{1}}\wedge\delta_{L_{2}},\mathbb{C}\xi)$ belongs to $\mathscr{I}_{Y}$ . Hence $\mathcal{F}$ is admissible.

Given $E\in\mathscr{I}_{X}$ , we can assume that $E\subseteq L\times L$ for some $L\in\mathbf{L}_{X}$ thanks to Proposition 2.9. Then for any $(x_{1},x_{2})\in E$ , it follows that $(f_{L}(x_{1}),f_{L}(x_{2}))\in F(E,\delta_{L},\mathbb{C}\xi)$ . Hence we have $\mathscr{I}_{X}\subseteq\mathcal{F}^{*}(\mathscr{I}_{Y})$ , which concludes that the family $\mathcal{F}$ is coarse. Similarly, we obtain that the family $\mathcal{G}$ is also coarse.

|\langle U(\delta_{g_{L^{\prime}}(y)}\otimes v_{y}),\delta_{y}\otimes\xi\rangle|\geq\delta_{L^{\prime}}

and there exists $w_{g_{L^{\prime}}(y)}\in p_{g_{L^{\prime}}(y)}\mathcal{H}$ such that

|\langle U(\delta_{g_{L^{\prime}}(y)}\otimes\xi),\delta_{f_{L^{\prime\prime}}(g_{L^{\prime}}(y))}\otimes w_{g_{L^{\prime}}(y)}\rangle|\geq\delta_{L^{\prime\prime}}.

This implies that

\{(y,f_{L^{\prime\prime}}(g_{L^{\prime}}(y)))\;|\;y\in L^{\prime}\}\subseteq F(\Delta_{L^{\prime\prime}},\delta_{L^{\prime}}\wedge\delta_{L^{\prime\prime}},\mathbb{C}\xi+p_{L^{\prime}}\mathcal{H})\in\mathscr{I}_{Y},

which concludes (3.2). Similarly, we obtain (3.1) and finish the proof. ∎

Combining Proposition 6.2 and Proposition 6.3, we conclude the proof of Theorem 6.1.

Finally, we study the $\sigma$ -unitalness of geometric ideals and prove the last sentence in Theorem E. Recall from [24, Definition 7.15] that an ideal $\mathbf{L}$ in a metric space $(X,d)$ is countably generated if there exists a countable subset $\mathcal{S}\subseteq\mathbf{L}$ such that $\mathbf{L}$ is generated by $\mathcal{S}$ .

Lemma 6.5.

Let $(X,d)$ be a discrete metric space of bounded geometry and $\mathbf{L}$ be a countably generated ideal in $(X,d)$ . Then the geometric ideal $C^{*}_{u}(X,\mathscr{I}(\mathbf{L}))$ is $\sigma$ -unital.

Proof.

Since $\mathbf{L}$ is countably generated, it follows from [24, Lemma 7.17] that there exists a countable subset $\{Y_{1},Y_{2},\cdots,Y_{n},\cdots\}$ of $\mathbf{L}$ such that $\mathbf{L}=\{Z\subseteq X~{}|~{}\exists~{}n\in\mathbb{N}\text{ such that }Z\subseteq Y_{n}\}$ . It is then easy to see that $\{\chi_{Y_{n}}~{}|~{}n\in\mathbb{N}\}$ is an approximate unit for $C^{*}_{u}(X,\mathscr{I}(\mathbf{L}))$ , i.e., $C^{*}_{u}(X,\mathscr{I}(\mathbf{L}))$ is $\sigma$ -unital. ∎

Combining Proposition 4.2, Theorem 6.1 and Lemma 6.5, we finally conclude the proof for Theorem E.

7. Discussions on ghostly ideals

In [24], Wang and the second-named author introduced a notion of ghostly ideals in uniform Roe algebras. Here we provide some discussions on the rigidity problem for ghostly ideals and pose some open questions. Recall the following:

Definition 7.1 ([24]).

Let $(X,d)$ be a discrete metric space of bounded geometry, and $\mathbf{L}$ be an ideal in $(X,d)$ . The associated ghostly ideal of $\mathbf{L}$ , denoted by $\tilde{I}(\mathbf{L})$ , is defined as follows:

\tilde{I}(\mathbf{L}):=\{T\in C^{*}_{u}(X)~{}|~{}\forall~{}\varepsilon>0,\mathrm{r}(\mathrm{supp}_{\varepsilon}(T))\in\mathbf{L}\}.

Similar to the situation for geometric ideals, we would also like to study isomorphisms between ghostly ideals. More precisely, similar to the discussions in Section 4 to Section 6, we pose the following questions:

Question 7.2.

