# Ghostly ideals in uniform Roe algebras

Qin Wang and Jiawen Zhang  Research Center for Operator Algebras, and Shanghai
Key Laboratory of Pure Mathematics and Mathematical Practice, School of
Mathematical Sciences, East China Normal University, Shanghai, 200241, China.
<EMAIL_ADDRESS>School of Mathematical Sciences, Fudan University, 220
Handan Road, Shanghai, 200433, China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we investigate the ideal structure of uniform Roe algebras for
general metric spaces beyond the scope of Yu’s property A. Inspired by the
ideal of ghost operators coming from expander graphs and in contrast to the
notion of geometric ideal, we introduce a notion of ghostly ideal in a uniform
Roe algebra, whose elements are locally invisible in certain directions at
infinity. We show that the geometric ideal and the ghostly ideal are
respectively the smallest and the largest element in the lattice of ideals
with a common invariant open subset of the unit space of the coarse groupoid
by Skandalis-Tu-Yu, and hence the study of ideal structure can be reduced to
classifying ideals between the geometric and the ghostly ones. As an
application, we provide a concrete description for the maximal ideals in a
uniform Roe algebra in terms of the minimal points in the Stone-Čech boundary
of the space. We also provide a criterion to ensure that the geometric and the
ghostly ideals have the same $K$-theory, which helps to recover
counterexamples to the Baum-Connes type conjectures. Moreover, we introduce a
notion of partial Property A for a metric space to characterise the situation
in which the geometric ideal coincides with the ghostly ideal.

QW is partially supported by NSFC (No. 11831006, 12171156), and the Science
and Technology Commission of Shanghai Municipality (No. 22DZ2229014). JZ is
supported by NSFC11871342.

Mathematics Subject Classification (2020): 47L20, 46L80, 51F30.
Keywords: Uniform Roe algebras, Coarse groupoids, Geometric and ghostly
ideals, Maximal ideals, Partial Property A

##  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 on higher index theory [31], where he discovered that the
$K$-theory of Roe algebras serves as a receptacle for higher indices of
elliptic differential operators on open manifolds. Hence the computation for
the $K$-theory of Roe algebras becomes crucial in the study of higher index
theory, and a pragmatic and practical approach is to consult the Baum-Connes
type conjectures [3, 4, 22]. There is also a uniform version of the Roe
algebra, which equally plays a key role in higher index theory (see [39, 41]).
Over the last four decades, there have been a number of excellent works around
this topic (_e.g._ , [17, 21, 24, 46, 47]), which lead to significant
progresses in topology, geometry and analysis (see, _e.g._ , [32, 33]).

On the other hand, the analytic structure of (uniform) Roe algebras reflects
the coarse geometry of the underlying spaces, and the rigidity problem asks
whether the coarse geometry of a metric space can be fully determined by the
associated (uniform) Roe algebra. This problem was initially studied by
Špakula and Willett in [42], followed by a series of works in the last decade
[5, 6, 7, 8, 25]. Recently this problem is completely solved in the uniform
case by the profound work [2], which again highlights the importance of
uniform Roe algebras in coarse geometry. Meanwhile, uniform Roe algebras have
also attained rapidly-growing interest from researchers in mathematical
physics, especially in the theory of topological materials and topological
insulators (see, _e.g._ , [18] and the references therein).

Due to their importance, Chen and the first-named author initiated the study
of the ideal structure for (uniform) Roe algebras [11, 12, 13, 14, 15, 44].
They succeeded in obtaining a full description for the ideal structure of the
uniform Roe algebra when the underlying space has Yu’s Property A (see [12,
14]). However, the general picture is far from clear beyond the scope of
Property A.

In the present paper, we aim to provide a systematic study on the ideal
structure of uniform Roe algebras for general discrete metric spaces. To
outline our main results, let us first explain some notions.

Let $(X,d)$ be a discrete metric space of bounded geometry (see Section 2.2
for precise definitions). Thinking of operators on $\ell^{2}(X)$ as $X$-by-$X$
matrices, we say that such an operator has _finite propagation_ if the non-
zero entries appear only in an entourage of finite width (measured by the
metric on $X$) around the main diagonal (see Section 2.3 for full details).
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 $C^{*}_{u}(X)$.

There is another viewpoint on the uniform Roe algebra based on groupoids.
Recall from [39] that Skandalis, Tu and Yu introduced a notion of coarse
groupoid $G(X)$ associated to a discrete metric space $X$, and they succeeded
in relating coarse geometry to the theory of groupoids. The coarse groupoid
$G(X)$ is a locally compact, Hausdorff, étale and principal groupoid (see
Section 2.5 for precise definitions), and the unit space of $G(X)$ coincides
with the Stone-Čech compactification $\beta X$ of $X$. Moreover, the uniform
Roe algebra $C^{*}_{u}(X)$ can be interpreted as the reduced groupoid
$C^{*}$-algebra of $G(X)$ (see also [34, Chapter 10]).

In [12], Chen and the first-named author concentrated on a class of ideals in
the uniform Roe algebra in which finite propagation operators therein are
dense, and they showed that these ideals can be described geometrically using
the coarse groupoid. More precisely, recall that a subset $U\subseteq\beta X$
is _invariant_ if any element $\gamma$ in $G(X)$ with source in $U$ also has
its range in $U$ (see Section 2.4). As shown in [12] (see also Section 4), for
any ideal $I$ in $C^{*}_{u}(X)$ one can associate an invariant open subset
$U(I)$ of $\beta X$, and conversely for any invariant open subset
$U\subseteq\beta X$ one can associate an ideal $I(U)$ in $C^{*}_{u}(X)$.
Furthermore, these two procedures provide a one-to-one correspondence between
invariant open subsets of $\beta X$ and ideals in $C^{*}_{u}(X)$ in which
finite propagation operators therein are dense.

Based on [12], the first-named author introduced the following notion in [44,
Definition 1.4]:

######  Definition A (Definition 4.4).

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

As explained above, [12, Theorem 6.3] (see also Proposition 4.9) indicates
that the geometric ideals in $C^{*}_{u}(X)$ can be fully determined by
invariant open subsets of $\beta X$, which explains the terminology.
Consequently, the geometric ideals in $C^{*}_{u}(X)$ are easy to handle and
they must have the form of $I(U)$ for some invariant open subset
$U\subseteq\beta X$, called the _geometric ideal associated to $U$_ (see
Definition 4.6). Moreover, it follows from [14, Theorem 4.4] that all ideals
in $C^{*}_{u}(X)$ are geometric when $X$ has Yu’s Property A.

However, things get complicated beyond the context of Property A. As noticed
in [12, Remark 6.5], when $X$ comes from a sequence of expander graphs then
the ideal $I_{G}$ consisting of all ghost operators are _not_ geometric (see
also [21]). Recall that an operator $T\in\mathfrak{B}(\ell^{2}(X))$ is a
_ghost_ if $T\in C_{0}(X\times X)$ when regarding $T$ as a function on
$X\times X$. Ghost operators are introduced by Yu, and they are crucial to
provide counterexamples to the coarse Baum-Connes conjecture ([21]).

Direct calculations show that the associated invariant open subsets for
$I_{G}$ and for the ideal of compact operators in $\mathfrak{B}(\ell^{2}(X))$
are the same, both of which equal $X$ (see also Example 4.8 and 5.5). Hence
for a general metric space $X$ and an invariant open subset $U\subseteq\beta
X$, there might be more than one ideal $I$ in the uniform Roe algebra
$C^{*}_{u}(X)$ satisfying $U(I)=U$. Therefore, the study of the ideal
structure for $C^{*}_{u}(X)$ can be reduced to analyse the lattice (where the
order is given by inclusion)

(1.1)
$\mathfrak{I}_{U}:=\\{I\mbox{~{}is~{}an~{}ideal~{}in~{}}C^{*}_{u}(X):U(I)=U\\}$

for each invariant open subset $U\subseteq\beta X$.

One of the main contributions of the present paper is to find the smallest and
the largest elements in the lattice $\mathfrak{I}_{U}$. Following the
discussions in [12], it is easy to see that $I(U(I))\subseteq I$ for any ideal
$I$ in $C^{*}_{u}(X)$, which implies that the geometric ideal $I(U)$ is the
smallest element in $\mathfrak{I}_{U}$ (see Proposition 4.10). To explore the
largest element, we have to include every ideal $I$ in $C^{*}_{u}(X)$ with
$U(I)=U$. Inspired by the definition of $U(I)$ (see Equality (4.1)), we
introduce the following key notion:

######  Definition B (Definition 5.1).

Let $(X,d)$ be a discrete metric space of bounded geometry and $U$ be an
invariant open subset of $\beta X$. The _ghostly ideal associated to $U$_ is
defined to be

$\tilde{I}(U):=\\{T\in
C^{*}_{u}(X):\overline{\mathrm{r}(\mathrm{supp}_{\varepsilon}(T))}\subseteq
U\mbox{~{}for~{}any~{}}\varepsilon>0\\},$

where $\mathrm{supp}_{\varepsilon}(T):=\\{(x,y)\in X\times
X:|T(x,y)|\geq\varepsilon\\}$ and $\mathrm{r}:X\times X\to X$ is the
projection onto the first coordinate.

We show that $\tilde{I}(U)$ is indeed an ideal in the uniform Roe algebra
$C^{*}_{u}(X)$ (see Lemma 5.2) and moreover, we obtain the following desired
result:

######  Theorem C (Theorem 5.4).

Let $(X,d)$ be a discrete metric space of bounded geometry and $U$ be an
invariant open subset of $\beta X$. Then any ideal $I$ in $C^{*}_{u}(X)$ with
$U(I)=U$ sits between $I(U)$ and $\tilde{I}(U)$. More precisely, the geometric
ideal $I(U)$ is the smallest element while the ghostly ideal $\tilde{I}(U)$ is
the largest element in the lattice $\mathfrak{I}_{U}$ in (1.1).

Theorem C draws the border of the lattice $\mathfrak{I}_{U}$ in (1.1), as an
important step to study the ideal structure of uniform Roe algebras for
general metric spaces. More precisely, once we can bust every ideal between
$I(U)$ and $\tilde{I}(U)$ for each invariant open subset $U\subseteq\beta X$,
then we will obtain a full description for the ideal structure of the uniform
Roe algebra $C^{*}_{u}(X)$. We pose it as an open question in Section 9 and
hope this will be done in some future work.

Concerning the ghostly ideal $\tilde{I}(U)$, we also provide an alternative
picture in terms of limit operators developed in [43], showing that
$\tilde{I}(U)$ consists of operators which vanish in the $(\beta X\setminus
U)$-direction (see Proposition 5.6). Note that ghost operators vanish in all
directions (see Corollary 5.7), and hence operators in $\tilde{I}(U)$ can be
regarded as “partial” ghosts, which clarifies its terminology. Thanks to this
viewpoint, we discover the deep reason behind the counterexample to the
conjecture in [12], constructed by the first-named author in [44, Section 3]
(see Example 5.10).

As an application, we manage to describe maximal ideals in the uniform Roe
algebra. More precisely, it follows directly from Theorem C that maximal
ideals correspond to minimal invariant closed subsets of the Stone-Čech
boundary $\partial_{\beta}X:=\beta X\setminus X$. Moreover using the theory of
limit spaces111Note that the theory of limit spaces and limit operators
developed in [43] only concerns strongly discrete metric spaces of bounded
geometry (see Section 2.2 for precise definitions). Although as noticed in
[43] this will not lose any generality, we put this assumption to simplify
proofs. developed in [43], we prove the following:

######  Proposition D (Proposition 6.2, Corollary 6.3 and Lemma 6.5).

Let $(X,d)$ be a strongly discrete metric space of bounded geometry and $I$ be
a maximal ideal in the uniform Roe algebra $C^{*}_{u}(X)$. Then there exists a
point $\omega\in\partial_{\beta}X$ such that $I$ coincides with the ghostly
ideal $\tilde{I}(\beta X\setminus\overline{X(\omega)})$, where $X(\omega)$ is
the limit space of $\omega$.

A point $\omega\in\partial_{\beta}X$ satisfying the condition in Proposition D
is called a _minimal point_ (see Definition 6.4). We show that there exist a
number of non-minimal points in the boundary even for the simple case of
$X=\mathbb{Z}$:

######  Theorem E (Theorem 6.8).

For the integer group $\mathbb{Z}$ with the usual metric, there exist non-
minimal points in the boundary $\partial_{\beta}\mathbb{Z}$. More precisely,
for any sequence $\\{h_{n}\\}_{n\in\mathbb{N}}$ in $\mathbb{Z}$ tending to
infinity such that $|h_{n}-h_{m}|\to+\infty$ when $n+m\to\infty$ and $n\neq
m$, and any $\omega\in\partial_{\beta}\mathbb{Z}$ with
$\omega(\\{h_{n}\\}_{n\in\mathbb{N}})=1$, then $\omega$ is _not_ a minimal
point.

We provide two approaches to prove Theorem E. One is topological, which makes
use of several constructions and properties of ultrafilters (recalled in
Appendix A). The other is $C^{*}$-algebraic, which replies on a description of
maximal ideals in terms of limit operators (see Lemma 6.13) together with a
recent result by Roch [30].

Returning to the lattice $\mathfrak{I}_{U}$ defined in (1.1), we already
notice that generally $\mathfrak{I}_{U}$ consists of more than one element.
Hence it will be interesting and important to explore when $\mathfrak{I}_{U}$
has only a single element, or equivalently (thanks to Theorem C), when the
geometric ideal $I(U)$ coincides with the ghostly ideal $\tilde{I}(U)$.

To study this problem, we start with an extra picture for geometric and
ghostly ideals using the associated groupoid $C^{*}$-algebras (see Lemma 4.7
and Proposition 5.9). Based on these descriptions, we show that the
amenability of the restriction $G(X)_{\partial_{\beta}X\setminus U}$ of the
coarse groupoid ensures that $I(U)=\tilde{I}(U)$ (see Proposition 7.2).
Meanwhile, we also discuss the $K$-theory of the geometric and ghostly ideals
and provide a criterion to ensure that $K_{\ast}(I(U))=K_{\ast}(\tilde{I}(U))$
for $\ast=0,1$:

######  Proposition F (Proposition 7.8).

Let $X$ be a discrete metric space of bounded geometry which can be coarsely
embedded into some Hilbert space. Then for any countably generated invariant
open subset $U\subseteq\beta X$, we have an isomorphism

$(\iota_{U})_{\ast}:K_{\ast}(I(U))\longrightarrow K_{\ast}(\tilde{I}(U))$

for $\ast=0,1$, where $\iota_{U}$ is the inclusion map.

Note that there is a technical condition of countable generatedness (see
Definition 7.4) used in Proposition F, which holds for a number of examples
(see Example 7.5) including $X$ itself. However as shown in Example 7.11,
there does exist an invariant open subset which is _not_ countably generated.
Applying Proposition F to the case of $U=X$, we partially recover [19,
Proposition 35]. This is crucial in the constructions of counterexamples to
the Baum-Connes type conjectures (see [39] for the coarse version and [20,
Section 5] for the boundary version, which is based on the example considered
in [44, Section 3]). Hence presumably Proposition F will find further
applications in higher index theory.

Conversely, it is natural to ask whether $I(U)=\tilde{I}(U)$ implies that the
restriction groupoid $G(X)_{\partial_{\beta}X\setminus U}$ is amenable. Note
that when $U=X$, [36, Theorem 1.3] implies that $I(X)=\tilde{I}(X)$ is
equivalent to that $X$ has Property A (see also Example 4.8 and 5.5), which is
further equivalent to that the coarse groupoid $G(X)$ is amenable thanks to
[39, Theorem 5.3]. Inspired by these works, we introduce the following partial
version of Property A:

######  Definition G (Definition 8.1).

Let $(X,d)$ be a discrete metric space of bounded geometry and
$U\subseteq\beta X$ be an invariant open subset. We say that $X$ has _partial
Property A towards $\partial_{\beta}X\setminus U$_ if
$G(X)_{\partial_{\beta}X\setminus U}$ is amenable.

Finally we reach the following, which recovers [36, Theorem 1.3] when $U=X$:

######  Theorem H (Theorem 8.16).

Let $(X,d)$ be a strongly discrete metric space of bounded geometry and
$U\subseteq\beta X$ be a countably generated invariant open subset. Then the
following are equivalent:

  1. (1)

$X$ has partial Property A towards $\beta X\setminus U$;

  2. (2)

$\tilde{I}(U)=I(U)$;

  3. (3)

the ideal $I_{G}$ of all ghost operators is contained in $I(U)$.

The proof of Theorem H follows the outline of the case that $U=X$ (cf. [36,
Theorem 1.3]), and is divided into several steps. Firstly, we unpack the
groupoid language of Definition G and provide a concrete geometric description
similar to the definition of Property A (see Proposition 8.2). Then we
introduce a notion of partial operator norm localisation property (Definition
8.9), which is a partial version of the operator norm localisation property
(ONL) introduced in [10]. Parallel to Sako’s result that Property A is
equivalent to ONL ([37]), we show that partial Property A is equivalent to
partial ONL (see Proposition 8.12). Finally thanks to the assumption of
countable generatedness, we conclude Theorem H.

We also remark that in the proof of Theorem H, we make use of the notion of
ideals in spaces introduced by Chen and the first-named author in [12] (see
also Definition 4.12) instead of using invariant open subsets of $\beta X$
directly. This has the advantage of playing within the given space rather than
going to the mysterious Stone-Čech boundary, which allows us to step over
several technical gaps (see, _e.g._ , Remark 8.18).

The paper is organised as follows. In Section 2, we recall necessary
background knowledge in coarse geometry and groupoid theory. In Section 3, we
recall the theory of limit spaces and limit operators developed in [43], which
will be an important tool used throughout the paper. Section 4 is devoted to
the notion of geometric ideals (Definition A) studied in [12, 44], and we also
discuss their minimality in the lattice of ideals $\mathfrak{I}_{U}$ from
(1.1). We introduce the key notion of ghostly ideals (Definition B) in Section
5, prove Theorem C and provide several characterisations for later use. Then
we discuss maximal ideals in uniform Roe algebras in Section 6, and prove
Proposition D and Theorem E. In Section 7, we study the problem when the
geometric ideal coincides with the ghostly ideal, discuss their $K$-theories
and prove Proposition F. Then in Section 8, we introduce the notion of partial
Property A (Definition G) and prove Theorem H. Finally, we list some open
questions in Section 9, and provide Appendix A to record the notion of
ultrafilters and their properties used throughout the paper.

Acknowledgement. We would like to thank Baojie Jiang and Ján Špakula for some
helpful discussions.

##  2\. Preliminaries

###  2.1. Standard notation

Here we collect the notation used throughout the paper.

For a set $X$, denote by $|X|$ the cardinality of $X$. For a subset
$A\subseteq X$, denote by $\chi_{A}$ the characteristic function of $A$, and
set $\delta_{x}:=\chi_{\\{x\\}}$ for $x\in X$.

When $X$ is a locally compact Hausdorff space, we denote by $C(X)$ the set of
complex-valued continuous functions on $X$, and by $C_{b}(X)$ the subset of
bounded continuous functions on $X$. Recall that the support of a function
$f\in C(X)$ is the closure of $\\{x\in X:f(x)\neq 0\\}$, written as
$\mathrm{supp}(f)$, and denote by $C_{c}(X)$ the set of continuous functions
with compact support. We also denote by $C_{0}(X)$ the set of continuous
functions vanishing at infinity, which is the closure of $C_{c}(X)$ with
respect to the supremum norm $\|f\|_{\infty}:=\sup\\{|f(x)|:x\in X\\}$.

When $X$ is discrete, denote $\ell^{\infty}(X):=C_{b}(X)$ and $\ell^{2}(X)$
the Hilbert space of complex-valued square-summable functions on $X$. Denote
by $\mathfrak{B}(\ell^{2}(X))$ the $C^{*}$-algebra of all bounded linear
operators on $\ell^{2}(X)$, and by $\mathfrak{K}(\ell^{2}(X))$ the
$C^{*}$-subalgebra of all compact operators on $\ell^{2}(X)$.

For a discrete space $X$, denote by $\beta X$ its Stone-Čech compactification
and $\partial_{\beta}X:=\beta X\setminus X$ the Stone-Čech boundary.

###  2.2. Notions from coarse geometry

Here we collect necessary notions from coarse geometry, and guide readers to
[27, 34] for more details.

For a discrete metric space $(X,d)$, denote the closed ball by
$B_{X}(x,r):=\\{y\in X:d(x,y)\leq r\\}$ for $x\in X$ and $r\geq 0$. For a
subset $A\subseteq X$ and $r>0$, denote the _$r$ -neighbourhood of $A$ in $X$_
by $\mathcal{N}_{r}(A):=\\{x\in X:d_{X}(x,A)\leq r\\}$. For $R>0$, denote the
_$R$ -entourage_ by $E_{R}:=\\{(x,y)\in X\times X:d(x,y)\leq R\\}$.

We say that $(X,d)$ has _bounded geometry_ if for any $r>0$, the number
$\sup_{x\in X}|B_{X}(x,r)|$ is finite. Also say that $(X,d)$ is _strongly
discrete_ if the set $\\{d(x,y):x,y\in X\\}$ is a discrete subset of
$\mathbb{R}$.

Convention. We say that “ _$X$ is a space_” as shorthand for “$X$ is a
strongly discrete metric space of bounded geometry” (as in [43]) throughout
the rest of this paper.

We remark that although our results hold without the assumption of strong
discreteness, we choose to add it so as to simplify the proofs. As discussed
in [43, Section 2], this will not lose any generality since one can always
modify a discrete metric space (using a coarse equivalence) to satisfy this
assumption.

Now we recall the notion of Property A introduced by Yu (see, _e.g._ , [45,
Proposition 1.2.4] for the equivalence to Yu’s original definition):

######  Definition 2.1 ([47]).

A space $(X,d)$ is said to have _Property A_ if for any $\varepsilon,R>0$
there exist an $S>0$ and a function $f:X\times X\to[0,+\infty)$ satisfying:

  1. (1)

$\mathrm{supp}(f)\subseteq E_{S}$;

  2. (2)

for any $x\in X$, we have $\sum_{z\in X}f(z,x)=1$;

  3. (3)

for any $x,y\in X$ with $d(x,y)\leq R$, then $\sum_{z\in
X}|f(z,x)-f(z,y)|\leq\varepsilon$.

Using a standard normalisation argument, we have the following:

######  Lemma 2.2.

A space $(X,d)$ has Property A _if and only if_ for any $\varepsilon,R>0$
there exist an $S>0$ and a function $f:X\times X\to[0,+\infty)$ satisfying:

  1. (1)

$\mathrm{supp}(f)\subseteq E_{S}$;

  2. (2)

for any $x\in X$, we have $|\sum_{z\in X}f(z,x)-1|\leq\varepsilon$;

  3. (3)

for any $x,y\in X$ with $d(x,y)\leq R$, then $\sum_{z\in
X}|f(z,x)-f(z,y)|\leq\varepsilon$.

We also need a characterisation for Property A using kernels. Recall that a
_kernel_ on $X$ is a function $k\colon X\times X\rightarrow\mathbb{R}$. We say
that $k$ is of positive type if for any $n\in\mathbb{N}$,
$x_{1},\ldots,x_{n}\in X$ and $\lambda_{1},\ldots,\lambda_{n}\in\mathbb{R}$,
we have:

$\sum_{i,j=1}^{n}\lambda_{i}\lambda_{j}k(x_{i},x_{j})\geq 0.$

The following is well-known (see, _e.g._ , [45, Proposition 1.2.4]):

######  Lemma 2.3.

A space $(X,d)$ has Property A if and only if for any $R>0$ and
$\varepsilon>0$, there exist $S>0$ and a kernel $k:X\times X\to\mathbb{R}$ of
positive type satisfying the following:

  1. (1)

for $x,y\in X$, we have $k(x,y)=k(y,x)$ and $k(x,x)=1$;

  2. (2)

for $x,y\in X$ with $d(x,y)\geq S$, we have $k(x,y)=0$;

  3. (3)

for $x,y\in X$ with $d(x,y)\leq R$, we have $|1-k(x,y)|\leq\varepsilon$.

We also recall the notion of coarse embedding:

######  Definition 2.4.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces and $f:X\to Y$ be a map. We
say that $f$ is a _coarse embedding_ if there exist functions
$\rho_{\pm}:[0,\infty)\to[0,\infty)$ with
$\lim_{t\to+\infty}\rho_{\pm}(t)=+\infty$ such that for any $x,y\in X$ we have

$\rho_{-}(d_{X}(x,y))\leq d_{Y}(f(x),f(y))\leq\rho_{+}(d_{X}(x,y)).$

If additionally there exists $C>0$ such that $Y=\mathcal{N}_{C}(f(X))$, then
we say that $f$ is a _coarse equivalence_ and $(X,d_{X}),(Y,d_{Y})$ are
_coarsely equivalent_.

###  2.3. Uniform Roe algebras

Let $(X,d)$ be a discrete metric space. 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 $T(x,y)=\langle
T\delta_{y},\delta_{x}\rangle\in\mathbb{C}$. We also regard
$T\in\mathfrak{B}(\ell^{2}(X))$ as a bounded function on $X\times X$, _i.e._ ,
an element in $\ell^{\infty}(X\times X)$. Denote by $\|T\|$ the operator norm
of $T$ in $\mathfrak{B}(\ell^{2}(X))$, and $\|T\|_{\infty}$ the supremum norm
when regarding $T$ as a function in $\ell^{\infty}(X\times X)$. It is clear
that $\|T\|_{\infty}\leq\|T\|$ for any $T\in\mathfrak{B}(\ell^{2}(X))$.

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

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

and the _propagation_ of $T$ to be

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

######  Definition 2.5.

Let $(X,d)$ be a space.

  1. (1)

The set of all finite propagation operators in $\mathfrak{B}(\ell^{2}(X))$
forms a $\ast$-algebra, called the _algebraic uniform Roe algebra of $X$_ and
denoted by $\mathbb{C}_{u}[X]$. For each $R\geq 0$, denote the subset

$\mathbb{C}_{u}^{R}[X]:=\\{T\in\mathfrak{B}(\ell^{2}(X)):\mathrm{prop}(T)\leq
R\\}.$

It is clear that $\mathbb{C}_{u}[X]=\bigcup_{R\geq 0}\mathbb{C}_{u}^{R}[X]$.

  2. (2)

The _uniform Roe algebra of $X$_ is defined to be the operator norm closure of
$\mathbb{C}_{u}[X]$ in $\mathfrak{B}(\ell^{2}(X))$, which forms a
$C^{*}$-algebra and is denoted by $C^{*}_{u}(X)$.

The following notion was originally introduced by Yu:

######  Definition 2.6.

An operator $T\in C^{*}_{u}(X)$ is called a _ghost_ if $T\in C_{0}(X\times X)$
when regarding $T$ as a function in $\ell^{\infty}(X\times X)$. In other
words, for any $\varepsilon$ there exists a finite subset $F\subseteq X$ such
that for any $(x,y)\notin F\times F$, we have $|T(x,y)|<\varepsilon$.

It is easy to see that all the ghost operators in $C^{*}_{u}(X)$ form an ideal
in $C^{*}_{u}(X)$, denoted by $I_{G}$. Intuitively speaking, a ghost operator
is locally invisible at infinity in all directions. This will be made more
precise in the sequel.

###  2.4. Groupoids and $C^{*}$-algebras

We collect here some basic notions and terminology on groupoids. Details can
be found in [28], or [38] in the étale case.

Recall that a _groupoid_ is a small category, in which every morphism is
invertible. More precisely, a groupoid consists of a set $\mathcal{G}$, a
subset $\mathcal{G}^{(0)}$ called the _unit space_ , two maps
$\mathrm{s},\mathrm{r}:\mathcal{G}\to\mathcal{G}^{(0)}$ called the _source_
and _range_ maps respectively, a _composition law_ :

$\mathcal{G}^{(2)}:=\\{(\gamma_{1},\gamma_{2})\in\mathcal{G}\times\mathcal{G}:\mathrm{s}(\gamma_{1})=\mathrm{r}(\gamma_{2})\\}\ni(\gamma_{1},\gamma_{2})\mapsto\gamma_{1}\gamma_{2}\in\mathcal{G},$

and an _inverse_ map $\gamma\mapsto\gamma^{-1}$. These operations satisfy a
couple of axioms, including associativity law and the fact that elements in
$\mathcal{G}^{(0)}$ act as units.

For $x\in\mathcal{G}^{(0)}$, denote $\mathcal{G}^{x}:=\mathrm{r}^{-1}(x)$ and
$\mathcal{G}_{x}:=\mathrm{s}^{-1}(x)$. For $Y\subseteq\mathcal{G}^{(0)}$,
denote $\mathcal{G}_{Y}^{Y}:=\mathrm{r}^{-1}(Y)\cap\mathrm{s}^{-1}(Y)$. Note
that $\mathcal{G}_{Y}^{Y}$ is a subgroupoid of $\mathcal{G}$ (in the sense
that it is stable under multiplication and inverse), called the _reduction of
$\mathcal{G}$ by $Y$_. A subset $Y$ is said to be _invariant_ if
$\mathrm{r}^{-1}(Y)=\mathrm{s}^{-1}(Y)$, and we write $\mathcal{G}_{Y}$
instead of $\mathcal{G}_{Y}^{Y}$ in this case.

A _locally compact Hausdorff_ groupoid is a groupoid equipped with a locally
compact and Hausdorff topology such that the structure maps (composition and
inverse) are continuous with respect to the induced topologies. Such a
groupoid is called _étale_ (also called _$r$ -discrete_) if the range (hence
the source) map is a local homeomorphism. Clearly in this case, each fibre
$\mathcal{G}^{x}$ (and $\mathcal{G}_{x}$) is discrete with the induced
topology, and $\mathcal{G}^{(0)}$ is clopen in $\mathcal{G}$. The notion of
étaleness for a groupoid can be regarded as an analogue of discreteness in the
group case.

######  Example 2.7.

Let $X$ be a set. The _pair groupoid_ of $X$ is $X\times X$ as a set, whose
unit space is $\\{(x,x)\in X\times X:x\in X\\}$ and identified with $X$ for
simplicity. The source map is the projection onto the second coordinate and
the range map is the projection onto the first coordinate. The composition is
given by $(x,y)\cdot(y,z)=(x,z)$ for $x,y,z\in X$. When $X$ is a discrete
Hausdorff space, then $X\times X$ is a locally compact Hausdorff étale
groupoid.

Now we introduce the algebras associated to groupoids. Here we only focus on
the case of étaleness, and guide readers to [28] for the general case.

Let $\mathcal{G}$ be a locally compact, Hausdorff and étale groupoid with unit
space $\mathcal{G}^{(0)}$. Note that the space $C_{c}(\mathcal{G})$ is a
$\ast$-involutive algebra with respect to the following operations: for
$f,g\in C_{c}(\mathcal{G})$,

$\displaystyle(f\ast g)(\gamma)$ $\displaystyle=$
$\displaystyle\sum_{\alpha\in\mathcal{G}_{\mathrm{s}(\gamma)}}f(\gamma\alpha^{-1})g(\alpha),$
$\displaystyle f^{*}(\gamma)$ $\displaystyle=$
$\displaystyle\overline{f(\gamma^{-1})}.$

Consider the following algebraic norm on $C_{c}(\mathcal{G})$ defined by:

$\|f\|_{I}:=\mathrm{max}\left\\{\sup_{x\in\mathcal{G}^{(0)}}\sum_{\gamma\in\mathcal{G}_{x}}|f(\gamma)|,~{}\sup_{x\in\mathcal{G}^{(0)}}\sum_{\gamma\in\mathcal{G}_{x}}|f^{*}(\gamma)|\right\\}.$

The completion of $C_{c}(\mathcal{G})$ with respect to the norm
$\|\cdot\|_{I}$ is denoted by $L^{1}(\mathcal{G})$.

The _maximal (full) groupoid $C^{*}$-algebra_
$C^{*}_{\mathrm{max}}(\mathcal{G})$ is defined to be the completion of
$C_{c}(\mathcal{G})$ with respect to the norm:

$\|f\|_{\mathrm{max}}:=\sup\|\pi(f)\|,$

where the supremum is taken over all $\ast$-representations $\pi$ of
$L^{1}(\mathcal{G})$.

In order to define the reduced counterpart, we recall that for each
$x\in\mathcal{G}^{(0)}$ the _regular representation at $x$_, denoted by
$\lambda_{x}:C_{c}(\mathcal{G})\to\mathfrak{B}(\ell^{2}(\mathcal{G}_{x}))$, is
defined as follows:

(2.1)
$\big{(}\lambda_{x}(f)\xi\big{)}(\gamma):=\sum_{\alpha\in\mathcal{G}_{x}}f(\gamma\alpha^{-1})\xi(\alpha),\quad\mbox{where}~{}f\in
C_{c}(\mathcal{G})\mbox{~{}and~{}}\xi\in\ell^{2}(\mathcal{G}_{x}).$

It is routine work to check that $\lambda_{x}$ is a well-defined
$\ast$-homomorphism. The _reduced norm_ on $C_{c}(\mathcal{G})$ is

$\|f\|_{r}:=\sup_{x\in\mathcal{G}^{(0)}}\|\lambda_{x}(f)\|,$

and the _reduced groupoid $C^{*}$-algebra_ $C^{*}_{r}(\mathcal{G})$ is defined
to be the completion of the $\ast$-algebra $C_{c}(\mathcal{G})$ with respect
to this norm. Clearly, each regular representation $\lambda_{x}$ can be
extended to a homomorphism
$\lambda_{x}:C^{*}_{r}(\mathcal{G})\to\mathfrak{B}(\ell^{2}(\mathcal{G}_{x}))$
automatically. It is also routine to check that there is a canonical
surjective homomorphism from $C^{*}_{\mathrm{max}}(\mathcal{G})$ to
$C^{*}_{r}(\mathcal{G})$.

###  2.5. Coarse groupoids

Let $(X,d)$ be a space as in Section 2.2. The coarse groupoid $G(X)$ on $X$
was introduced by Skandalis, Tu and Yu in [39] (see also [34, Chapter 10]) to
relate coarse geometry to the theory of groupoids. As a topological space,

$G(X):=\bigcup_{r>0}{\overline{E_{r}}}^{\beta(X\times
X)}\subseteq\beta(X\times X).$

Recall from Example 2.7 that $X\times X$ is the pair groupoid with source and
range maps $\mathrm{s}(x,y)=y$ and $\mathrm{r}(x,y)=x$. These maps extend to
maps $G(X)\to\beta X$, still denoted by $\mathrm{r}$ and $\mathrm{s}$.

Now consider $(\mathrm{r},\mathrm{s}):G(X)\to\beta X\times\beta X$. It was
shown in [39, Lemma 2.7] that the map $(\mathrm{r},\mathrm{s})$ is injective,
and hence $G(X)$ can be endowed with a groupoid structure induced by the pair
groupoid $\beta X\times\beta X$, called the _coarse groupoid_ of $X$.
Therefore, $G(X)$ can also be equivalently defined by

$G(X):=\bigcup_{r>0}{\overline{E_{r}}}^{\beta X\times\beta X}\subseteq\beta
X\times\beta X,$

with the weak topology. It was also shown in [39, Proposition 3.2] that the
coarse groupoid $G(X)$ is locally compact, Hausdorff, étale and principal.
Clearly, the unit space of $G(X)$ can be identified with $\beta X$.

Given $f\in C_{c}(G(X))$, then $f$ is a continuous function supported on
$\overline{E_{r}}$ for some $r>0$; equivalently, we can interpret $f$ as a
bounded function on $E_{r}$. Hence we define an operator $\theta(f)$ on
$\ell^{2}(X)$ by setting its matrix coefficients to be
$\theta(f)(x,y):=f(x,y)$ for $x,y\in X$. We have the following:

######  Proposition 2.8 ([34, Proposition 10.29]).

The map $\theta$ provides a $\ast$-isomorphism from $C_{c}(G(X))$ to
$\mathbb{C}_{u}[X]$, and extends to a $C^{*}$-isomorphism
$\Theta:C^{*}_{r}(G(X))\to C^{*}_{u}(X)$. Note that $\Theta$ maps the
$C^{*}$-subalgebra $C^{*}_{r}(X\times X)$ onto the compact operators
$\mathfrak{K}(\ell^{2}(X))$.

Recall from Section 2.3 that an operator $T\in\mathfrak{B}(\ell^{2}(X))$ can
be regarded as an element in $\ell^{\infty}(X\times X)$. Following the
notation from [12], we denote by $\overline{T}$ the continuous extension of
$T$ on $\beta(X\times X)$ when regarding $T\in\ell^{\infty}(X\times X)$. Then

$\mathrm{supp}(\overline{T})=\overline{\mathrm{supp}(T)}.$

Note that $G(X)$ is open in $\beta(X\times X)$, hence $C_{0}(G(X))$ is a
subalgebra in $C(\beta(X\times X))$. Restricting to $G(X)$, we also regard
$\overline{T}$ as a function on $G(X)$ and hence we can talk of the value
$\overline{T}(\alpha,\gamma)$ for $(\alpha,\gamma)\in G(X)$. Moreover, we have
the following:

######  Lemma 2.9.

For $T\in\mathbb{C}_{u}[X]$, we have $\overline{T}\in C_{c}(G(X))$ and
$\theta(\overline{T})=T$. For $T\in C^{*}_{u}(X)$, we have $\overline{T}\in
C_{0}(G(X))$ and $\Theta(\overline{T})=T$.

###### Proof.

The first statement is a direct corollary of Proposition 2.8. Since
$\|T\|_{\infty}\leq\|T\|$ for any $T\in\mathfrak{B}(\ell^{2}(X))$, the second
follows from the first. ∎

###  2.6. Amenability and a-T-menability for groupoids

Amenable groupoids comprise a large class of groupoids with relatively nice
properties, which are literally the analogue of amenable groups in the world
of groupoids. Here we only focus on the case of étaleness, in which the notion
of amenability behaves quite well. A standard reference is [1] and another
reference for just étale groupoids is [9, Chapter 5.6].