Let $(X,d_{X}),(Y,d_{Y})$ be discrete metric spaces of bounded geometry, and $\mathbf{L}_{X},\mathbf{L}_{Y}$ be ideals in $(X,d_{X})$ and $(Y,d_{Y})$ , respectively.

(1)

If $(X,\mathscr{I}(\mathbf{L}_{X}))$ and $(Y,\mathscr{I}(\mathbf{L}_{X}))$ are coarsely equivalent (in the sense of Definition C), are $\tilde{I}(\mathbf{L}_{X})$ and $\tilde{I}(\mathbf{L}_{Y})$ Morita equivalent?
(2)

Conversely, if $\tilde{I}(\mathbf{L}_{X})$ and $\tilde{I}(\mathbf{L}_{Y})$ are isomorphic (or stably isomorphic), are $(X,\mathscr{I}(\mathbf{L}_{X}))$ and $(Y,\mathscr{I}(\mathbf{L}_{X}))$ coarsely equivalent?

Unfortunately, currently we are unable to completely answer either of these questions. We merely manage to provide a partial answer to Question 7.2(1). To state our result, we need some preparations.

Let $f:(X,d_{X})\to(Y,d_{Y})$ be a coarse equivalence with a coarse inverse $g:Y\to X$ , and $\mathbf{L}$ be an ideal in $(X,d_{X})$ . Denote $f_{\ast}(\mathbf{L})$ the ideal in $(Y,d_{Y})$ generated by the set $\{f(L)~{}|~{}L\in\mathbf{L}\}$ . Then we have:

Lemma 7.3.

With the same notation as above, we have $g_{\ast}f_{\ast}(\mathbf{L})=\mathbf{L}$ . Hence in this case, the maps $f_{\ast}$ and $g_{\ast}$ provide a one-to-one correspondence between ideals in $(X,d_{X})$ and those in $(Y,d_{Y})$ .

Proof.

Firstly, note that $gf(L)\in\mathbf{L}$ for any $L\in\mathbf{L}$ , which implies that $\mathbf{L}\subseteq g_{\ast}f_{\ast}(\mathbf{L})$ . Conversely, it suffices to show that $g(L^{\prime})\in\mathbf{L}$ for any $L^{\prime}\in f_{\ast}(\mathbf{L})$ . Given $L^{\prime}\in f_{\ast}(\mathbf{L})$ , It follows from [24, Lemma 7.18] that there exist $R>0$ and $L\in\mathbf{L}$ such that $L^{\prime}\subseteq\mathcal{N}_{R}(f(L))$ . Hence $g(L^{\prime})\subseteq g(\mathcal{N}_{R}(f(L)))$ , which belongs to $\mathbf{L}$ since $g$ is coarse. So we conclude the proof. ∎

As a direct corollary, we obtain the following:

Corollary 7.4.

Let $f:(X,d_{X})\to(Y,d_{Y})$ be a coarse equivalence and $\mathbf{L}$ be an ideal in $(X,d_{X})$ . Then the family $\mathcal{F}(f):=\{f|_{L}:L\to Y~{}|~{}L\in\mathbf{L}\}$ is a coarse equivalence (in the sense of Definition 3.23) between $(X,\mathscr{I}(\mathbf{L}))$ and $(Y,\mathscr{I}(f_{\ast}(\mathbf{L})))$ .

Now we present our partial answer to Question 7.2(1):

Proposition 7.5.

Let $(X,d_{X}),(Y,d_{Y})$ be discrete metric spaces of bounded geometry, and $\mathbf{L}_{X},\mathbf{L}_{Y}$ be ideals in $(X,d_{X})$ and $(Y,d_{Y})$ , respectively. If $f:(X,d_{X})\to(Y,d_{Y})$ is a coarse equivalence with $f_{\ast}(\mathbf{L}_{X})=\mathbf{L}_{Y}$ , then $\tilde{I}(\mathbf{L}_{X})$ and $\tilde{I}(\mathbf{L}_{Y})$ are Morita equivalent.

Remark 7.6.

By Corollary 7.4, the assumption in Proposition 7.5 implies that $(X,\mathscr{I}(\mathbf{L}_{X}))$ and $(Y,\mathscr{I}(\mathbf{L}_{Y}))$ are coarsely equivalent, which is the assumption considered in Question 7.2(1). However, it is unclear to us whether the converse holds or not.

Proof of Proposition 7.5.

Let $f:(X,d_{X})\to(Y,d_{Y})$ be a coarse equivalence and $\mathscr{E}(f)$ the associated coarse correspondence. Similar to the proof of Proposition 4.2, we define $\mathcal{M}(f)$ to be the norm closure of

\{T\in\mathbb{C}_{u}[X\sqcup Y,\mathscr{E}(f)]~{}|~{}\mathrm{supp}(T)\subseteq X\times Y\}

in $\mathfrak{B}(\ell^{2}(X\sqcup Y))$ , which is a left $C^{*}_{u}(X)$ -module as well as a right $C^{*}_{u}(Y)$ -module under composition of operators. Writing $\tilde{I}_{X}:=\tilde{I}(\mathbf{L}_{X})$ and $\tilde{I}_{Y}:=\tilde{I}(\mathbf{L}_{Y})$ , we set $\mathcal{M}:=\tilde{I}_{X}\mathcal{M}(f)\tilde{I}_{Y}$ . Also define the inner products

{}_{\tilde{I}_{X}}\langle\cdot,\cdot\rangle:\mathcal{M}\times\mathcal{M}\longrightarrow\tilde{I}_{X}\quad\text{by}\quad_{\tilde{I}_{X}}\langle T,S\rangle:=TS^{*}\quad\text{for}\quad T,S\in\mathcal{M},

and

\langle\cdot,\cdot\rangle_{\tilde{I}_{Y}}:\mathcal{M}\times\mathcal{M}\longrightarrow\tilde{I}_{Y}\quad\text{by}\quad\langle T,S\rangle_{\tilde{I}_{Y}}:=T^{*}S\quad\text{for}\quad T,S\in\mathcal{M}.

It is straightforward to check that $\mathcal{M}$ is a right Hilbert $\tilde{I}_{Y}$ -module as well as a left Hilbert $\tilde{I}_{X}$ -module with commuting actions of $\tilde{I}_{X}$ and $\tilde{I}_{Y}$ on $\mathcal{M}$ . We will show that $\mathcal{M}$ provides a Morita equivalence between $\tilde{I}_{X}$ and $\tilde{I}_{Y}$ .

Firstly, we claim that $\mathcal{M}(f)\tilde{I}_{Y}\mathcal{M}(f)^{*}\subseteq\tilde{I}_{X}$ where $\mathcal{M}(f)^{*}:=\{T^{*}~{}|~{}T\in\mathcal{M}(f)\}$ . Given $T_{1},T_{2}\in\mathbb{C}_{u}[X\sqcup Y,\mathscr{E}(f)]$ with support in $X\times Y$ and $S\in\tilde{I}_{Y}$ , there exists $R>0$ such that

\mathrm{supp}(T_{i})\subseteq\{(x,y)\in X\times Y~{}|~{}d_{Y}(f(x),y)\leq R\}\quad\text{for}\quad i=1,2.

Hence for $x_{1},x_{2}\in X$ , we have

(T_{1}ST^{*}_{2})(x_{1},x_{2})=\sum_{\begin{subarray}{c}y_{1}\in B_{Y}(f(x_{1}),R)\\ y_{2}\in B_{Y}(f(x_{2},R))\end{subarray}}T_{1}(x_{1},y_{1})S(y_{1},y_{2})T_{2}^{*}(y_{2},x_{2}).

Denote $N_{R}:=\sup_{y\in Y}|B_{Y}(y,R)|<\infty$ . Then for any $\varepsilon>0$ and $(x_{1},x_{2})\in\mathrm{supp}_{\varepsilon}(T_{1}ST^{*}_{2})$ , there exist $y_{1}\in B_{Y}(f(x_{1}),R)$ and $y_{2}\in B_{Y}(f(x_{2}),R)$ such that

(y_{1},y_{2})\in\mathrm{supp}_{\varepsilon^{\prime}}(S)\quad\text{for}\quad\varepsilon^{\prime}:=\frac{\varepsilon}{\|T_{1}\|\cdot\|T_{2}\|\cdot N_{R}^{2}}.

Since $S\in\tilde{I}_{Y}$ , there exists $L\in\mathbf{L}_{Y}$ such that $\mathrm{r}(\mathrm{supp}_{\varepsilon^{\prime}}(S))\subseteq L$ . Hence we obtain

f\big{(}\mathrm{r}(\mathrm{supp}_{\varepsilon}(T_{1}ST^{*}_{2}))\big{)}\subseteq\mathcal{N}_{R}\big{(}\mathrm{r}(\mathrm{supp}_{\varepsilon^{\prime}}(S))\big{)}\subseteq\mathcal{N}_{R}(L)\in\mathbf{L}_{Y}.