######  Definition 2.10 ([1]).

A locally compact, Hausdorff and étale groupoid $\mathcal{G}$ is said to be
_(topologically) amenable_ if for any $\varepsilon>0$ and compact
$K\subseteq\mathcal{G}$, there exists $f\in C_{c}(\mathcal{G})$ with range in
$[0,1]$ such that for any $\gamma\in K$ we have

$\sum_{\alpha\in\mathcal{G}_{\mathrm{r}(\gamma)}}f(\alpha)=1\quad\mbox{and}\quad\sum_{\alpha\in\mathcal{G}_{\mathrm{r}(\gamma)}}|f(\alpha)-f(\alpha\gamma)|<\varepsilon.$

Similar to the case of Property A, we have the following by a standard
normalisation argument:

######  Lemma 2.11.

A locally compact, Hausdorff and étale groupoid $\mathcal{G}$ is amenable _if
and only if_ for any $\varepsilon>0$ and compact $K\subseteq\mathcal{G}$,
there exists $f\in C_{c}(\mathcal{G})$ with range in $[0,+\infty)$ such that
for any $\gamma\in K$ we have

$\big{|}\sum_{\alpha\in\mathcal{G}_{\mathrm{r}(\gamma)}}f(\alpha)-1\big{|}<\varepsilon\quad\mbox{and}\quad\sum_{\alpha\in\mathcal{G}_{\mathrm{r}(\gamma)}}|f(\alpha)-f(\alpha\gamma)|<\varepsilon.$

Amenability for étale groupoids enjoy similar permanence properties as in the
case of groups. For example, open or closed subgroupoids of amenable étale
groupoids are amenable, and amenability is preserved under taking groupoid
extensions. See [1, Section 5] for details. Also recall that we have the
following:

######  Proposition 2.12 ([9, Corollary 5.6.17]).

Let $\mathcal{G}$ be a locally compact, Hausdorff, étale and amenable
groupoid. Then the natural quotient $C^{*}_{\mathrm{max}}(\mathcal{G})\to
C^{*}_{r}(\mathcal{G})$ is an isomorphism.

Now we recall the notion of a-T-menability for groupoids introduced by Tu
[40]. Let $\mathcal{G}$ be a locally compact, Hausdorff and étale groupoid. A
continuous function $f:\mathcal{G}\to\mathbb{R}$ is said to be of _negative
type_ if

  1. (1)

$f|_{\mathcal{G}^{(0)}}=0$;

  2. (2)

for any $\gamma\in\mathcal{G}$, $f(\gamma)=f(\gamma^{-1})$;

  3. (3)

Given $\gamma_{1},\cdots,\gamma_{n}\in\mathcal{G}$ with the same range and
$\lambda_{1},\cdots,\lambda_{n}\in\mathbb{R}$ with
$\sum_{i=1}^{n}\lambda_{i}=0$, we have
$\sum_{i,j}\lambda_{i}\lambda_{j}f(\gamma_{i}^{-1}\gamma_{j})\leq 0$.

A continuous function $f:\mathcal{G}\to\mathbb{R}$ is called _locally proper_
if for any compact subset $K\subseteq\mathcal{G}^{(0)}$, the restriction of
$f$ on $\mathcal{G}_{K}^{K}$ is proper.

######  Definition 2.13 ([40, Section 3.3]).

A locally compact, Hausdorff and étale groupoid $\mathcal{G}$ is said to be
_a-T-menable_ if there exists a continuous locally proper function
$f:\mathcal{G}\to\mathbb{R}$ of negative type on $\mathcal{G}$.

Analogous to the case of groups, Tu [40] proved that a locally compact,
$\sigma$-compact, Hausdorff and étale groupoid $\mathcal{G}$ is a-T-menable
_if and only if_ there exists a continuous field of Hilbert spaces over
$\mathcal{G}^{(0)}$ with a proper affine action of $\mathcal{G}$. We also need
the following significant result by Tu:

######  Proposition 2.14 ([40, Théorème 0.1]).

Let $\mathcal{G}$ be a locally compact, $\sigma$-compact, Hausdorff, étale and
a-T-menable groupoid. Then $\mathcal{G}$ is _$K$ -amenable_, _i.e._ , the
quotient map induces an isomorphism
$K_{\ast}(C^{*}_{\mathrm{max}}(\mathcal{G}))\to
K_{\ast}(C^{*}_{r}(\mathcal{G}))$ for $\ast=0,1$.

Finally, we record the following result for coarse groupoids:

######  Proposition 2.15 ([39, Theorem 5.3 and 5.4]).

Let $(X,d)$ be a space and $G(X)$ be the associated coarse groupoid. Then:

  1. (1)

$X$ has Property A _if and only if_ $G(X)$ is amenable;

  2. (2)

$X$ can be coarsely embedded into Hilbert space _if and only if_ $G(X)$ is
a-T-menable.

##  3\. Limit spaces and limit operators

In this section, we recall the theory of limit spaces and limit operators for
metric spaces developed by Špakula and Willett in [43], which becomes an
important tool for later use.

Throughout the section, we always assume that $(X,d)$ is a space (see
“Convention” in Section 2.2) and $G(X)$ is its coarse groupoid. We will freely
use the notion of ultrafilters on $X$, and related materials are recalled in
Appendix A.

###  3.1. Limit spaces

First recall that a function $t:D\to R$ with $D,R\subseteq X$ is called a
_partial translation_ if $t$ is a bijection from $D$ to $R$, and $\sup_{x\in
X}d(x,t(x))$ is finite. The _graph_ of $t$ is $\\{(t(x),x):x\in D\\}$, denoted
by $\mathrm{gr}(t)$. It is well-known that each entourage $E$ on $X$ can be
decomposed into finitely many graphs of partial translations (see, _e.g._ ,
[34, Lemma 4.10]) thanks to the bounded geometry of $X$.

######  Definition 3.1 ([43, Definition 3.2 and 3.6]).

Fix an ultrafilter $\omega\in\beta X$. A partial translation $t:D\to R$ on $X$
is _compatible with $\omega$_ if $\omega(D)=1$. In this case, regarding $t$ as
a function from $D$ to $\beta X$, we define the following thanks to Lemma A.2:

$t(\omega):=\lim_{\omega}t\in\beta X.$

In other words, consider the extension
$\overline{t}:\overline{D}\to\overline{R}$ then
$t(\omega)=\overline{t}(\omega)$.

An ultrafilter $\alpha\in\beta X$ is _compatible with $\omega$_ if there
exists a partial translation $t$ compatible with $\omega$ and
$t(\omega)=\alpha$. Denote by $X(\omega)$ the collection of all ultrafilters
on $X$ compatible with $\omega$. A _compatible family for $\omega$_ is a
collection of partial translations $\\{t_{\alpha}\\}_{\alpha\in X(\omega)}$
such that each $t_{\alpha}$ is compatible with $\omega$ and
$t_{\alpha}(\omega)=\alpha$.

Fix an ultrafilter $\omega$ on $X$, and a compatible family
$\\{t_{\alpha}\\}_{\alpha\in X(\omega)}$. Define a function
$d_{\omega}:X(\omega)\times X(\omega)\to[0,\infty)$ by

$d_{\omega}(\alpha,\beta):=\lim_{x\to\omega}d(t_{\alpha}(x),t_{\beta}(x)).$

It is shown in [43, Proposition 3.7] that $d_{\omega}$ is a uniformly discrete
metric of bounded geometry on $X(\omega)$ which does not depend on the choice
of $\\{t_{\alpha}\\}$.

This leads to the following:

######  Definition 3.2 ([43, Definition 3.8]).

For each non-principal ultrafilter $\omega$ on $X$, the metric space
$(X(\omega),d_{\omega})$ is called the _limit space_ of $X$ at $\omega$, which
is a space in the sense of “Convention” in Section 2.2.

It is shown in [43, Proposition 3.9] that for any $\alpha\in X(\omega)$, we
have $X(\alpha)=X(\omega)$ as metric spaces. Also note that when $\omega$ is
principal, _i.e._ , $\omega\in X$, then it is clear that
$(X(\omega),d_{\omega})=(X,d)$.

We recall the following result, which reveals that the local geometry of $X$
can be recaptured by those of the limit spaces.

######  Proposition 3.3 ([43, Proposition 3.10]).

Let $\omega$ be a non-principal ultrafilter on $X$, and
$\\{t_{\alpha}:D_{\alpha}\to R_{\alpha}\\}$ a compatible family for $\omega$.
Then for each finite $F\subseteq X(\omega)$, there exists a subset $Y\subseteq
X$ with $\omega(Y)=1$ such that for each $y\in Y$, there is a finite subset
$G(y)\subseteq X$ such that the map

$f_{y}:F\to G(y),\alpha\mapsto t_{\alpha}(y)$

is a surjective isometry. Such a collection $\\{f_{y}\\}_{y\in Y}$ is called a
_local coordinate systerm_ for $F$, and the maps $f_{y}$ are called _local
coordinates_.

Furthermore, if $F$ is a metric ball $B(\omega,r)$, then there exist
$Y\subseteq X$ with $\omega(Y)=1$ and a local coordinate system $\\{f_{y}:F\to
G(y)\\}_{y\in Y}$ such that each $G(y)$ is the ball $B(y,r)$.

As shown in [43, Appendix C], limit spaces can be described in terms of the
coarse groupoid $G(X)$:

######  Lemma 3.4 ([43, Lemma C.3]).

Given a non-principal ultrafilter $\omega\in\beta X$, the map

$F:X(\omega)\rightarrow G(X)_{\omega},\quad\alpha\mapsto(\alpha,\omega)$

is a bijection. Hence $X(\omega)$ is the smallest invariant subset of $\beta
X$ containing $\omega$. Here we consider $G(X)$ as a subset of $\beta
X\times\beta X$, as explained in Section 2.5.

Consequently, we obtain the following:

######  Corollary 3.5.

As a set, we have

$G(X)=\big{(}X\times
X\big{)}\sqcup\bigcup_{\omega\in\partial_{\beta}X}\big{(}X(\omega)\times
X(\omega)\big{)}.$

Now we would like to provide a quantitative version for Corollary 3.5. First
we record the following observation, whose proof is straightforward and almost
identical to that of Lemma 3.4 (originally from [34, discussion in
10.18-10.24]):

######  Lemma 3.6.

Let $t:D\to R$ be a partial translation on $X$. Then we have:

$\overline{\mathrm{gr}(t)}^{\beta X\times\beta
X}=\mathrm{gr}(t)\sqcup\bigcup_{\omega\in\partial_{\beta}X}\big{\\{}(\alpha,\omega):\omega(D)=1\text{
and }\alpha=t(\omega)\big{\\}}.$

In general, we have the following:

######  Lemma 3.7.

For any $S\geq 0$, we have:

$\overline{E_{S}}^{\beta X\times\beta
X}=E_{S}\sqcup\bigcup_{\omega\in\partial_{\beta}X}\big{\\{}(\alpha,\gamma)\in
X(\omega)\times X(\omega):d_{\omega}(\alpha,\gamma)\leq S\big{\\}}.$

###### Proof.

As explained at the beginning of this subsection, we can decompose

$E_{S}=\mathrm{gr}(t_{1})\sqcup\cdots\sqcup\mathrm{gr}(t_{N})$

where each $t_{i}:D_{i}\to R_{i}$ is a partial translation. Hence we have

$\overline{E_{S}}=\overline{\mathrm{gr}(t_{1})}\cup\cdots\cup\overline{\mathrm{gr}(t_{N})}.$

Applying Lemma 3.6, we obtain that $\overline{E_{S}}$ is contained in the
right hand side in the statement.

On the other hand, given $\omega\in\partial_{\beta}X$ and $\alpha,\gamma\in
X(\omega)$ with $d_{\omega}(\alpha,\gamma)\leq S$, then we have
$(X(\omega),d_{\omega})=(X(\gamma),d_{\gamma})$. Take a partial translation
$t:D\to R$ such that $\gamma(D)=1$ and $\alpha=t(\gamma)$. Note that

$d_{\gamma}(\alpha,\gamma)=\lim_{x\to\gamma}d(t(x),x)\leq S.$

Hence for $D^{\prime}:=\\{x\in D:d(t(x),x)\leq S\\}$, we have
$\gamma(D^{\prime})=1$. Consider the restriction of $t$ on $D^{\prime}$,
denoted by $t^{\prime}$. Then $t^{\prime}$ is also a partial translation and
$\alpha=t^{\prime}(\gamma)$. By Lemma 3.6, we obtain that
$(\alpha,\gamma)\in\overline{\mathrm{gr}(t^{\prime})}$, which is contained in
$\overline{E_{S}}$ as desired. ∎

Now we compute concrete examples of limit spaces. First we recall the case of
groups from [43, Appendix B]. Let $\Gamma$ be a countable discrete group,
equipped with a left-invariant bounded geometry and strongly discrete metric
$d$. For each $g\in\Gamma$, denote

$\rho_{g}:\Gamma\to\Gamma,h\mapsto hg$

the right translation map. Each $\rho_{g}$ is a partial translation with full
domain, and hence is compatible with every $\omega\in\beta\Gamma$. Moreover,
we have the following:

######  Lemma 3.8 ([43, Lemma B.1]).

For each non-principal ultrafilter $\omega\in\beta\Gamma$, the map

$b_{\omega}:\Gamma\longrightarrow\Gamma(\omega),g\mapsto\rho_{g}(\omega)$

is an isometric bijection.

Inspired by Lemma 3.8, we provide the following general method:

######  Lemma 3.9.

Let $\\{t_{\lambda}:D_{\lambda}\to R_{\lambda}\\}_{\lambda\in\Lambda}$ be a
family of partial translations on $X$ satisfying the following: for each
$S>0$, there exists a finite subset $\Lambda_{S}\subseteq\Lambda$ such that
$\mathrm{gr}(t_{\lambda})\cap\mathrm{gr}(t_{\mu})$ is finite for
$\lambda\neq\mu$ in $\Lambda_{S}$ and
$E_{S}\setminus\big{(}\bigcup_{\lambda\in\Lambda_{S}}\mathrm{gr}(t_{\lambda})\big{)}$
is finite. Then for any non-principal ultrafilter $\omega$ on $X$ and
$\alpha\in X(\omega)$, there exists $\lambda\in\Lambda$ such that
$\omega(D_{\lambda})=1$ and $\alpha=t_{\lambda}(\omega)$.

###### Proof.

By definition, we assume that $\alpha=t(\omega)$ for some partial translation
$t:D\to R$ with $\omega(D)=1$. For $\lambda\in\Lambda$, set
$\widetilde{D}_{\lambda}:=\\{x\in D\cap D_{\lambda}:t(x)=t_{\lambda}(x)\\}$.
Choose $S>0$ such that $\mathrm{gr}(t)\subseteq E_{S}$, and hence there exists
a finite subset $F\subseteq E_{S}$ such that

$\mathrm{gr}(t)\subseteq\big{(}\bigcup_{\lambda\in\Lambda_{S}}\mathrm{gr}(t_{\lambda})\big{)}\sqcup
F.$

This implies that
$D\subseteq\big{(}\bigcup_{\lambda\in\Lambda_{S}}\widetilde{D}_{\lambda}\big{)}\sqcup
F^{\prime}$ for some finite $F^{\prime}\subseteq X$. Since $\omega$ is non-
principal, $\Lambda_{S}$ is finite and
$\widetilde{D}_{\lambda}\cap\widetilde{D}_{\mu}$ is finite for any
$\lambda,\mu\in\Lambda_{S}$, there exists a unique $\lambda\in\Lambda_{S}$
such that $\omega(\widetilde{D}_{\lambda})=1$. This implies that
$\omega(D_{\lambda})=1$ and $\alpha=t_{\lambda}(\omega)$, which concludes the
proof. ∎

Back to the case of the group $\Gamma$, the set
$\\{\rho_{g}:\Gamma\to\Gamma\\}_{g\in\Gamma}$ satisfies the condition in Lemma
3.9, and hence the map $b_{\omega}$ in Lemma 3.8 is surjective. It is
straightforward to check that $b_{\omega}$ is isometric, which recovers the
proof for Lemma 3.8.

######  Example 3.10.

Consider $X=\mathbb{N}$ with the usual metric. For each $k\in\mathbb{Z}$ with
$k\geq 0$, define a partial translation

$\rho_{k}:\mathbb{N}\longrightarrow\mathbb{N},\quad n\mapsto n+k.$

For $k\in\mathbb{Z}$ with $k<0$, define a partial translation

$\rho_{k}:[-k,\infty)\cap\mathbb{N}\longrightarrow\mathbb{N},\quad n\mapsto
n+k.$

Then it is clear that the set $\\{\rho_{k}\\}_{k\in\mathbb{Z}}$ satisfies the
condition in Lemma 3.9. Now for a non-principal ultrafilter $\omega$ on
$\mathbb{N}$, consider the map

$b_{\omega}:\mathbb{Z}\longrightarrow\mathbb{N}(\omega),\quad
k\mapsto\rho_{k}(\omega).$

Note that for $k,l\in\mathbb{Z}$, we have

$d_{\omega}(\rho_{k}(\omega),\rho_{l}(\omega))=\lim_{n\to\omega}|(k+n)-(l+n)|=|k-l|,$

which implies that $b_{\omega}$ is isometric. Moreover, Lemma 3.9 shows that
$b_{\omega}$ is surjective. Therefore, every limit space of $\mathbb{N}$ is
isometric to $\mathbb{Z}$. This provides a detailed proof for [43, Example
3.14(2)].

Similar to the analysis in Example 3.10, we can also apply Lemma 3.9 to obtain
proofs for [43, Example 3.14(3)-(5)]. Details are left to readers.

###  3.2. Limit operators

Now we recall the notion of limit operators for metric spaces introduced by
Špakula and Willett:

######  Definition 3.11 ([43, Definition 4.4]).

For a non-principal ultrafilter $\omega$ on $X$, fix a compatible family
$\\{t_{\alpha}\\}_{\alpha\in X(\omega)}$ for $\omega$ and let $T\in
C^{*}_{u}(X)$. The _limit operator of $T$ at $\omega$_, denoted by
$\Phi_{\omega}(T)$, is an $X(\omega)$-by-$X(\omega)$ indexed matrix defined by

$\Phi_{\omega}(T)_{\alpha\gamma}:=\lim_{x\to\omega}T_{t_{\alpha}(x)t_{\gamma}(x)}\quad\text{for}\quad\alpha,\gamma\in
X(\omega).$

It was studied in [43, Chapter 4] that the above definition does not depend on
the choice of the compatible family $\\{t_{\alpha}\\}_{\alpha\in X(\omega)}$
for $\omega$. Furthermore, the limit operator $\Phi_{\omega}(T)$ is indeed a
bounded operator on $\ell^{2}(X(\omega))$, and belongs to the uniform Roe
algebra $C^{*}_{u}(X(\omega))$.

Recall from Lemma 2.9 that for an operator $T\in C^{*}_{u}(X)$, the continuous
extension $\overline{T}\in C_{0}(G(X))$. We have the following, which was
implicitly mentioned in the proof of [43, Lemma C.3].

######  Lemma 3.12.

For a non-principal ultrafilter $\omega$ on $X$ and $T\in C^{*}_{u}(X)$, we
have

$\Phi_{\omega}(T)_{\alpha\gamma}=\overline{T}(\alpha,\gamma)\quad\text{for}\quad\alpha,\gamma\in
X(\omega).$

###### Proof.

Choose partial translations $t_{\alpha},t_{\gamma}$ compatible with
$\alpha,\gamma$ such that $t_{\alpha}(\omega)=\alpha$,
$t_{\gamma}(\omega)=\gamma$. By definition, we have

$\Phi_{\omega}(T)_{\alpha\gamma}=\lim_{x\to\omega}T_{t_{\alpha}(x)t_{\gamma}(x)}=\lim_{x\to\omega}T(t_{\alpha}(x),t_{\gamma}(x))=\overline{T}(\alpha,\gamma)$

where the last equality comes from the discussion before Lemma 2.9. ∎

Since the limit operator $\Phi_{\omega}(T)$ contains the information of the
asymptotic behaviour of $T$ “in the $\omega$-direction”, we introduce the
following:

######  Definition 3.13.

For an $\omega\in\partial_{\beta}X$, we say that an operator $T\in
C^{*}_{u}(X)$ is _locally invisible (_or_ vanishes) in the $\omega$-direction_
if $\Phi_{\omega}(T)=0$. For a subset $V\subseteq\partial_{\beta}X$, we say
that $T$ is _locally invisible (_or_ vanishes) in the $V$-direction_ if
$\Phi_{\omega}(T)=0$ for any $\omega\in V$.