Therefore by Lemma 7.3 and the assumption that $f_{\ast}(\mathbf{L}_{X})=\mathbf{L}_{Y}$ , we obtain that

\mathrm{r}(\mathrm{supp}_{\varepsilon}(T_{1}ST^{*}_{2}))\in\mathbf{L}_{X},

which implies that $T_{1}ST^{*}_{2}\in\tilde{I}_{X}$ and concludes the claim.

Now we claim that $\tilde{I}_{X}\mathcal{M}(f)=\mathcal{M}(f)\tilde{I}_{Y}$ . Taking a coarse inverse $g:Y\to X$ to $f$ , we decompose $\mathrm{Gr}(g)$ into a finite union of $\mathrm{Gr}(g_{i})$ for $i=1,\cdots,n$ such that each $g_{i}:D_{i}\to X$ is an injective map with $\{D_{i}\}_{i=1}^{n}$ mutually disjoint. For each $i=1,\cdots,n$ , define $T^{\prime}_{i}:=\chi_{\mathrm{Gr}(g_{i})}$ , which is regarded as a bounded operator from $\ell^{2}(Y)$ to $\ell^{2}(X)$ . Clearly, we have $T^{\prime}_{i}\in\mathcal{M}(f)$ , $(T^{\prime}_{i})^{*}T^{\prime}_{i}=\mathrm{Id}_{\ell^{2}(D_{i})}$ and $\sum_{i=1}^{n}(T^{\prime}_{i})^{*}T^{\prime}_{i}=\mathrm{Id}_{\ell^{2}(Y)}$ . Hence for $T\in\mathcal{M}(f)$ and $S\in\tilde{I}_{Y}$ , we have

TS=TS\sum_{i=1}^{n}(T^{\prime}_{i})^{*}T^{\prime}_{i}\in\big{(}\mathcal{M}(f)\tilde{I}_{Y}\mathcal{M}(f)^{*}\big{)}\cdot\mathcal{M}(f)\subseteq\tilde{I}_{X}\mathcal{M}(f),

where the last containment follows from the previous paragraph. By symmetry, we conclude $\tilde{I}_{X}\mathcal{M}(f)=\mathcal{M}(f)\tilde{I}_{Y}$ . Consequently, we obtain $\mathcal{M}=\tilde{I}_{X}\mathcal{M}(f)=\mathcal{M}(f)\tilde{I}_{Y}$ .

From the proof of Proposition 4.2 together with Corollary 7.4, we know that $\langle\mathcal{M}(f),\mathcal{M}(f)\rangle_{C^{*}_{u}(Y)}=C^{*}_{u}(Y)$ . Hence we have

\langle\mathcal{M},\mathcal{M}\rangle_{\tilde{I}_{Y}}=\langle\mathcal{M}(f)\tilde{I}_{Y},\mathcal{M}(f)\tilde{I}_{Y}\rangle_{\tilde{I}_{Y}}=\tilde{I}_{Y}\cdot\langle\mathcal{M}(f),\mathcal{M}(f)\rangle_{C^{*}_{u}(Y)}\cdot\tilde{I}_{Y}=\tilde{I}_{Y}.

Hence the inner $\tilde{I}_{Y}$ -product on $\mathcal{M}$ is full. Similarly, we obtain that the inner $\tilde{I}_{X}$ -product on $\mathcal{M}$ is full. Therefore, $\mathcal{M}$ is a $\tilde{I}_{X}$ - $\tilde{I}_{Y}$ imprimitivity bimodule, which concludes that $\tilde{I}_{X}$ and $\tilde{I}_{Y}$ are Morita equivalent. ∎

Finally, we remark that it seems that the proof of Proposition 7.5 does not work if we only know that $(X,\mathscr{I}(\mathbf{L}_{X}))$ and $(Y,\mathscr{I}(\mathbf{L}_{Y}))$ are coarsely equivalent instead of requiring a global coarse equivalence from $(X,d_{X})$ to $(Y,d_{Y})$ .

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(5.4)		$\displaystyle\left\\|\mu(A)-\frac{\mu(M^{\prime}(x,\delta))}{2}\right\\|$	$\displaystyle=\left\\|\pi\left(\left(\Phi^{-1}(\chi_{A})-\frac{1}{2}\Phi^{-1}(\chi_{M^{\prime}(x,\delta)})\right)\delta_{x}\right)\right\\|$
		$\displaystyle=\left\\|\pi\left(\left(\frac{1}{2}\Phi^{-1}(\chi_{A})-\frac{1}{2}\Phi^{-1}(\chi_{M^{\prime}(x,\delta)\backslash A})\right)\delta_{x}\right)\right\\|.$