Finally, we recall from Proposition 2.8 that there is a $C^{*}$-isomorphism
$\Theta:C^{*}_{r}(G(X))\to C^{*}_{u}(X)$. This allows us to relate limit
operators to left regular representations of $C^{*}_{r}(G(X))$:

######  Lemma 3.14 ([43, Lemma C.3]).

For a non-principal ultrafilter $\omega$ on $X$, let
$W_{\omega}:\ell^{2}(G(X)_{\omega})\to\ell^{2}(X(\omega))$ be the unitary
representation induced by $F$ in Lemma 3.4. Then we have the following
commutative diagram:

$\textstyle{C^{*}_{r}(G(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textstyle\lambda_{\omega}}$$\scriptstyle{\textstyle\Theta}$$\scriptstyle{\textstyle\cong}$$\textstyle{\mathfrak{B}(\ell^{2}(G(X)_{\omega}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textstyle\mathrm{Ad}_{W_{\omega}}}$$\scriptstyle{\textstyle\cong}$$\textstyle{C^{*}_{u}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textstyle\Phi_{\omega}}$$\textstyle{\mathfrak{B}(\ell^{2}(X(\omega))),}$

where $\lambda_{\omega}$ is the left regular representation from (2.1).

##  4\. Geometric ideals

In this section, we recall the notion of geometric ideals, which was
originally introduced by the first-named author in [44] (see also [12]).
Throughout the section, let $X$ be a space in the sense of “Convention” in
Section 2.2.

######  Definition 4.1 ([12, Definition 3.1, 3.3]).

For an operator $T\in C^{*}_{u}(X)$ and $\varepsilon>0$, the _$\varepsilon$
-support of $T$_ is defined to be

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

Also define the _$\varepsilon$ -truncation of $T$_ to be

$T_{\varepsilon}(x,y):=\begin{cases}~{}T(x,y),&\mbox{if~{}}|T(x,y)|\geq\varepsilon;\\\
~{}0,&\mbox{otherwise}.\end{cases}$

It is clear that
$\mathrm{supp}(T_{\varepsilon})=\mathrm{supp}_{\varepsilon}(T)$. We also
record the following elementary result for later use. The proof is
straightforward, hence omitted.

######  Lemma 4.2.

Given $T\in C^{*}_{u}(X)$ and $\varepsilon>0$, we have

$\overline{\mathrm{supp}(T_{\varepsilon})}\subseteq\\{\tilde{\omega}\in\beta(X\times
X):|\overline{T}(\tilde{\omega})|\geq\varepsilon\\}\subseteq\overline{\mathrm{supp}(T_{\varepsilon/2})}.$

The following is a key result in [12]:

######  Proposition 4.3 ([12, Theorem 3.5]).

Let $I$ be an ideal in the uniform Roe algebra $C^{*}_{u}(X)$. For each $T\in
I$ and $\varepsilon>0$, we have $T_{\varepsilon}\in I\cap\mathbb{C}_{u}[X]$.
Moreover, we have

$\overline{I\cap\mathbb{C}_{u}[X]}=\overline{\\{T_{\varepsilon}:T\in
I,\varepsilon>0\\}}$

where the closures are taken with respect to the operator norm.

Now we recall the notion of geometric ideals from [44]:

######  Definition 4.4.

An ideal $I$ in the uniform Roe algebra $C^{*}_{u}(X)$ is called _geometric_
if $I\cap\mathbb{C}_{u}[X]$ is dense in $I$.

In [12], Chen and the first-named author provide a full description for
geometric ideals in $C^{*}_{u}(X)$ in terms of invariant open subsets of
$G(X)^{(0)}=\beta X$. To outline their work, let us start with the following
elementary observation:

######  Lemma 4.5.

Let $U$ be a non-empty invariant open subset of $\beta X$, then $X\subseteq
U$.

###### Proof.

Since $X$ is dense in $\beta X$ and $U$ is open and non-empty, we obtain that
$U\cap X\neq\emptyset$. Take an $x\in U\cap X$, then the pair $(x,y)\in G(X)$
for any $y\in X$. Thanks to the invariance of $U$, we obtain that $y\in U$.
This implies that $X\subseteq U$. ∎

Given an invariant open subset $U\subseteq\beta X$, denote $G(X)_{U}:=G(X)\cap
s^{-1}(U)$. Following [12], we define

$\displaystyle I_{c}(U):$ $\displaystyle=\\{f\in
C_{c}(G(X)):f(\tilde{\omega})=0\mbox{~{}for~{}any~{}}\tilde{\omega}\notin
G(X)_{U}\\}$
$\displaystyle=\\{T\in\mathbb{C}_{u}[X]:\overline{T}(\tilde{\omega})=0\mbox{~{}for~{}any~{}}\tilde{\omega}\notin
G(X)_{U}\\}.$

Obviously, $I_{c}(U)$ is a two-sided ideal in $C_{c}(G(X))$. Denote its
closure in $C^{*}_{r}(G(X))$ by $I(U)$, which is a geometric ideal in
$C^{*}_{r}(G(X))\cong C^{*}_{u}(X)$ from the definition (see also [12, Lemma
5.1]). This leads to the following:

######  Definition 4.6.

For an invariant open subset $U\subseteq\beta X$, the ideal $I(U)$ is called
the _geometric ideal associated to $U$_.

For later use, we record the following alternative description for the
geometric ideal $I(U)$:

######  Lemma 4.7.

Let $U$ be an invariant open subset of $\beta X$. Then the ideal $I(U)$ is
isomorphic to the reduced groupoid $C^{*}$-algebra $C^{*}_{r}(G(X)_{U})$.

###### Proof.

This was implicitly contained in the proof of [12, Proposition 5.5]. For
convenience to the readers, we include a proof here. By definition,
$C^{*}_{r}(G(X)_{U})$ is isomorphic to the norm closure of $C_{c}(G(X)_{U})$
in $C^{*}_{r}(G(X))$. Note that

$C_{c}(G(X)_{U})=\\{T\in\mathbb{C}_{u}[X]:\mathrm{supp}(\overline{T})\subseteq
G(X)_{U}\\}\subseteq I_{c}(U).$

On the other hand, for $T\in I_{c}(U)$ we have $T=\lim_{\varepsilon\to
0}T_{\varepsilon}$ since $T$ has finite propagation. Note that
$\mathrm{supp}(\overline{T_{\varepsilon}})\subseteq G(X)_{U}$, which implies
$T_{\varepsilon}\in C_{c}(G(X)_{U})$. Hence we obtain that $C_{c}(G(X)_{U})$
and $I_{c}(U)$ have the same closure in $C^{*}_{r}(G(X))$, which concludes the
proof. ∎

######  Example 4.8.

For $U=X$, it follows directly from definition that $G(X)_{X}=X\times X$.
Hence combining Proposition 2.8 and Lemma 4.7, we obtain that the geometric
ideal associated to $X$ is $I(X)=\mathfrak{K}(\ell^{2}(X))$. On the other
hand, for $U=\beta X$ it is clear that $I(\beta X)=C^{*}_{u}(X)$.

Conversely, following [12, Section 4 and 5] we can associate an invariant open
subset of $\beta X$ to any ideal in the uniform Roe algebra. More precisely,
let $I$ be an ideal in the uniform Roe algebra $C^{*}_{u}(X)$. Define:

(4.1) $U(I):=\bigcup_{T\in
I,\varepsilon>0}\overline{\mathrm{r}(\mathrm{supp}_{\varepsilon}(T))}=\bigcup_{T\in
I\cap\mathbb{C}_{u}[X],\varepsilon>0}\overline{\mathrm{r}(\mathrm{supp}_{\varepsilon}(T))},$

where the second equality follows directly from Proposition 4.3. Also [12,
Lemma 5.2] implies that $U(I)$ is an invariant open subset of $\beta X$.
Furthermore, as a special case of [12, Theorem 6.3], we have the following:

######  Proposition 4.9.

For a space $(X,d)$, the map $I\mapsto U(I)$ provides an isomorphism between
the lattice of all geometric ideals in $C^{*}_{u}(X)$ and the lattice of all
invariant open subsets of $\beta X$, with the inverse map given by $U\mapsto
I(U)$.

Proposition 4.9 shows that geometric ideals in $C^{*}_{u}(X)$ can be fully
determined by invariant open subsets of $\beta X$. In contrast, general ideals
in $C^{*}_{u}(X)$ cannot be characterised merely by the associated subsets of
$\beta X$. For example, direct calculations show that the associated invariant
open subsets for the ideal $I_{G}$ defined in Section 2.3 and for the ideal of
compact operators in $\mathfrak{B}(\ell^{2}(X))$ are the same, both of which
equal $X$ (see also Example 4.8 and 5.5 below).

Hence as pointed out in Section 1, the study of the ideal structure for the
uniform Roe algebra can be reduced to analyse the lattice (where the order is
given by inclusion)

$\mathfrak{I}_{U}=\\{I\mbox{~{}is~{}an~{}ideal~{}in~{}}C^{*}_{u}(X):U(I)=U\\}$

in (1.1) for each invariant open subset $U\subseteq\beta X$. The following
result busts the smallest element in $\mathfrak{I}_{U}$:

######  Proposition 4.10.

Let $U$ be an invariant open subset of $\beta X$. Then the geometric ideal
$I(U)$ is the smallest element in the lattice $\mathfrak{I}_{U}$ in (1.1).

The proof of Proposition 4.10 follows directly from the following lemma:

######  Lemma 4.11.

Let $(X,d)$ be a space and $I$ an ideal in $C^{*}_{u}(X)$. Then we have

$I(U(I))=\overline{I\cap\mathbb{C}_{u}[X]},$

where the closure is taken in $C^{*}_{u}(X)$. Hence we have $I(U(I))\subseteq
I$.

###### Proof.

Denoting $\mathring{I}:=\overline{I\cap\mathbb{C}_{u}[X]}$, it is clear that
$\mathring{I}\cap\mathbb{C}_{u}[X]=I\cap\mathbb{C}_{u}[X]$. This implies that
$\mathring{I}$ is a geometric ideal in $C^{*}_{u}(X)$, and hence
$\mathring{I}=I(U(\mathring{I}))$ by Proposition 4.9. By definition, we have

$U(\mathring{I})=\bigcup_{T\in\mathring{I}\cap\mathbb{C}_{u}[X],\varepsilon>0}\overline{\mathrm{r}(\mathrm{supp}_{\varepsilon}(T))}=\bigcup_{T\in
I\cap\mathbb{C}_{u}[X],\varepsilon>0}\overline{\mathrm{r}(\mathrm{supp}_{\varepsilon}(T))}=U(I).$

Therefore, we obtain that
$I(U(I))=I(U(\mathring{I}))=\mathring{I}=\overline{I\cap\mathbb{C}_{u}[X]}$ as
required. ∎

We would like to recall another description for geometric ideals based on the
notion of ideals in spaces introduced in [12]. It has the advantage of playing
within the given metric space, rather than going to the _mysterious_ Stone-
Čech boundary, and hence will help us to step over several technical gaps in
Section 8.

######  Definition 4.12 ([12, Definition 6.1]).

An _ideal_ in a space $(X,d)$ is a collection ${\bf L}$ of subsets of $X$
satisfying the following:

  1. (1)

if $Y\in{\bf L}$ and $Z\subseteq Y$, then $Z\in{\bf L}$;

  2. (2)

if $R\geq 0$ and $Y\in{\bf L}$, then $\mathcal{N}_{R}(Y)\in{\bf L}$;

  3. (3)

if $Y,Z\in{\bf L}$, then $Y\cup Z\in{\bf L}$.

For an ideal ${\bf L}$ in $X$, we define

$U({\bf L}):=\bigcup_{Y\in{\bf L}}\overline{Y}^{\beta X}.$

Conversely, given an invariant open subset $U$ of $\beta X$, we define

${\bf L}(U):=\\{Y\subseteq X:\overline{Y}^{\beta X}\subseteq U\\}.$

As a special case of [12, Theorem 6.3], we have the following:

######  Proposition 4.13.

For a space $(X,d)$, the map ${\bf L}\mapsto U({\bf L})$ provides an
isomorphism between the lattice of all ideals in $X$ and the lattice of all
invariant open subsets of $\beta X$, with the inverse map given by
$U\mapsto{\bf L}(U)$.

Combining Proposition 4.9 and 4.13, we obtain an isomorphism between the
lattice of all ideals in $X$ and the lattice of all geometric ideals in
$C^{*}_{u}(X)$. Direct calculation shows (see also [12, Theorem 6.4]):

(4.2) $I(U({\bf
L}))=\overline{\\{T\in\mathbb{C}_{u}[X]:\mathrm{supp}(T)\subseteq Y\times
Y\mbox{~{}for~{}some~{}}Y\in{\bf L}\\}},$

where the closure is taken in $C^{*}_{u}(X)$.

Now we consider a special class of geometric ideals coming from subspaces.
Given a subspace $A\subseteq X$, recall from [23, Section 5] that there is an
associated ideal $I_{A}$ in $C^{*}_{u}(X)$ whose $K$-theory is isomorphic to
that of the uniform Roe algebra $C^{*}(A)$. More precisely, recall that an
operator $T\in\mathfrak{B}(\ell^{2}(X))$ is _near $A$_ if there exists $R>0$
such that
$\mathrm{supp}(T)\subseteq\mathcal{N}_{R}(A)\times\mathcal{N}_{R}(A)$, and the
ideal $I_{A}$ is defined to be the operator norm closure of all operators in
$\mathbb{C}_{u}[X]$ near $A$.

The ideal $I_{A}$ is called _spatial_ in [13] since it is related to a
subspace in $X$. However, as shown in [13, Example 2.1], there exist non-
spatial ideals in general. On the other hand, spatial ideals play an important
role in the computation of the $K$-theory of Roe algebras via the Mayer-
Vietoris sequence argument (see [23]).

To show that spatial ideals are geometric, we observe that the smallest ideal
in $X$ containing $A$ is ${\bf
L}_{A}:=\\{Z\subseteq\mathcal{N}_{R}(A):R>0\\}$. Hence applying Proposition
4.13, we immediately obtain the following:

######  Lemma 4.14.

Let $A$ be a subset of $X$. Then the set

(4.3) $U_{A}:=U({\bf L}_{A})=\bigcup_{R>0}\overline{\mathcal{N}_{R}(A)}$

is an invariant open subset of $\beta X$. Moreover, if $U$ is an invariant
open subset of $\beta X$ containing $A$, then $U_{A}\subseteq U$.

Consequently, combining with (4.2) we reach the following:

######  Corollary 4.15.

Let $A$ be a subset of $X$, then $I(U_{A})=I_{A}$. Hence the spatial ideal
$I_{A}$ is geometric.

For later use, we record the following result concerning the set $U_{A}$
defined in (4.3). Recall from Corollary A.8 that for a subset $Z\subseteq X$,
the closure $\overline{Z}$ in $\beta X$ is homeomorphic to $\beta Z$.

######  Lemma 4.16.

Let $A$ be a subset of $X$. Then we have:

$G(X)_{U_{A}}=\bigcup_{R>0}G(\mathcal{N}_{R}(A)).$

###### Proof.

By definition, we have
$G(X)_{U_{A}}=\bigcup_{R>0}G(X)_{\overline{\mathcal{N}_{R}(A)}}$. Note that
for each $R>0$ and $(\alpha,\omega)\in G(X)_{\overline{\mathcal{N}_{R}(A)}}$,
we have $\omega\in\overline{\mathcal{N}_{R}(A)}$ and there exists $S>0$ such
that $(\alpha,\omega)\in\overline{E_{S}}$ due to Lemma 3.7. Hence the pair
$(\alpha,\omega)$ in
$\overline{\mathcal{N}_{R+S}(A)}\times\overline{\mathcal{N}_{R+S}(A)}$ belongs
to $G(\mathcal{N}_{R+S}(A))$, which concludes the proof. ∎

To end this section, we remark that for a given ideal $I$ in $C^{*}_{u}(X)$,
it is usually hard to compute the associated $U(I)$ directly from definition.
However, this is always achievable for principal ideals:

######  Lemma 4.17.

Let $I=\langle T\rangle$ be the principal ideal in $C^{*}_{u}(X)$ generated by
$T\in C^{*}_{u}(X)$. Denote

$U:=\bigcup_{\varepsilon>0,R>0}\overline{\mathcal{N}_{R}(\mathrm{r}(\mathrm{supp}_{\varepsilon}(T)))}.$

Then $U$ is an invariant open subset of $\beta X$, and we have $U(I)=U$.

###### Proof.

By Lemma 4.14, it is clear that $U$ is an invariant open subset of $\beta X$.
By (4.1), $U(I)$ contains
$\overline{\mathrm{r}(\mathrm{supp}_{\varepsilon}(T))}$ for any
$\varepsilon>0$. Since $U(I)$ is invariant, we obtain that $U(I)$ contains $U$
again by Lemma 4.14.

On the other hand, we consider $S=\sum_{i=1}^{n}a_{i}Tb_{i}$ where
$a_{i},b_{i}$ are non-zero with supports being partial translations contained
in $E_{R}$ for some $R>0$. Hence for any $\varepsilon>0$, we have

$\mathrm{r}\left(\mathrm{supp}_{\varepsilon}(S)\right)\subseteq\bigcup_{i=1}^{n}\mathrm{r}\left(\mathrm{supp}_{\frac{\varepsilon}{n}}(a_{i}Tb_{i})\right)\subseteq\bigcup_{i=1}^{n}\mathcal{N}_{R}\left(\mathrm{r}\left(\mathrm{supp}_{\frac{\varepsilon}{n\|a_{i}\|\cdot\|b_{i}\|}}(T)\right)\right),$

which implies that
$\overline{\mathrm{r}\left(\mathrm{supp}_{\varepsilon}(S)\right)}\subseteq U$.
Note that operators of the form $\sum_{i=1}^{n}a_{i}Tb_{i}$ as above are dense
in $I$. Hence for a general element $\tilde{S}\in I$ and $\varepsilon>0$,
there exists $S=\sum_{i=1}^{n}a_{i}Tb_{i}$ where $a_{i},b_{i}$ are non-zero
with supports being partial translations such that
$\|\tilde{S}-S\|<\varepsilon/2$. Hence
$\mathrm{supp}_{\varepsilon}(\tilde{S})\subseteq\mathrm{supp}_{\varepsilon/2}(S)$,
which concludes the proof. ∎

##  5\. Ghostly ideals

In the previous section, we observe that for an invariant open subset $U$ of
$\beta X$, the associated geometric ideal $I(U)$ is the smallest element in
the lattice

$\mathfrak{I}_{U}=\\{I\mbox{~{}is~{}an~{}ideal~{}in~{}}C^{*}_{u}(X):U(I)=U\\}.$

In this section, we would like to explore the largest element in
$\mathfrak{I}_{U}$. As revealed in Section 1, a natural idea is to include all
operators $T\in C^{*}_{u}(X)$ sitting in some ideal $I$ with $U(I)=U$, which
leads to the following:

######  Definition 5.1.

For a space $(X,d)$ and an invariant open subset $U$ of $\beta X$, denote

$\tilde{I}(U):=\\{T\in
C^{*}_{u}(X):\overline{\mathrm{r}(\mathrm{supp}_{\varepsilon}(T))}\subseteq
U\mbox{~{}for~{}any~{}}\varepsilon>0\\}.$

We call $\tilde{I}(U)$ the _ghostly ideal associated to $U$_.

The terminology will become clear later (see Proposition 5.6 and Remark 5.8
below). First let us verify that $\tilde{I}(U)$ is indeed an ideal in
$C^{*}_{u}(X)$.

######  Lemma 5.2.

For an invariant open subset $U$ of $\beta X$, $\tilde{I}(U)$ is an ideal in
$C^{*}_{u}(X)$.

###### Proof.

For $T,S\in C^{*}_{u}(X)$ and $\varepsilon>0$, we have
$\mathrm{supp}_{\varepsilon}(T+S)\subseteq\mathrm{supp}_{\varepsilon/2}(T)\cup\mathrm{supp}_{\varepsilon/2}(S)$.
It follows that $\tilde{I}(U)$ is a linear space in $C^{*}_{u}(X)$. Given
$T\in\tilde{I}(U)$ and $\varepsilon>0$, note that
$\mathrm{r}(\mathrm{supp}_{\varepsilon}(T^{*}))=\mathrm{s}(\mathrm{supp}_{\varepsilon}(T))$
and hence $T^{*}\in\tilde{I}(U)$ since $U$ is invariant. Now given a sequence
$\\{T_{n}\\}_{n}$ in $\tilde{I}(U)$ converging to $T\in C^{*}_{u}(X)$ and an
$\varepsilon>0$, there exists $n\in\mathbb{N}$ such that
$\mathrm{supp}_{\varepsilon}(T)\subseteq\mathrm{supp}_{\varepsilon/2}(T_{n})$.
Hence we obtain that $T\in\tilde{I}(U)$.

Finally, given $T\in\tilde{I}(U)$ and $S\in C^{*}_{u}(X)$ we need show that
$TS$ and $ST$ belong to $\tilde{I}(U)$. Note that $ST=(T^{*}S^{*})^{*}$ and
$\tilde{I}(U)$ is closed under taking the $\ast$-operation, hence it suffices
to show that $TS\in\tilde{I}(U)$. Since $\tilde{I}(U)$ is closed in
$C^{*}_{u}(X)$ with respect to the operator norm, it suffices to consider the
case that $S\in\mathbb{C}_{u}[X]$. We can further assume that $S$ is a partial
translation since $\tilde{I}(U)$ is closed under taking addition. In this
case, we have
$\mathrm{r}(\mathrm{supp}_{\varepsilon}(TS))\subseteq\mathrm{r}(\mathrm{supp}_{\varepsilon/\|S\|}(T))$
for any $\varepsilon>0$. Hence we conclude the proof. ∎

Using the language of ideals in $X$, we record that for an ideal ${\bf L}$ in
$X$ and $T\in C^{*}_{u}(X)$, then $T\in\tilde{I}(U({\bf L}))$ _if and only if_
for any $\varepsilon>0$ there exist $R>0$ and $Y\in{\bf L}$ such that for any
$(x,y)\notin E_{R}\cap(Y\times Y)$ then $|T(x,y)|<\varepsilon$.

The following result shows that the ghostly ideal $\tilde{I}(U)$ is indeed the
largest element in the lattice $\mathfrak{I}_{U}$:

######  Lemma 5.3.

Given an invariant open subset $U$ of $\beta X$, we have the following:

  1. (1)

for an ideal $I$ in $C^{*}_{u}(X)$ with $U(I)=U$, then
$I\subseteq\tilde{I}(U)$.

  2. (2)

$U(\tilde{I}(U))=U$.

###### Proof.

(1). Given $T\in I$, the condition $U(I)=U$ implies that
$\overline{\mathrm{r}(\mathrm{supp}_{\varepsilon}(T))}\subseteq U$ for each
$\varepsilon>0$. Hence by definition, we have $T\in\tilde{I}(U)$.

(2). Note that

$U(\tilde{I}(U))=\bigcup_{T\in\tilde{I}(U),\varepsilon>0}\overline{\mathrm{r}(\mathrm{supp}_{\varepsilon}(T))}\subseteq
U.$

On the other hand, Proposition 4.9 shows that $U(I(U))=U$, which implies that
$\tilde{I}(U)\supseteq I(U)$ thanks to (1). Hence we have
$U(\tilde{I}(U))\supseteq U(I(U))=U$ again by Proposition 4.9, which concludes
the proof. ∎

Combining with Proposition 4.10, we reach the following desired result:

######  Theorem 5.4.

Let $(X,d)$ be a space as in Section 2.2, and $U$ be an invariant open subset
of $\beta X$. Then any ideal $I$ in $C^{*}_{u}(X)$ with $U(I)=U$ sits between
$I(U)$ and $\tilde{I}(U)$. More precisely, the geometric ideal $I(U)$ is the
smallest element while the ghostly ideal $\tilde{I}(U)$ is the largest element
in the lattice $\mathfrak{I}_{U}$ in (1.1).

As mentioned in Section 1, Theorem 5.4 draws the border of the lattice
$\mathfrak{I}_{U}$. Therefore, once we can bust every ideal between $I(U)$ and
$\tilde{I}(U)$ for each invariant open subset $U\subseteq\beta X$, then we
will obtain a full description for the ideal structure of the uniform Roe
algebra $C^{*}_{u}(X)$ (see Question 9.1).

Now we aim to provide a geometric description for ghostly ideals, which helps
to explain the terminology. Let us start with an easy example.

######  Example 5.5.

Taking $U=X$, then $\tilde{I}(X)$ is the ideal $I_{G}$ defined in Section 2.3.
Indeed, $T\in\tilde{I}(X)$ if and only if for any $\varepsilon>0$,
$\mathrm{r}(\mathrm{supp}_{\varepsilon}(T))$ is finite. This is equivalent to
that $T\in C_{0}(X\times X)$ since $T\in C^{*}_{u}(X)$. On the other hand, it
is clear that $\tilde{I}(\beta X)=C^{*}_{u}(X)$.

More generally, we have the following result. Note that for any invariant open
subset $U$ of $\beta X$, $G(X)_{U}$ is an open subset of $\beta(X\times X)$.
Hence both $C_{0}(G(X)_{U})$ and $I_{c}(U)$ can be regarded as subalgebras in
$C(\beta(X\times X))\cong\ell^{\infty}(X\times X)$.

######  Proposition 5.6.

For an invariant open subset $U\subseteq\beta X$ and $T\in C^{*}_{u}(X)$, the
following are equivalent:

  1. (1)

$T\in\tilde{I}(U)$;

  2. (2)

$\overline{T}\in C_{0}(G(X)_{U})$;

  3. (3)

$\overline{T}\in\overline{I_{c}(U)}^{\|\cdot\|_{\infty}}$;

  4. (4)

$T$ vanishes in the $(\beta X\setminus U)$-direction, _i.e._ ,
$\Phi_{\omega}(T)=0$ for any $\omega\in\beta X\setminus U$.

###### Proof.

“(1) $\Rightarrow$ (2)”: By definition, for any $\varepsilon>0$ we have

$\mathrm{r}(\mathrm{supp}(\overline{T_{\varepsilon}}))=\mathrm{r}(\overline{\mathrm{supp}(T_{\varepsilon})})=\overline{\mathrm{r}(\mathrm{supp}_{\varepsilon}(T))}\subseteq
U.$

Consider the compact set $K:=\\{\tilde{\omega}\in\beta(X\times
X):|\overline{T}(\tilde{\omega})|\geq 2\varepsilon\\}$. By Lemma 4.2, we have
$\mathrm{r}(K)\subseteq\mathrm{r}(\mathrm{supp}(\overline{T_{\varepsilon}}))\subseteq
U$, which implies that $K\subseteq G(X)_{U}$. Moreover, we have
$|\overline{T}(\tilde{\omega})|<2\varepsilon$ for any
$\tilde{\omega}\in\beta(X\times X)\setminus K$, which concludes that
$\overline{T}\in C_{0}(G(X)_{U})$.

“(2) $\Rightarrow$ (1)”: Assume that $\overline{T}\in C_{0}(G(X)_{U})\subseteq
C(\beta(X\times X))$. Then for any $\varepsilon>0$ there exists a compact
subset $K\subseteq G(X)_{U}$ such that for any $\tilde{\omega}\in\beta(X\times
X)\setminus K$, we have $|\overline{T}(\tilde{\omega})|<\varepsilon$. This
implies that $\\{\tilde{\omega}\in\beta(X\times
X):|\overline{T}(\tilde{\omega})|\geq\varepsilon\\}\subseteq K$. Using Lemma
4.2, we obtain that $\overline{\mathrm{supp}(T_{\varepsilon})}\subseteq K$,
which implies that
$\overline{\mathrm{r}(\mathrm{supp}_{\varepsilon}(T))}\subseteq U$. Hence
$T\in\tilde{I}(U)$.

“(2) $\Leftrightarrow$ (3)”: This is due to the fact that

$\overline{I_{c}(U)}^{\|\cdot\|_{\infty}}=\\{f\in
C_{0}(G(X)):f(\tilde{\omega})=0\text{ for }\tilde{\omega}\notin G(X)_{U}\\}.$

“(3) $\Leftrightarrow$ (4)”: By Lemma 3.12, (4) holds if and only if
$\overline{T}(\alpha,\gamma)=0$ for any $\alpha,\gamma\in X(\omega)$ and
$\omega\in\beta X\setminus U$. Applying Lemma 2.9 and Lemma 3.4, this holds if
and only if $\overline{T}(\tilde{\omega})=0$ whenever $\tilde{\omega}\notin
G(X)_{U}$, which describes elements in
$\overline{I_{c}(U)}^{\|\cdot\|_{\infty}}$. Hence we conclude the proof. ∎

Note that $G(X)_{X}=X\times X$. Hence as a direct corollary, we recover the
following characterisation for ghost operators:

######  Corollary 5.7 ([43, Proposition 8.2]).

An operator $T\in C^{*}_{u}(X)$ is a ghost _if and only if_
$\Phi_{\omega}(T)=0$ for any non-principal ultrafilter $\omega$ on $X$.

######  Remark 5.8.

In other words, Corollary 5.7 shows that a ghost in $C^{*}_{u}(X)$ is locally
invisible in all directions. This suggests us to consider operators in
$\tilde{I}(U)$ as “partial” ghosts, which clarifies the terminology of
“ghostly ideals”.

As an application of Proposition 5.6, we now provide another description for
ghostly ideals in terms of operator algebras, which will be used later in
Section 7. Let us start with the short exact sequences studied in [21] (see
also [20, Section 2]).

Given an invariant open subset $U\subseteq\beta X$, notice that $U^{c}=\beta
X\setminus U$ is also invariant. Denote by
$G(X)_{U^{c}}:=G(X)\cap\mathrm{s}^{-1}(U^{c})$ and clearly, we have a
decomposition:

$G(X)=G(X)_{U}\sqcup G(X)_{U^{c}}.$

Note that $G(X)_{U}$ is open in $G(X)$, hence the above induces the following
short exact sequence of $\ast$-algebras:

(5.1) $0\longrightarrow C_{c}(G(X)_{U})\longrightarrow
C_{c}(G(X))\longrightarrow C_{c}(G(X)_{U^{c}})\longrightarrow 0$

where the map $C_{c}(G(X)_{U})\longrightarrow C_{c}(G(X))$ is the inclusion
and the map $C_{c}(G(X))\longrightarrow C_{c}(G(X)_{U^{c}})$ is the
restriction.

We may complete the sequence (5.1) with respect to the maximal groupoid
$C^{*}$-norms and obtain the following sequence:

(5.2) $0\longrightarrow C^{*}_{\rm max}(G(X)_{U})\longrightarrow C^{*}_{\rm
max}(G(X))\longrightarrow C^{*}_{\rm max}(G(X)_{U^{c}})\longrightarrow 0,$

which is easy to check by definition to be automatically exact (see, _e.g._ ,
[26, Lemma 2.10]). We may also complete this sequence with respect to the
reduced groupoid $C^{*}$-norms and obtain the following sequence:

(5.3) $0\longrightarrow C^{*}_{r}(G(X)_{U})\stackrel{{\scriptstyle
i_{U}}}{{\longrightarrow}}C^{*}_{r}(G(X))\stackrel{{\scriptstyle
q_{U}}}{{\longrightarrow}}C^{*}_{r}(G(X)_{U^{c}})\longrightarrow 0.$

By construction, $i_{U}$ is injective, $q_{U}$ is surjective and $q_{U}\circ
i_{U}=0$. Also recall from Lemma 4.7 that $i_{U}(C^{*}_{r}(G(X)_{U}))=I(U)$,
the geometric ideal associated to $U$. However in general, (5.3) fails to be
exact at the middle item. This is crucial in [21] to provide a counterexample
to the Baum-Connes conjecture with coefficients.

More precisely when $U=X$, it is proved in [35, Proposition 2.11] for the
group case and [20, Proposition 4.4] for the Roe algebraic case (see also [43,
Proposition 8.2]) that ${\rm Ker}(q_{X})=I_{G}$, _i.e._ , the ideal consisting
of all ghost operators in $C^{*}_{u}(X)$. Hence from Example 4.8 and Example
5.5, the sequence (5.3) is exact for $U=X$ _if and only if_
$I(X)=\tilde{I}(X)$. More generally, we have the following:

######  Proposition 5.9.

Given an invariant open subset $U\subseteq\beta X$, the kernel of
$q_{U}:C^{*}_{r}(G(X))\to C^{*}_{r}(G(X)_{U^{c}})$ coincides with the ghostly
ideal $\tilde{I}(U)$. Hence the sequence (5.3) is exact _if and only if_
$I(U)=\tilde{I}(U)$.

###### Proof.

It is easy to see that an operator $T\in C^{*}_{u}(X)\cong C^{*}_{r}(G(X))$
belongs to the kernel of $q_{U}$ if and only if $\lambda_{\omega}(T)=0$ for
any $\omega\in\beta X\setminus U$, where $\lambda_{\omega}$ is the left
regular representation from (2.1). Now Lemma 3.14 implies that
$\Phi_{\omega}(T)=0$ for any $\omega\in\beta X\setminus U$. Finally, we
conclude the proof thanks to Proposition 5.6. ∎

We end this section with an illuminating example from [44, Section 3] (see
also [20, Section 5]), which is important to construct counterexamples to
Baum-Connes type conjectures:

######  Example 5.10.

Let $\\{X_{i}\\}_{i\in\mathbb{N}}$ be a sequence of expander graphs or
pertubed expander graphs (see [44] for the precise definition). Let
$Y_{i,j}=X_{i}$ for all $j\in\mathbb{N}$ and set $Y:=\bigsqcup_{i,j}Y_{i,j}$.
We endow $Y$ with a metric $d$ such that it is the graph metric on each
$Y_{i,j}$ and satisfies $d(Y_{i,j},Y_{k,l})\to\infty$ as $i+j+k+l\to\infty$.

Let $P_{i,j}\in\mathfrak{B}(\ell^{2}(Y_{i,j}))$ be the orthogonal projection
onto constant functions on $Y_{i,j}$, and we set $P$ to be the direct sum of
$P_{i,j}$ in the strong operator topology. By the assumption on the expansion
of $\\{X_{i}\\}_{i\in\mathbb{N}}$, it is clear that $P\in C^{*}_{u}(Y)$. It is
explained in [44, Section 3] (see also [20, Lemma 5.1]) that $P$ is _not_ a
ghost, _i.e._ , $P\notin\tilde{I}(X)$. However intuitively, $P$ should vanish
“in the $i$-direction”. We will make it more precisely in the following.

Recall from [20, Section 5.1] that we have a surjective map $\beta Y\to\beta
X\times\beta\mathbb{N}$ induced by the bijection of $Y$ with
$X\times\mathbb{N}$ and the universal property of $\beta Y$. Define:

$f:\beta Y\longrightarrow\beta X\times\beta\mathbb{N}\longrightarrow\beta X$

where the second map is just the projection onto the first coordinate. Denote
$U=f^{-1}(X)$, which is open in $\beta Y$. Note that
$U=\bigsqcup_{i}f^{-1}(X_{i})$, where each $f^{-1}(X_{i})$ is homeomorphic to
$X_{i}\times\beta\mathbb{N}$. On the other hand, note that

$U=\bigsqcup_{i}f^{-1}(X_{i})=\bigcup_{i}\overline{\bigsqcup_{j}Y_{i,j}}=\bigcup_{\varepsilon>0}\overline{\mathrm{r}(\mathrm{supp}_{\varepsilon}(P))}=\bigcup_{\varepsilon>0,R>0}\overline{\mathcal{N}_{R}(\mathrm{r}(\mathrm{supp}_{\varepsilon}(P)))}.$

Hence it follows from Lemma 4.17 that $U$ is invariant (comparing with [20,
Lemma 5.2]), and $U(\langle P\rangle)=U$. (Note that $P\in\tilde{I}(U)$ was
already implicitly proved in [20, Theorem 5.5], thanks to Proposition 5.9.)
Since $U$ contains $Y$ as a proper subset, we reprove that $P$ is not a ghost.

Moreover, it follows from Proposition 5.6 that $P$ vanishes in the
$(\partial_{\beta}Y\setminus U)$-direction. In particular, fixing an index
$j_{0}\in\mathbb{N}$ and taking a sequence $\\{x_{i}\in
Y_{i,j_{0}}\\}_{i\in\mathbb{N}}$, we choose a cluster point
$\omega\in\overline{\\{x_{i}\\}_{i}}$. It is clear that $\omega\notin U$.
Intuitively speaking, this means that $P$ vanishes “in the $i$-direction”.

We remark that the first-named author proved in [44, Section 3] that the
principal ideal $\langle P\rangle$ cannot be decomposed into
$I(U)+(I_{G}\cap\langle P\rangle)$, which provided a couterexample to the
conjecture at the end of [12]. Our explanation above reveals that the reason
behind this counterexample is that the ghostly part of $\langle P\rangle$
could not be “exhausted” merely by ghostly elements associated to $X$ (rather
than $U$).

Finally, we remark that the groupoid $G(Y)_{U}$ also plays a key role in
constructing a counterexample to the boundary coarse Baum-Connes conjecture
introduced in [20] (see Section 5.2 therein).

##  6\. Maximal ideals

In this section, we would like to study maximal ideals in uniform Roe algebras
using the tools developed in Section 5. Throughout this section, let $(X,d)$
be a space as in Section 2.2.

###  6.1. Minimal points in the boundary

Recall from previous sections that ideals are closely related to invariant
open subsets of the unit space $\beta X$. Hence we introduce the following:

######  Definition 6.1.

An invariant open subset $U\subseteq\beta X$ is called _maximal_ if
$U\neq\beta X$ and $U$ is not properly contained in any proper invariant open
subset of $\beta X$. Similarly, an invariant closed subset $K\subseteq\beta X$
is called _minimal_ if $K\neq\emptyset$ and $K$ does not properly contain any
non-empty invariant closed subset of $\beta X$.

######  Proposition 6.2.

For any maximal invariant open subset $U\subset\beta X$, the ghostly ideal
$\tilde{I}(U)$ is a maximal ideal in the uniform Roe algebra $C^{*}_{u}(X)$.
Conversely for any maximal ideal $I$ in $C^{*}_{u}(X)$, the associated
invariant open subset $U(I)$ is maximal and we have $I=\tilde{I}(U(I))$.

###### Proof.

For any ideal $J$ in $C^{*}_{u}(X)$ containing $\tilde{I}(U)$, we have
$U(J)\supseteq U(\tilde{I}(U))=U$ by Lemma 5.3(2). Since $U$ is maximal, then
either $U(J)=\beta X$ or $U(J)=U$. If $U(J)=\beta X$, it follows from Lemma
4.11 that $J$ contains $I(U(J))=C^{*}_{u}(X)$, which implies that
$J=C^{*}_{u}(X)$. If $U(J)=U$, then it follows from Lemma 5.3(1) that
$J\subseteq\tilde{I}(U(J))=\tilde{I}(U)$, which implies that $J=\tilde{I}(U)$.
This concludes that $\tilde{I}(U)$ is maximal.

Conversely for any maximal ideal $I$ in $C^{*}_{u}(X)$, we have $U(I)\neq\beta
X$. For any open invariant subset $V\neq\beta X$ containing $U$, we have
$I\subseteq\tilde{I}(U)\subseteq\tilde{I}(V)$ by Theorem 5.4, and
$\tilde{I}(V)\neq C^{*}_{u}(X)$. Hence due to the maximality of $I$, we obtain
that $I=\tilde{I}(U)=\tilde{I}(V)$. This implies that $U=V$ and also
$I=\tilde{I}(U)$ as required. ∎

Taking complements, we obtain the following:

######  Corollary 6.3.

For any minimal invariant closed subset $K\subseteq\beta X$, the ghostly ideal
$\tilde{I}(\beta X\setminus K)$ is a maximal ideal in the uniform Roe algebra
$C^{*}_{u}(X)$. Moreover, every maximal ideal in $C^{*}_{u}(X)$ arises in this
form.

Therefore, in order to describe maximal ideals in the uniform Roe algebra, it
suffices to study minimal invariant closed subsets of the unit space $\beta
X$. Recall from Lemma 3.4 that for each $\omega\in\partial_{\beta}X$, the
limit space $X(\omega)$ is the smallest invariant subset of $\beta X$
containing $\omega$. However, note that $X(\omega)$ might not be closed in
general.

######  Definition 6.4.

A point $\omega\in\partial_{\beta}X$ is called _minimal_ if the closure of the
limit space $X(\omega)$ in $\beta X$ is minimal in the sense of Definition
6.1.

The following result is straightforward, hence we omit the proof. It suggests
us to study minimal points in the boundary.

######  Lemma 6.5.

For a minimal invariant closed subset $K\subseteq\beta X$, there exists a
minimal point $\omega\in\partial_{\beta}X$ such that $K=\overline{X(\omega)}$.
Conversely for any minimal point $\omega\in\partial_{\beta}X$, the set
$\overline{X(\omega)}$ is minimal.

One might wonder whether every $\omega\in\partial_{\beta}X$ is minimal.
However, things become very complicated after taking closures and we will show
later that this does not hold even in the case of $X=\mathbb{Z}$. Firstly, we
notice the following:

######  Lemma 6.6.

Let $K$ be an invariant closed subset of $\beta X$. Then $K$ contains a
minimal point. In particular, for any $\omega\in\partial_{\beta}X$ there
exists a minimal point $\omega^{\prime}$ such that
$\omega^{\prime}\in\overline{X(\omega)}$.

###### Proof.

This follows directly from the Zorn’s lemma together with the fact that $\beta
X$ is compact. Details are left to readers. ∎

Consequently, we obtain:

######  Corollary 6.7.

For $\omega\in\partial_{\beta}X$, $\omega$ is minimal _if and only if_ for any
$\alpha\in\overline{X(\omega)}$, we have
$\overline{X(\alpha)}=\overline{X(\omega)}$. Writing
$\overline{X(\omega)}=\bigsqcup_{\lambda\in\Lambda}X(\omega_{\lambda})$ for
certain $\omega_{\lambda}\in\partial_{\beta}X$, then $\omega$ is minimal _if
and only if_ $\overline{X(\omega_{\lambda})}=\overline{X(\omega)}$ for any
$\lambda\in\Lambda$.

As promised, now we study the case of $X=\mathbb{Z}$ and show that it admits a
number of non-minimal points. The following is the main result:

######  Theorem 6.8.

For the integer group $\mathbb{Z}$ with the usual metric, there exist non-
minimal points in the boundary $\partial_{\beta}\mathbb{Z}$. More precisely,
for any sequence $\\{h_{n}\\}_{n\in\mathbb{N}}$ in $\mathbb{Z}$ tending to
infinity such that $|h_{n}-h_{m}|\to+\infty$ when $n+m\to\infty$ and $n\neq
m$, and any $\omega\in\partial_{\beta}\mathbb{Z}$ with
$\omega(\\{h_{n}\\}_{n\in\mathbb{N}})=1$, then $\omega$ is _not_ a minimal
point.

To prove Theorem 6.8, we need some preparations. For future use, we record the
following result in the context of a countable discrete group $\Gamma$
equipped with a left-invariant word length metric.

######  Lemma 6.9.

For $\omega,\alpha\in\partial_{\beta}\Gamma$, then
$\alpha\in\overline{\Gamma(\omega)}$ _if and only if_ for any
$S\subseteq\Gamma$ with $\alpha(S)=1$, there exists $g_{S}\in\Gamma$ such that
$\omega(S\cdot g_{S}^{-1})=1$.

###### Proof.

Recall from Lemma 3.8 that $\Gamma(\omega)=\\{\rho_{g}(\omega):g\in\Gamma\\}$,
and from Appendix A that $\\{\overline{S}:S\subseteq X\\}$ forms a basis for
$\beta X$. Hence by definition, we obtain that $\alpha\in\overline{X(\omega)}$
if and only if for any $S\subseteq X$ with $\alpha\in\overline{S}$, there
exists $g_{S}\in\Gamma$ such that $\rho_{g_{S}}(\omega)\in\overline{S}$.
Equivalently, this means that for any $S\subseteq X$ with $\alpha(S)=1$, there
exists $g_{S}\in\Gamma$ such that $\rho_{g_{S}}(\omega)(S)=\omega(S\cdot
g_{S}^{-1})=1$, which concludes the proof. ∎

Now we return to the case of $\Gamma=\mathbb{Z}$ and prove Theorem 6.8:

###### Proof of Theorem 6.8.

Fix a subset $H=\\{h_{n}\\}_{n\in\mathbb{N}}\subseteq\mathbb{Z}$ tending to
infinity such that $|h_{n}-h_{m}|\to+\infty$ when $n+m\to\infty$ and $n\neq
m$. For any non-zero $g\in\mathbb{Z}$, note that $h\in(g+H)\cap H$ if and only
if there exists $h^{\prime}\in H$ such that $h-h^{\prime}=g$. Since $g$ is
fixed and distances between different points in $H$ tend to infinity, we
obtain that $(g+H)\cap H$ is finite. Hence for any $g_{1}\neq g_{2}$ in
$\mathbb{Z}$, $(g_{1}+H)\cap(g_{2}+H)$ is finite.

Fixing a non-principal ultrafilter $\omega\in\partial_{\beta}\mathbb{Z}$ with
$\omega(H)=1$, we denote

$\mathcal{U}:=\\{B\subseteq H:\omega(B)=1\\}.$

We claim that for each $n\in\mathbb{N}$, there exists $g_{n}\in\mathbb{Z}$ and
$B_{n}\in\mathcal{U}$ such that $\\{B_{n}+g_{n}\\}_{n\in\mathbb{N}}$ are
mutually disjoint. Indeed, we take $g_{0}=0$ and $B_{0}=H$. Set $g_{1}=1$ and
$B_{1}:=H\setminus(H-g_{1})$. Since $H\cap(H-g_{1})$ is finite by the previous
paragraph, then $\omega(B_{1})=1$, _i.e._ , $B_{1}\in\mathcal{U}$. Similarly
for each $n\in\mathbb{N}$, we take $g_{n}=n$ and
$B_{n}:=H\setminus\big{(}(H-g_{1})\cup(H-g_{2})\cup\cdots\cup(H-g_{n})\big{)}$,
which concludes the claim.

Consider $\widetilde{H}:=\bigsqcup_{n\in\mathbb{N}}(B_{n}+g_{n})$, and denote
$\mathcal{U}_{n}:=\\{B\subseteq B_{n}:\omega(B)=1\\}$ for each
$n\in\mathbb{N}$. By Lemma A.4, $\mathcal{U}_{n}$ is an ultrafilter on
$B_{n}$. Choose a non-principal ultrafilter $\omega_{0}$ on $\mathbb{N}$, and
we consider:

$\widetilde{\mathcal{U}}:=\big{\\{}\bigsqcup_{n\in\mathbb{N}}(A_{n}+g_{n})\subseteq\bigsqcup_{n\in\mathbb{N}}(B_{n}+g_{n})=\widetilde{H}:\exists~{}J\subseteq\mathbb{N}\text{
with }\omega_{0}(J)=1\text{ s.t. }\forall n\in
J,A_{n}\in\mathcal{U}_{n}\big{\\}}.$

Lemma A.6 implies that $\widetilde{\mathcal{U}}$ is an ultrafilter on
$\widetilde{H}$. We define a function
$\alpha:\mathcal{P}(\mathbb{Z})\to\\{0,1\\}$ by setting $\alpha(S)=1$ if and
only if $S\cap\widetilde{H}\in\widetilde{\mathcal{U}}$, which is indeed an
ultrafilter on $\mathbb{Z}$ thanks to Lemma A.5. Also note that $\alpha$ is
non-principal and $\alpha(\widetilde{H})=1$.

Now we show that $\alpha\in\overline{\mathbb{Z}(\omega)}$ while
$\omega\notin\overline{\mathbb{Z}(\alpha)}$, and hence conclude the proof. To
see that $\alpha\in\overline{\mathbb{Z}(\omega)}$, we will consult Lemma 6.9.
For any $S\subseteq\mathbb{Z}$ with $\alpha(S)=1$, by definition we have
$S\cap\widetilde{H}\in\widetilde{\mathcal{U}}$. Writing
$S\cap\widetilde{H}=\bigsqcup_{n\in\mathbb{N}}(A_{n}+g_{n})$ with
$A_{n}\subseteq B_{n}$, then there exists $J\in\omega_{0}$ such that for any
$n\in J$ we have $A_{n}\in\mathcal{U}_{n}$. Hence for any $n\in J$, we have
$S\supseteq S\cap\widetilde{H}\supseteq A_{n}+g_{n}$ and $\omega(A_{n})=1$,
which implies that $\omega(S-g_{n})=1$. Applying Lemma 6.9, we conclude that
$\alpha\in\overline{\mathbb{Z}(\omega)}$.

Finally, it remains to check that $\omega\notin\overline{\mathbb{Z}(\alpha)}$.
Assume the contrary, then Lemma 6.9 implies that there exists $g\in\mathbb{Z}$
and $\widetilde{B}\subseteq\widetilde{H}$ with $\alpha(\widetilde{B})=1$ such
that $H\supseteq\widetilde{B}-g$. Writing
$\widetilde{B}=\bigsqcup_{n\in\mathbb{N}}(A_{n}+g_{n})$ with $A_{n}\subseteq
B_{n}$, then there exists $J\in\omega_{0}$ such that for any $n\in J$,
$\omega(A_{n})=1$. This implies that

$H+g\supseteq\widetilde{B}\supseteq\bigsqcup_{n\in J}(A_{n}+g_{n})\supseteq
A_{n_{0}}+g_{n_{0}}$

for some $n_{0}\in J$ with $g_{n_{0}}\neq g$ (this can be achieved since $J$
is infinite). However, it follows from the first paragraph that
$(H+g)\cap(H+g_{n_{0}})$ is finite. While this intersection contains
$A_{n_{0}}+g_{n_{0}}$, which is infinite since $\omega(A_{n_{0}})=1$.
Therefore, we reach a contradiction and conclude the proof. ∎

###  6.2. Maximal ideals via limit operators

We already see that maximal ideals in the uniform Roe algebra correspond to
minimal points in the Stone-Čech boundary of the underlying space and in
Section 6.1, we use topological methods to show the existence of non-minimal
points. Now we turn to a $C^{*}$-algebraic viewpoint, and use the tool of
limit operators to provide an alternative description for these ideals.

First recall from Corollary 6.3 and Lemma 6.5 that maximal ideals in
$C^{*}_{u}(X)$ arise in the form of $\tilde{I}(\beta
X\setminus\overline{X(\omega)})$ for some boundary point
$\omega\in\partial_{\beta}X$. Moreover according to Proposition 5.9,
$\tilde{I}(\beta X\setminus\overline{X(\omega)})$ is the kernel of the
following surjective homomorphism:

$q_{\beta X\setminus\overline{X(\omega)}}:C^{*}_{r}(G(X))\longrightarrow
C^{*}_{r}(G(X)_{\overline{X(\omega)}}).$

Hence we obtain the following:

######  Corollary 6.10.

A point $\omega\in\partial_{\beta}X$ is minimal _if and only if_
$C^{*}_{r}(G(X)_{\overline{X(\omega)}})$ is simple.

######  Example 6.11.

Consider the case of a countable discrete group $\Gamma$. For a point
$\omega\in\partial_{\beta}\Gamma$, it follows from Lemma 3.8 that the limit
space $\Gamma(\omega)$ is identical to $\Gamma\omega$, and hence
$C^{*}_{r}(G(\Gamma)_{\overline{\Gamma(\omega)}})$ is $C^{*}$-isomorphic to
the reduced crossed product $C(\overline{\Gamma\omega})\rtimes\Gamma$. Thanks
to Corollary 6.10, we obtain that $\omega$ is minimal if and only if
$C(\overline{\Gamma\omega})\rtimes\Gamma$ is simple. Moreover, note that the
action of $\Gamma$ on $\beta\Gamma$ is free (which is also a consequence of
Lemma 3.8). Hence it follows from [29, Corollary 4.6] that
$C(\overline{\Gamma\omega})\rtimes\Gamma$ is simple if and only if the action
of $\Gamma$ on $\overline{\Gamma\omega}$ is minimal. In conclusion, we reach
the following:

######  Corollary 6.12.

In the case of a countable discrete group $\Gamma$, a point
$\omega\in\partial_{\beta}\Gamma$ is minimal _if and only if_ the action of
$\Gamma$ on $\overline{\Gamma\omega}$ is minimal.

Corollary 6.10 suggests an approach to distinguish minimal points via the
simplicity of the reduced groupoid $C^{*}$-algebra. However, this
$C^{*}$-algebra is still not easy to handle since it requires to consider all
points in the $\overline{X(\omega)}$-direction (see Proposition 5.6). Now we
show that this can be simplified by merely considering the $\omega$-direction:

######  Lemma 6.13.

For $\omega\in\partial_{\beta}X$, an operator $T\in C^{*}_{u}(X)$ belongs to
the ideal $\tilde{I}(\beta X\setminus\overline{X(\omega)})$ _if and only if_
$T$ vanishes in the $\omega$-direction, _i.e._ , $\Phi_{\omega}(T)=0$.

###### Proof.

We assume that $\Phi_{\omega}(T)=0$, and it suffices to show that
$\Phi_{\alpha}(T)=0$ for any $\alpha\in\overline{X(\omega)}$. Fixing such an
$\alpha$, we take a net $\\{\omega_{\lambda}\\}_{\lambda\in\Lambda}$ in
$X(\omega)$ such that $\omega_{\lambda}\to\alpha$ and it follows that
$\Phi_{\omega_{\lambda}}(T)=0$. For any $\gamma_{1},\gamma_{2}\in X(\alpha)$,
Lemma 3.4 and the fact that the coarse groupoid $G(X)$ is étale imply that
there exist $\gamma_{1,\lambda}$ and $\gamma_{2,\lambda}$ in
$X(\omega_{\lambda})$ for each $\lambda\in\Lambda$ such that
$\gamma_{1,\lambda}\to\gamma_{1}$ and $\gamma_{2,\lambda}\to\gamma_{2}$. Now
Lemma 3.12 implies that

$\Phi_{\alpha}(T)_{\gamma_{1}\gamma_{2}}=\overline{T}((\gamma_{1},\gamma_{2}))=\lim_{\lambda\in\Lambda}\overline{T}((\gamma_{1,\lambda},\gamma_{2,\lambda}))=\lim_{\lambda\in\Lambda}\Phi_{\omega_{\lambda}}(T)_{\gamma_{1,\lambda}\gamma_{2,\lambda}}=0,$

which concludes the proof. ∎

Hence for $\omega\in\partial_{\beta}X$, Lemma 6.13 implies that the associated
ideal $\tilde{I}(\beta\setminus\overline{X(\omega)})$ coincides with the
kernel of the following limit operator homomorphism (see also [43, Theorem
4.10]):

(6.1) $\Phi_{\omega}:C^{*}_{u}(X)\longrightarrow C^{*}_{u}(X(\omega)),\quad
T\mapsto\Phi_{\omega}(T).$

Consequently, we reach the following:

######  Corollary 6.14.

A point $\omega\in\partial_{\beta}X$ is minimal _if and only if_ the image
$\mathrm{Im}(\Phi_{\omega})$ is simple. Hence when $X(\omega)$ is infinite and
$\Phi_{\omega}$ is surjective, the point $\omega$ is _not_ minimal.

###### Proof.

For the last statement, it suffices to note that the ideal of compact
operators is always contained in $C^{*}_{u}(X(\omega))$, which concludes the
proof. ∎

Thanks to Corollary 6.14, a special case of Theorem 6.8 can also be deduced
from a recent work by Roch [30]. More precisely, combining [43, Proposition
B.6] with [30, Lemma 2.1], we have the following:

######  Proposition 6.15.

Let $\\{h_{n}\\}_{n\in\mathbb{N}}$ be a sequence in $\mathbb{Z}^{N}$ tending
to infinity such that

$\|h_{n}-h_{k}\|_{\infty}\geq 3k\quad\text{for any}\quad k>n.$

Then for any $\omega\in\partial_{\beta}\mathbb{Z}^{N}$ with
$\omega(\\{h_{n}\\}_{n\in\mathbb{N}})=1$, the map
$\Phi_{\omega}:C^{*}_{u}(\mathbb{Z}^{N})\longrightarrow
C^{*}_{u}(\mathbb{Z}^{N}(\omega))$ is surjective.

Note that the limit space $\mathbb{Z}^{N}(\omega)$ is bijective to
$\mathbb{Z}^{N}$ by Lemma 3.8, and hence infinite. Therefore applying
Corollary 6.14, we obtain the following (when $N=1$, it partially recovers
Theorem 6.8).

######  Corollary 6.16.

For any sequence $\\{h_{n}\\}_{n\in\mathbb{N}}$ in $\mathbb{Z}^{N}$ tending to
infinity such that $\|h_{n}-h_{k}\|_{\infty}\geq 3k$ for any $k>n$, and any
$\omega\in\partial_{\beta}\mathbb{Z}^{N}$ with
$\omega(\\{h_{n}\\}_{n\in\mathbb{N}})=1$, then $\omega$ is _not_ a minimal
point.

######  Remark 6.17.

We notice from the discussion above that for $\omega\in\partial_{\beta}X$,
there is a $C^{*}$-monomorphism:

(6.2) $C^{*}_{r}(G(X)_{\overline{X(\omega)}})\cong
C^{*}_{u}(X)/\tilde{I}(\beta X\setminus\overline{X(\omega)})\longrightarrow
C^{*}_{u}(X(\omega))$

where the first comes from Proposition 5.9 and the second comes from (6.1)
together with Lemma 6.13.

Now we provide another explanation for this map in terms of groupoids.
Firstly, Corollary 3.5 implies that $G(X)_{X(\omega)}=X(\omega)\times
X(\omega)$ and hence $G(X)_{\overline{X(\omega)}}=\overline{X(\omega)\times
X(\omega)}^{G(X)}$. Now we have

$G(X)_{\overline{X(\omega)}}=\bigcup_{S>0}\big{(}\overline{E_{S}}^{G(X)}\cap\overline{X(\omega)\times
X(\omega)}^{G(X)}\big{)}=\bigcup_{S>0}\overline{\overline{E_{S}}^{G(X)}\cap(X(\omega)\times
X(\omega))}^{G(X)},$

where the last inequality is due to the fact that $\overline{E_{S}}^{G(X)}$ is
clopen in $G(X)$. By Lemma 3.7, we have

$\overline{E_{S}}^{G(X)}\cap(X(\omega)\times X(\omega))=\\{(\alpha,\gamma)\in
X(\omega)\times X(\omega):d_{\omega}(\alpha,\gamma)\leq
S\\}=:E_{S}(X(\omega),d_{\omega})$

_i.e._ , the $S$-entourage in the limit space $(X(\omega),d_{\omega})$.
Therefore, we conclude that

$G(X)_{\overline{X(\omega)}}=\bigcup_{S>0}\overline{E_{S}(X(\omega),d_{\omega})}^{G(X)}.$

On the other hand, we have (by definition) that

$G(X(\omega))=\bigcup_{S>0}\overline{E_{S}(X(\omega),d_{\omega})}^{\beta(X(\omega)\times
X(\omega))}.$

By the universal property of the Stone-Čech compactification, there is a
surjective continuous map
$\overline{E_{S}(X(\omega),d_{\omega})}^{\beta(X(\omega)\times
X(\omega))}\longrightarrow\overline{E_{S}(X(\omega),d_{\omega})}^{G(X)}$,
which induces an injective map
$C_{c}(G(X)_{\overline{X(\omega)}})\longrightarrow C_{c}(G(X(\omega)))$.
Moreover, it is routine to check that it induces a $C^{*}$-monomorphism

$C^{*}_{r}(G(X)_{\overline{X(\omega)}})\longrightarrow
C^{*}_{r}(G(X(\omega)))\cong C^{*}_{u}(X(\omega)),$

which can be verified to coincide with the map (6.2). Details are left to
readers.

In particular, we consider a countable discrete group $X=\Gamma$. Fixing a
point $\omega\in\partial_{\beta}\Gamma$, we mentioned in Example 6.11 that
$C^{*}_{r}(G(\Gamma)_{\overline{\Gamma(\omega)}})\cong
C(\overline{\Gamma\omega})\rtimes\Gamma$. On the other hand, we know from
Lemma 3.8 that
$C^{*}_{u}(\Gamma(\omega))\cong\ell^{\infty}(\Gamma\omega)\rtimes\Gamma$. In
this case, one can check that the map (6.2) is induced by the natural
embedding
$C(\overline{\Gamma\omega})\hookrightarrow\ell^{\infty}(\Gamma\omega)$.

##  7\. Geometric ideals vs ghostly ideals

In this section, we return to the lattice $\mathfrak{I}_{U}$ in (1.1). Recall
from Theorem 5.4 that for an invariant open subset $U\subseteq\beta X$, the
geometric ideal $I(U)$ and the ghostly ideal $\tilde{I}(U)$ are the smallest
and the largest elements in $\mathfrak{I}_{U}$. Also as noticed before,
generally $\mathfrak{I}_{U}$ consists of more than one element.

Now we would like to study when $I(U)=\tilde{I}(U)$ or equivalently, when
$\mathfrak{I}_{U}$ consists of a single element. Moreover, we will discuss
their $K$-theories and provide a sufficient condition to ensure
$K_{\ast}(I(U))=K_{\ast}(\tilde{I}(U))$ for $\ast=0,1$.

First recall from Proposition 5.9 that we already have a characterisation for
$I(U)=\tilde{I}(U)$ using short exact sequences, while the condition therein
still seems hard to check. Now we aim to search for a more practical criterion
to ensure $I(U)=\tilde{I}(U)$. We start with the following result, which
combines [14, Theorem 4.4] and [36, Theorem 1.3].

######  Proposition 7.1.

For a space $(X,d)$, the following are equivalent:

  1. (1)

$X$ has Property A;

  2. (2)

$G(X)$ is amenable;

  3. (3)

$G(X)_{\partial_{\beta}X}$ is amenable;

  4. (4)

$I(U)=\tilde{I}(U)$ for any invariant open subset $U\subseteq\beta X$;

  5. (5)

$I(X)=\tilde{I}(X)$.

We remark that “(1) $\Rightarrow$ (4)” was originally proved in [14] using
approximations by kernels. Here we will take a shortcut, and the idea will
also be used later.

###### Proof of Proposition 7.1.

“(1) $\Leftrightarrow$ (2)” was proved in [39, Theorem 5.3]. “(2)
$\Leftrightarrow$ (3)” is due to the permanence properties of amenability (see
Section 2.6) together with the fact that $G(X)_{X}\cong X\times X$ is always
amenable.

“(2) $\Rightarrow$ (4)”: Let $U\subseteq\beta X$ be an invariant open subset.
As open/closed subgroupoids, both $G(X)_{U}$ and $G(X)_{U^{c}}$ are amenable
as well. Consider the following commutative diagram coming from (5.2) and
(5.3):

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}_{{\rm
max}}(G(X)_{U})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}_{{\rm
max}}(G(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}_{{\rm
max}}(G(X)_{U^{c}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}_{r}(G(X)_{U})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}_{r}(G(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}_{r}(G(X)_{U^{c}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

By Proposition 2.12, all three vertical lines are isomorphisms. Hence the
exactness of the first row implies that the second row is exact as well.
Therefore, we conclude (4) thanks to Proposition 5.9.

“(4) $\Rightarrow$ (5)” holds trivially, and “(5) $\Rightarrow$ (1)” comes
from [36, Theorem 1.3] together with Example 4.8 and Example 5.5. Hence we
conclude the proof. ∎

Proposition 7.1 provides a coarse geometric characterisation for
$I(X)=\tilde{I}(X)$ using Property A. However, we notice that assuming
Property A is often too strong to ensure that $I(U)=\tilde{I}(U)$ for merely a
_specific_ invariant open subset $U\subseteq\beta X$. A trivial example is
that $I(\beta X)=\tilde{I}(\beta X)$ holds for any space $X$. This suggests us
to explore a weaker criterion for $I(U)=\tilde{I}(U)$, and we reach the
following:

######  Proposition 7.2.

Let $(X,d)$ be a space and $U$ be an invariant open subset of $\beta X$. If
the canonical quotient map $C^{*}_{{\rm max}}(G(X)_{U^{c}})\to
C^{*}_{r}(G(X)_{U^{c}})$ is an isomorphism, then $I(U)=\tilde{I}(U)$. In
particular, if the groupoid $G(X)_{U^{c}}$ is amenable then
$I(U)=\tilde{I}(U)$.

###### Proof.

We consider the following commutative diagram:

(7.1)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}_{{\rm
max}}(G(X)_{U})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{U}}$$\textstyle{C^{*}_{{\rm
max}}(G(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}_{{\rm
max}}(G(X)_{U^{c}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{I}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}_{r}(G(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}_{r}(G(X)_{U^{c}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Here the map $\pi_{U}$ is the composition:

(7.2) $C^{*}_{{\rm max}}(G(X)_{U})\to C^{*}_{r}(G(X)_{U})\cong
I(U)\hookrightarrow\tilde{I}(U),$

where the middle isomorphism comes from Lemma 4.7. Note that the top
horizontal line is automatically exact, while the bottom one is also exact
thanks to Proposition 5.9. Also note that the middle vertical map is always
surjective and by assumption, the right vertical map is an isomorphism. Hence
via a diagram chasing argument, we obtain that the left vertical map is
surjective. This concludes that $I(U)=\tilde{I}(U)$ thanks to (7.2). ∎

######  Remark 7.3.

When $U=X$, Proposition 7.2 recovers “(3) $\Rightarrow$ (5)” in Proposition
7.1. Readers might wonder whether the converse of Proposition 7.2 holds as in
the case of $U=X$. We manage to provide a partial answer in Section 8.3 below.

Now we move to discuss the $K$-theory of geometric and ghostly ideals.
Firstly, we need an extra notion:

######  Definition 7.4.

For a set $\mathcal{S}$ of subsets of $X$, denote ${\bf L}(\mathcal{S})$ the
smallest ideal in $X$ containing $\mathcal{S}$, and we say that ${\bf
L}(\mathcal{S})$ is _generated by $\mathcal{S}$_. An ideal ${\bf L}$ in $X$ is
called _countably generated_ if there exists a countable set $\mathcal{S}$
such that ${\bf L}={\bf L}(\mathcal{S})$.

An invariant open subset $U\subseteq\beta X$ is called _countably generated_
if the associated ideal ${\bf L}(U)$ is countably generated.

######  Example 7.5.

For a space $X$ and a subspace $A\subseteq X$, it follows from Lemma 4.14 that
the spatial ideal $I_{A}$ is countably generated. On the other hand, it
follows from Lemma 4.17 that principal ideals are always countably generated.
In particular, the ideal $\langle P\rangle$ considered in [44, Section 3] (see
also Example 5.10) is countably generated.

The property of countable generatedness leads to the following:

######  Lemma 7.6.

Let ${\bf L}$ be a countably generated ideal in $X$. Then there exists a
countable subset $\\{Y_{1},Y_{2},\cdots,Y_{n},\cdots\\}$ in ${\bf L}$ such
that

${\bf L}=\\{Z\subseteq X:~{}\exists~{}n\in\mathbb{N}\text{ such that
}Z\subseteq Y_{n}\\}.$

Consequently for any countably generated open invariant subset $U$ of $\beta
X$, the subgroupoid $G(X)_{U}$ is $\sigma$-compact.

To prove Lemma 7.6, we need an auxiliary result on the structure of ${\bf
L}(\mathcal{S})$:

######  Lemma 7.7.

For a set $\mathcal{S}$ of subsets of $X$, denote
$\mathcal{S}^{(1)}:=\\{A_{1}\cup\cdots\cup
A_{n}:A_{i}\in\mathcal{S},n\in\mathbb{N}\\}$ and
$\mathcal{S}^{(2)}:=\\{\mathcal{N}_{k}(\tilde{A}):\tilde{A}\in\mathcal{S}^{(1)},k\in\mathbb{N}\\}$.
Then we have:

${\bf L}(\mathcal{S})=\\{Z:\exists~{}Y\in\mathcal{S}^{(2)}\text{ such that
}Z\subseteq Y\\}.$

###### Proof.

Denote $\mathcal{S}^{(3)}:=\\{Z:\exists~{}Y\in\mathcal{S}^{(2)}\text{ such
that }Z\subseteq Y\\}$. It is clear that $\mathcal{S}^{(3)}\subseteq{\bf
L}(\mathcal{S})$, and any ideal containing $\mathcal{S}$ must contain
$\mathcal{S}^{(3)}$. Hence it remains to check that $\mathcal{S}^{(3)}$ is an
ideal.

For $Z\in\mathcal{S}^{(3)}$ and $W\subseteq Z$, there exists
$Y\in\mathcal{S}^{(2)}$ such that $Z\subseteq Y$. Hence $W\subseteq Y$, which
implies that $W\in\mathcal{S}^{(3)}$. Also note that $Y\in\mathcal{S}^{(2)}$
implies that there exists $\tilde{A}\in\mathcal{S}^{(1)}$ and
$k^{\prime}\in\mathbb{N}$ such that $Y=\mathcal{N}_{k^{\prime}}(\tilde{A})$.
Hence for any $k\in\mathbb{N}$, we have
$\mathcal{N}_{k}(Z)\subseteq\mathcal{N}_{k}(Y)\subseteq\mathcal{N}_{k+k^{\prime}}(\tilde{A})$,
which implies that $\mathcal{N}_{k}(Z)\in\mathcal{S}^{(3)}$.

Finally for $Z_{1},Z_{2}\in\mathcal{S}^{(3)}$, there exist
$\tilde{A}_{1},\tilde{A}_{2}\in\mathcal{S}^{(1)}$ and
$k_{1},k_{2}\in\mathbb{N}$ such that
$Z_{i}\subseteq\mathcal{N}_{k_{i}}(\tilde{A}_{i})$ for $i=1,2$. Hence

$Z_{1}\cup
Z_{2}\subseteq\mathcal{N}_{k_{1}}(\tilde{A}_{1})\cup\mathcal{N}_{k_{2}}(\tilde{A}_{2})\subseteq\mathcal{N}_{k_{1}+k_{2}}(\tilde{A}_{1}\cup\tilde{A}_{2})\in\mathcal{S}^{(2)},$

which implies that $Z_{1}\cup Z_{2}\in\mathcal{S}^{(3)}$. Therefore, we
conclude the proof. ∎

###### Proof of Lemma 7.6.

By assumption, there exists a countable $\mathcal{S}$ such that ${\bf L}={\bf
L}(\mathcal{S})$. Using the notation of Lemma 7.7, the set $\mathcal{S}^{(2)}$
is countable as well. Hence the first statement follows directly from Lemma
7.7.

For the second, note that $U=\bigcup_{n\in\mathbb{N}}\overline{Y_{n}}$ and
hence $G(X)_{U}=\bigcup_{n\in\mathbb{N}}G(X)_{\overline{Y_{n}}}$. Since each
$\overline{Y_{n}}$ is closed, we obtain that $G(X)_{U}$ is $\sigma$-compact as
desired. ∎

Now we are in the position to discuss the $K$-theory of geometric and ghostly
ideals:

######  Proposition 7.8.

Let $X$ be a space which can be coarsely embedded into some Hilbert space.
Then for any countably generated invariant open subset $U\subseteq\beta X$, we
have an isomorphism

$(\iota_{U})_{\ast}:K_{\ast}(I(U))\longrightarrow K_{\ast}(\tilde{I}(U))$

for $\ast=0,1$, where $\iota_{U}$ is the inclusion map. Therefore for any
ideal $I$ in $C^{*}_{u}(X)$ with $U(I)$ countably generated, we have an
injective homomorphism

$(\iota_{I})_{\ast}:K_{\ast}(\overline{I\cap\mathbb{C}_{u}[X]})\longrightarrow
K_{\ast}(I)$

for $\ast=0,1$, where $\iota_{I}$ is the inclusion map.

###### Proof.

Fixing such a $U\subseteq\beta X$, we consider the commutative diagram (7.1)
from the proof of Proposition 7.2, where both of the horizontal lines are
exact. This implies the following commutative diagram in $K$-theories:

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast}(C^{*}_{{\rm
max}}(G(X)_{U}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\pi_{U})_{\ast}}$$\textstyle{K_{\ast}(C^{*}_{{\rm
max}}(G(X)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast}(C^{*}_{{\rm
max}}(G(X)_{U^{c}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast+1}(C^{*}_{{\rm
max}}(G(X)_{U}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\pi_{U})_{\ast+1}}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast}(\tilde{I}(U))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast}(C^{*}_{r}(G(X)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast}(C^{*}_{r}(G(X)_{U^{c}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\ast+1}(\tilde{I}(U))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

where both horizontal lines are exact. Recall from [39, Theorem 5.4] that $X$
is coarsely embeddable if and only if $G(X)$ is a-T-menable. It follows
directly from Definition 2.13 that as subgroupoids, both $G(X)_{U}$ and
$G(X)_{U^{c}}$ are a-T-menable. Moreover, it is clear that $G(X)$ and
$G(X)_{U^{c}}$ are $\sigma$-compact by definition. Hence Proposition 2.14
implies that $G(X)$ and $G(X)_{U^{c}}$ are $K$-amenable. This shows that the
middle two vertical maps in the above diagram are isomorphisms, which implies
that $(\pi_{U})_{\ast}$ is an isomorphism by the Five Lemma. Therefore from
(7.2), we obtain that the composition

$K_{\ast}(C^{*}_{{\rm max}}(G(X)_{U}))\longrightarrow
K_{\ast}(C^{*}_{r}(G(X)_{U}))\cong
K_{\ast}(I(U))\stackrel{{\scriptstyle(\iota_{U})_{\ast}}}{{\longrightarrow}}K_{\ast}(\tilde{I}(U))$

is an isomorphism for $\ast=0,1$.

On the other hand, it follows from Lemma 7.6 that $G(X)_{U}$ is
$\sigma$-compact, which implies that $G(X)_{U}$ is $K$-amenable again by
Proposition 2.14. Hence we conclude that the map $(\iota_{U})_{\ast}$ is an
isomorphism for $\ast=0,1$.

For the last statement, we assume that $U=U(I)$. By Lemma 4.11, we have that
$\overline{I\cap\mathbb{C}_{u}[X]}=I(U)$. Also Lemma 5.3(1) shows that
$I\subseteq\tilde{I}(U)$. Hence the inclusion map $\iota_{U}$ can be
decomposed as follows:

$I(U)=\overline{I\cap\mathbb{C}_{u}[X]}\stackrel{{\scriptstyle\iota_{I}}}{{\hookrightarrow}}I\hookrightarrow\tilde{I}(U).$

Therefore, $(\iota_{U})_{\ast}$ being an isomorphism implies that
$(\iota_{I})_{\ast}$ is injective for $\ast=0,1$. ∎

Applying Proposition 7.8 to the case of $U=X$, we partially recover the
following result by Finn-Sell (see [19, Proposition 35]), which is crucial for
the counterexamples to the coarse Baum-Connes conjecture:

######  Corollary 7.9.

Let $X$ be a space which can be coarsely embedded into some Hilbert space.
Then the inclusion of $\mathfrak{K}(\ell^{2}(X))$ into $I_{G}$ induces an
isomorphism on the $K$-theory level.

######  Remark 7.10.

For a general ideal $I$ in $C^{*}_{u}(X)$, our method in the proof of
Proposition 7.8 only provides the injectivity of the induced map
$(\iota_{I})_{\ast}$. We wonder whether this map is indeed an isomorphism
under the same assumption (see Question 9.4).

We end this section with an example, which shows that _not every_ ideal in a
space is countably generated.

######  Example 7.11.

Let $X=\mathbb{N}\times\mathbb{N}$, equipped with the metric induced from the
Euclidean metric $d_{E}$ on $\mathbb{R}^{2}$. For each
$\theta\in[0,\frac{\pi}{2}]$ and $k\in\mathbb{N}$, we define

$\ell_{\theta}:=\\{(x,y)\in\mathbb{R}\times\mathbb{R}:y=\tan(\theta)x\\}$

and

$S_{\theta,k}:=\\{(x,y)\in X:d_{E}((x,y),\ell_{\theta})\leq
k\\}=\mathcal{N}_{k}(\ell_{\theta})\cap X.$

Consider
$\mathcal{S}:=\\{S_{\theta,k}:\theta\in[0,\frac{\pi}{2}],k\in\mathbb{N}\\}$
and set $\mathcal{S}^{(1)}:=\\{A_{1}\cup\cdots\cup
A_{n}:A_{i}\in\mathcal{S},n\in\mathbb{N}\\}$ as in Lemma 7.7. For any $R>0$
and $A=A_{1}\cup\cdots\cup A_{n}\in\mathcal{S}^{(1)}$ where
$A_{i}\in\mathcal{S}$, we have
$\mathcal{N}_{R}(A)=\mathcal{N}_{R}(A_{1})\cup\cdots\cup\mathcal{N}_{R}(A_{n})$,
which is contained in some element in $\mathcal{S}$. Hence applying Lemma 7.7,
the ideal ${\bf L}(\mathcal{S})$ generated by $\mathcal{S}$ is:

(7.3) ${\bf L}(\mathcal{S})=\\{Z:\exists~{}Y\in\mathcal{S}^{(1)}\text{ such
that }Z\subseteq Y\\}.$

We claim that ${\bf L}(\mathcal{S})$ is _not_ countably generated. Otherwise,
there exists a countable subset $\mathcal{S}^{\prime}$ generating ${\bf
L}(\mathcal{S})$. Moreover, according to (7.3) we can assume that

$\mathcal{S}^{\prime}=\\{Y_{n}:n\in\mathbb{N}\\}\quad\text{where}\quad
Y_{n}=S_{\theta_{n,1},k_{n}}\cup\cdots\cup
S_{\theta_{n,p_{n}},k_{n}}\in\mathcal{S}^{(1)}.$

Choose
$\theta\in[0,\frac{\pi}{2}]\setminus\\{\theta_{n,i}:i=1,2,\cdots,p_{n};n\in\mathbb{N}\\}$,
and consider $Y=S_{\theta,1}$. Since $\mathcal{S}^{\prime}$ generates ${\bf
L}(\mathcal{S})$, it follows from Lemma 7.7 that there exist $R>0$ and
$Y_{m_{1}},\cdots,Y_{m_{l}}\in\mathcal{S}^{\prime}$ such that

$S_{\theta,1}=Y\subseteq\mathcal{N}_{R}(Y_{m_{1}}\cup\cdots\cup Y_{m_{l}}).$

Note that the right hand side is contained in a finite union of some
$R^{\prime}$-neighbourhood of lines (in $\mathbb{R}^{2}$ crossing the origin)
with slopes in the set

$\big{\\{}\tan(\theta_{n,i}):i=1,2,\cdots,p_{n};n\in\mathbb{N}\big{\\}}.$

This leads to a contradiction due to the choice of $\theta$, which concludes
that ${\bf L}(\mathcal{S})$ cannot be countably generated.

##  8\. Partial Property A and partial operator norm localisation property

In the previous section, we find a sufficient condition (Proposition 7.2) to
ensure $\tilde{I}(U)=I(U)$ for a given invariant open subset $U\subseteq\beta
X$. As promised in Remark 7.3, now we study its converse and show that this is
indeed an equivalent condition under the assumption of countable
generatedness.

Our strategy is to follow the outline of the case that $U=X$. More precisely,
we introduce a notion called _partial Property A towards invariant subsets of
the boundary $\partial_{\beta}X$_, and then consider its counterpart in the
context of operator norm localisation property to provide the desired
characterisation.

###  8.1. Partial Property A

Recall from Proposition 7.1 that a space $X$ has Property A if and only if the
groupoid $G(X)_{\partial_{\beta}X}$ is amenable, which characterises
$I(X)=\tilde{I}(X)$. Together with Proposition 7.2, this inspires us to
introduce the following:

######  Definition 8.1.

Let $(X,d)$ be a space and $U\subseteq\beta X$ be an invariant open subset. We
say that $X$ has _partial Property A towards $\partial_{\beta}X\setminus U$_
if $G(X)_{\partial_{\beta}X\setminus U}$ is amenable.

It is clear from definition that $X$ has Property A if and only if it has
partial Property A towards the whole boundary $\partial_{\beta}X$. On the
other hand, it follows from Proposition 7.2 that if $X$ has partial Property A
towards $\partial_{\beta}X\setminus U$, then we have $I(U)=\tilde{I}(U)$. The
rest of this section is devoted to studying the converse.

Firstly, we aim to unpack the groupoid language and provide a concrete
geometric description for partial Property A, which resembles the definition
of Property A (see Definition 2.1). The following is the main result:

######  Proposition 8.2.

Let $(X,d)$ be a space and $U\subseteq\beta X$ an invariant open subset. Then
$X$ has partial Property A towards $U^{c}=\partial_{\beta}X\setminus U$ _if
and only if_ for any $\varepsilon,R>0$, there exist $S>0$, a subset
$D\subseteq X$ with $\overline{D}\supseteq U^{c}$ and a function $f:X\times
X\to[0,1]$ satisfying:

  1. (1)

$\mathrm{supp}(f)\subseteq E_{S}$;

  2. (2)

for any $x\in D$, we have $\sum_{z\in X}f(z,x)=1$;

  3. (3)

for any $x,y\in D$ with $d(x,y)\leq R$, we have $\sum_{z\in
X}|f(z,x)-f(z,y)|\leq\varepsilon$.

Comparing Proposition 8.2 with Definition 2.1, it is clear that Property A
implies partial Property A towards any invariant closed subset of
$\partial_{\beta}X$.

To prove Proposition 8.2, we start with the following lemma:

######  Lemma 8.3.

With the same notation as above, $X$ has partial Property A towards $U^{c}$
_if and only if_ for any $\varepsilon,R>0$, there exist $S>0$ and a function
$f:X\times X\to[0,1]$ satisfying:

  1. (1)

$\mathrm{supp}(f)\subseteq E_{S}$;

  2. (2)

for any $\omega\in U^{c}$, we have $\sum_{\alpha\in
X(\omega)}\overline{f}(\alpha,\omega)=1$;

  3. (3)

for $\omega\in U^{c}$ and $\alpha\in X(\omega)$ with
$d_{\omega}(\alpha,\omega)\leq R$, then $\sum_{\gamma\in
X(\omega)}|\overline{f}(\gamma,\alpha)-\overline{f}(\gamma,\omega)|\leq\varepsilon$,

where $\overline{f}\in C_{0}(G(X))$ is the continuous extension from Lemma
2.9.

###### Proof.

By definition, $X$ has partial Property A towards $U^{c}$ if and only if for
any $\varepsilon>0$ and compact $K\subseteq G(X)_{U^{c}}$, there exists $g\in
C_{c}(G(X)_{U^{c}})$ with range in $[0,1]$ such that for any $\gamma\in K$ we
have

$\sum_{\alpha\in\mathcal{G}_{\mathrm{r}(\gamma)}}g(\alpha)=1\quad\mbox{and}\quad\sum_{\alpha\in\mathcal{G}_{\mathrm{r}(\gamma)}}|g(\alpha)-g(\alpha\gamma)|<\varepsilon.$

Recall from (5.1) that the restriction map $C_{c}(G(X))\to
C_{c}(G(X)_{U^{c}})$ is surjective, and hence $g$ can be regarded as a
function in $C_{c}(G(X))$. Taking $f$ to be the restriction of $g$ on $X\times
X$, then $f\in\ell^{\infty}(X\times X)$ and there exists $S>0$ such that
$\mathrm{supp}(f)\subseteq E_{S}$ for some $S>0$. Using the notation from
Lemma 2.9, we have $g=\overline{f}$. Note that
$G(X)_{U^{c}}=\bigcup_{R>0}(\overline{E_{R}}\cap G(X)_{U^{c}})$, and hence
compact subsets of $G(X)_{U^{c}}$ are always contained in those of the form
$\overline{E_{R}}\cap G(X)_{U^{c}}$. Furthermore, Lemma 3.7 implies

$\overline{E_{R}}\cap G(X)_{U^{c}}=\bigcup_{\omega\in
U^{c}}\\{(\alpha,\gamma)\in X(\omega)\times
X(\omega):d_{\omega}(\alpha,\gamma)\leq R\\}.$

Combining with Lemma 3.4, we conclude the proof. ∎

As a direct corollary (together with Lemma 3.7), we obtain:

######  Corollary 8.4.

Assume that $X$ has partial Property A towards $U^{c}$. Then the family of
metric spaces $\\{(X(\omega),d_{\omega})\\}_{\omega\in U^{c}}$ has uniform
Property A in the sense that the parameters in Definition 2.1 can be chosen
uniformly.

######  Remark 8.5.

It is unclear to us whether the converse of Corollary 8.4 holds. Note that
$\overline{X(\omega)}$ might contain points outside $X(\omega)$ as discussed
in Theorem 6.8, hence we do not know whether functions on $X(\omega)\times
X(\omega)$ can be glued together to provide a continuous function on
$G(X)_{U^{c}}$.

###### Proof of Proposition 8.2.

_Sufficiency:_ For any $\varepsilon,R>0$, choose $S>0$, $D\subseteq X$ and a
function $g:X\times X\to[0,1]$ satisfying the conditions (1)-(3) for
$\varepsilon$ and $3R$. Take a map $p:\mathcal{N}_{R}(D)\to D$ such that the
restriction of $p$ on $D$ is the identity map and $d(p(x),x)\leq R$. Now we
define:

$f(x,y)=\begin{cases}~{}g(x,p(y)),&y\in\mathcal{N}_{R}(D);\\\
~{}g(x,y),&\text{otherwise}.\end{cases}$

It is clear that $\mathrm{supp}(f)\subseteq E_{R+S}$. Moreover for any
$y_{1},y_{2}\in\mathcal{N}_{R}(D)$ with $d(y_{1},y_{2})\leq R$, we have
$d(p(y_{1}),p(y_{2}))\leq 3R$ and hence

$\sum_{x\in X}|f(x,y_{1})-f(x,y_{2})|=\sum_{x\in
X}|g(x,p(y_{1}))-g(x,p(y_{2}))|\leq\varepsilon.$

Therefore (enlarging $S$ to $S+R$) we obtain that for any $\varepsilon,R>0$,
there exist $S>0$, a subset $D\subseteq X$ with $\overline{D}\supseteq U^{c}$
and a function $f:X\times X\to[0,1]$ satisfying:

  1. (1)

$\mathrm{supp}(f)\subseteq E_{S}$;

  2. (2)

for any $x\in D$, we have $\sum_{z\in X}f(z,x)=1$;

  3. (3)

for any $x\in D$ and $y\in X$ with $d(x,y)\leq R$, we have $\sum_{z\in
X}|f(z,x)-f(z,y)|\leq\varepsilon$.

Now we fix $\varepsilon,R>0$ and take such $S,D$ and function $f$.

Given $\omega\in U^{c}$, we have $\omega(D)=1$. Choose
$\\{t_{\alpha}:D_{\alpha}\to R_{\alpha}\\}$ to be a compatible family for
$\omega$. Applying Proposition 3.3, there exists $Y_{\omega}\subseteq X$ with
$\omega(Y)=1$ and a local coordinate system $\\{h_{y}:B(\omega,R+S)\to
B(y,R+S)\\}_{y\in Y_{\omega}}$ such that the map

$h_{y}:B(\omega,R+S)\to B(y,R+S),\quad\alpha\mapsto t_{\alpha}(y)$

is a surjective isometry for each $y\in Y_{\omega}$. Replacing $Y$ by
$Y_{\omega}\cap D$, we assume that $Y_{\omega}\subseteq D$. Note that
$\mathrm{supp}(f)\subseteq E_{S}$, and hence applying Lemma 3.7 we have

$\displaystyle\sum_{\alpha\in X(\omega)}\overline{f}(\alpha,\omega)$
$\displaystyle=\sum_{\alpha\in
B(\omega,S)}\overline{f}(\alpha,\omega)=\sum_{\alpha\in
B(\omega,S)}\lim_{x\to\omega}f(t_{\alpha}(x),x)=\lim_{x\to\omega,x\in
Y_{\omega}}\sum_{\alpha\in B(\omega,S)}f(t_{\alpha}(x),x)$ (8.1)
$\displaystyle=\lim_{x\to\omega,x\in Y_{\omega}}\sum_{z\in
B(x,S)}f(z,x)=\lim_{x\to\omega,x\in Y_{\omega}}\sum_{z\in X}f(z,x),$

where the last item equals $1$ thanks to the assumption.

On the other hand, for any $\alpha\in B(\omega,R)$ we have
$\lim_{x\to\omega}d(t_{\alpha}(x),x)\leq R$. Hence shrinking $Y_{\omega}$ if
necessary, we can assume that $d(t_{\alpha}(x),x)\leq R$ for any $x\in
Y_{\omega}$. Therefore, we have

(8.2) $\displaystyle\sum_{\gamma\in
X(\omega)}|\overline{f}(\gamma,\alpha)-\overline{f}(\gamma,\omega)|$
$\displaystyle=\sum_{\gamma\in
B(\omega,R+S)}|\overline{f}(\gamma,\alpha)-\overline{f}(\gamma,\omega)|$
$\displaystyle=\lim_{x\to\omega,x\in Y_{\omega}}\sum_{\gamma\in
B(\omega,R+S)}|f(t_{\gamma}(x),t_{\alpha}(x))-f(t_{\gamma}(x),x)|$
$\displaystyle=\lim_{x\to\omega,x\in Y_{\omega}}\sum_{z\in
B(x,R+S)}|f(z,t_{\alpha}(x))-f(z,x)|$ $\displaystyle=\lim_{x\to\omega,x\in
Y_{\omega}}\sum_{z\in X}|f(z,t_{\alpha}(x))-f(z,x)|,$

where the last item is no more than $\varepsilon$ by assumption. Therefore
applying Lemma 8.3, we conclude the sufficiency.

_Necessity:_ Given $\varepsilon,R>0$, Lemma 8.3 provides $S>0$ and a function
$f:X\times X\to[0,1]$ satisfying the conditions (1)-(3) therein. Fix an
$\omega\in U^{c}$ and we choose $\\{t_{\alpha}:D_{\alpha}\to R_{\alpha}\\}$ to
be a compatible family for $\omega$. Applying Proposition 3.3, there exists
$Y_{\omega}\subseteq X$ with $\omega(Y)=1$ and a local coordinate system
$\\{h_{y}:B(\omega,R+2S)\to B(y,R+S)\\}_{y\in Y_{\omega}}$ such that the map

$h_{y}:B(\omega,R+S)\to B(y,R+S),\quad\alpha\mapsto t_{\alpha}(y)$

is a surjective isometry for each $y\in Y_{\omega}$. By the calculations in
(8.1), we obtain that

$\lim_{x\to\omega,x\in Y_{\omega}}\sum_{z\in X}f(z,x)=1.$

Hence for the given $\varepsilon$, there exists $Y^{\prime}_{\omega}\subseteq
Y_{\omega}$ with $\omega(Y^{\prime}_{\omega})=1$ such that for any $x\in
Y^{\prime}_{\omega}$ we have

$\sum_{z\in X}f(z,x)\in(1-\varepsilon,1+\varepsilon).$

On the other hand, for any $\alpha\in B(\omega,R)$ we apply the calculations
in (8.2) and obtain:

$\lim_{x\to\omega,x\in Y^{\prime}_{\omega}}\sum_{z\in
X}|f(z,t_{\alpha}(x))-f(z,x)|=\sum_{\gamma\in
X(\omega)}|\overline{f}(\gamma,\alpha)-\overline{f}(\gamma,\omega)|\leq\varepsilon.$

Note that for $x\in Y^{\prime}_{\omega}$, Proposition 3.3 implies that
$\\{t_{\alpha}(x):\alpha\in B(\omega,R)\\}=B(x,R)$. Hence there exists
$Y^{\prime\prime}_{\omega}\subseteq Y^{\prime}_{\omega}$ with
$\omega(Y^{\prime\prime}_{\omega})=1$ such that for any $x\in
Y^{\prime\prime}_{\omega}$ and $y\in B(x,R)$, we have

$\sum_{z\in X}|f(z,y)-f(z,x)|<2\varepsilon.$

Taking $D:=\bigcup_{\omega\in U^{c}}Y^{\prime\prime}_{\omega}$, then it is
clear that $\overline{D}\supseteq U^{c}$. Moreover, the analysis above shows
that:

  * •

for any $x\in D$ we have $\sum_{z\in
X}f(z,x)\in(1-\varepsilon,1+\varepsilon)$;

  * •

for any $x\in D$ and $y\in B(x,R)$, we have $\sum_{z\in
X}|f(z,y)-f(z,x)|<2\varepsilon$.

Finally using a standard normalisation argument (or equivalently, applying
Lemma 2.11 and modifying Lemma 8.3 accordingly), we conclude the proof. ∎

Setting $\xi_{y}(x):=f(x,y)$ for the function $f$ in Proposition 8.2, we can
rewrite Proposition 8.2 as follows:

######  Proposition 8.2′.

Let $(X,d)$ be a space and $U\subseteq\beta X$ be an invariant open subset.
Then $X$ has partial Property A towards $U^{c}$ _if and only if_ for any
$\varepsilon,R>0$, there exist $S>0$, a subset $D\subseteq X$ with
$\overline{D}\supseteq U^{c}$ and a function
$\xi:D\to\ell^{1}(X)_{1,+},x\mapsto\xi_{x}$ satisfying:

  1. (1)

$\mathrm{supp}(\xi_{x})\subseteq B(x,S)$ for any $x\in D$;

  2. (2)

for any $x,y\in D$ with $d(x,y)\leq R$, we have
$\|\xi_{x}-\xi_{y}\|_{1}\leq\varepsilon$.

######  Remark 8.6.

We remark that the function $\xi$ in Proposition 8.2′ can be made such that
$\xi_{x}\in\ell^{1}(D)_{1,+}$. In fact, this is the same trick as in the case
of Property A (see, _e.g._ , [27, Proposition 4.2.5]). Moreover, we can
further replace $\ell^{1}(D)_{1,+}$ by $\ell^{2}(D)_{1,+}$ using the Mazur map
(see, _e.g._ , [45, Proposition 1.2.4] for the same trick).

Now we provide an alternative picture for Proposition 8.2 using the notion of
ideals in spaces (see Definition 4.12). Recall that for an ideal ${\bf L}$ in
$X$, we denote $U({\bf L}):=\bigcup_{Y\in{\bf L}}\overline{Y}$. We need the
following auxiliary lemma:

######  Lemma 8.7.

Let ${\bf L}$ be an ideal in $X$ and $D\subseteq X$. Then
$\overline{D}\supseteq U({\bf L})^{c}$ _if and only if_ there exists $Y\in{\bf
L}$ such that $D\supseteq Y^{c}$.

###### Proof.

Assume $Y\in{\bf L}$ such that $D\supseteq Y^{c}$. Note that
$\overline{Y}\cap\overline{Y^{c}}=\emptyset$ and
$\overline{Y}\cup\overline{Y^{c}}=\beta X$. Hence we have
$\overline{D}\supseteq\overline{Y^{c}}=\beta
X\setminus\overline{Y}\supseteq\beta X\setminus U({\bf L})=U({\bf L})^{c}$.

Conversely, assume that $\overline{D}\supseteq U({\bf L})^{c}$. Then
$(\overline{D})^{c}\subseteq U({\bf L})=\bigcup_{Y\in{\bf L}}\overline{Y}$.
Since $\overline{D}$ is clopen in the compact space $\beta X$, the set
$(\overline{D})^{c}$ is compact as well. Hence there exists
$Y_{1},\cdots,Y_{n}\in{\bf L}$ such that
$(\overline{D})^{c}\subseteq\overline{Y_{1}}\cup\cdots\cup\overline{Y_{n}}=\overline{Y_{1}\cup\cdots\cup
Y_{n}}$. Since ${\bf L}$ is an ideal, the set $Y:=Y_{1}\cup\cdots\cup
Y_{n}\in{\bf L}$. Then we have $(\overline{D})^{c}\subseteq\overline{Y}$,
which implies that $\overline{D}\supseteq(\overline{Y})^{c}=\overline{Y^{c}}$.
Finally we obtain $D=\overline{D}\cap X\supseteq\overline{Y^{c}}\cap X=Y^{c}$,
which conclude the proof. ∎

Thanks to Lemma 8.7, now we can rewrite Proposition 8.2 (combining with Remark
8.6) as follows:

######  Proposition 8.2′′.

Let $(X,d)$ be a space, $U\subseteq\beta X$ an invariant open subset and ${\bf
L}={\bf L}(U)$ the associated ideal in $X$. Then $X$ has partial Property A
towards $U^{c}$ _if and only if_ for any $\varepsilon,R>0$, there exist $S>0$,