diff --git "a/K9E4T4oBgHgl3EQfJgxB/content/tmp_files/2301.04921v1.pdf.txt" "b/K9E4T4oBgHgl3EQfJgxB/content/tmp_files/2301.04921v1.pdf.txt"
new file mode 100644--- /dev/null
+++ "b/K9E4T4oBgHgl3EQfJgxB/content/tmp_files/2301.04921v1.pdf.txt"
@@ -0,0 +1,2734 @@
+arXiv:2301.04921v1  [math.OA]  12 Jan 2023
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+QIN WANG AND JIAWEN ZHANG
+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- ˇCech 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.
+Mathematics Subject Classification (2020): 47L20, 46L80, 51F30.
+Keywords: Uniform Roe algebras, Coarse groupoids, Geometric and ghostly ideals, Max-
+imal 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 pi-
+oneering 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 oper-
+ators 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 prac-
+tical 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 ˇSpakula 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
+Date: January 13, 2023.
+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.
+1
+
+2
+QIN WANG AND JIAWEN ZHANG
+[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 ℓ2(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 ∗-subalgebra of B(ℓ2(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 lo-
+cally compact, Hausdorff, ´etale and principal groupoid (see Section 2.5 for precise
+definitions), and the unit space of G(X) coincides with the Stone- ˇCech compacti-
+fication βX 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 ⊆ βX is invariant if any element
+γ 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 βX, and conversely for any invariant open subset U ⊆ βX 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 βX 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 βX, which explains the terminology. Consequently, the geometric ideals in
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+3
+C∗
+u(X) are easy to handle and they must have the form of I(U) for some invariant
+open subset U ⊆ βX, 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 IG consisting of all ghost operators are not geometric (see also [21]). Recall
+that an operator T ∈ B(ℓ2(X)) is a ghost if T ∈ C0(X × X) when regarding T as a
+function on X × 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 IG and
+for the ideal of compact operators in B(ℓ2(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 ⊆ βX, 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)
+IU := {I is an ideal in C∗
+u(X) : U(I) = U}
+for each invariant open subset U ⊆ βX.
+One of the main contributions of the present paper is to find the smallest and
+the largest elements in the lattice IU. Following the discussions in [12], it is easy to
+see that I(U(I)) ⊆ I for any ideal I in C∗
+u(X), which implies that the geometric ideal
+I(U) is the smallest element in IU (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 βX. The ghostly ideal associated to
+U is defined to be
+˜I(U) := {T ∈ C∗
+u(X) : r(suppε(T)) ⊆ U for any ε > 0},
+where suppε(T) := {(x, y) ∈ X ×X : |T(x, y)| ≥ ε} and r : X ×X → X is the projection
+onto the first coordinate.
+We show that ˜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 βX. Then any ideal I in C∗
+u(X) with U(I) = U sits
+between I(U) and ˜I(U). More precisely, the geometric ideal I(U) is the smallest element
+while the ghostly ideal ˜I(U) is the largest element in the lattice IU in (1.1).
+Theorem C draws the border of the lattice IU 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 ˜I(U) for each invariant
+open subset U ⊆ βX, 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 ˜I(U), we also provide an alternative picture in
+terms of limit operators developed in [43], showing that ˜I(U) consists of operators
+
+4
+QIN WANG AND JIAWEN ZHANG
+which vanish in the (βX \ U)-direction (see Proposition 5.6).
+Note that ghost
+operators vanish in all directions (see Corollary 5.7), and hence operators in ˜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- ˇCech boundary
+∂βX := βX \ X. Moreover using the theory of limit spaces1 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 ω ∈ ∂βX such that I coincides with
+the ghostly ideal ˜I(βX \ X(ω)), where X(ω) is the limit space of ω.
+A point ω ∈ ∂βX 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 = Z:
+Theorem E (Theorem 6.8). For the integer group Z with the usual metric, there exist
+non-minimal points in the boundary ∂βZ. More precisely, for any sequence {hn}n∈N in Z
+tending to infinity such that |hn − hm| → +∞ when n + m → ∞ and n � m, and any
+ω ∈ ∂βZ with ω({hn}n∈N) = 1, then ω 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 Ap-
+pendix 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 IU defined in (1.1), we already notice that generally IU
+consists of more than one element. Hence it will be interesting and important to
+explore when IU has only a single element, or equivalently (thanks to Theorem
+C), when the geometric ideal I(U) coincides with the ghostly ideal ˜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)∂βX\U of the coarse groupoid ensures that I(U) = ˜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∗(I(U)) = K∗(˜I(U)) for ∗ = 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 ⊆ βX, we have an isomorphism
+(ιU)∗ : K∗(I(U)) −→ K∗(˜I(U))
+1Note 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.
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+5
+for ∗ = 0, 1, where ιU is the inclusion map.
+Note that there is a technical condition of countable generatedness (see Def-
+inition 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 Propo-
+sition 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 conjec-
+tures (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) = ˜I(U) implies that the restriction
+groupoid G(X)∂βX\U is amenable. Note that when U = X, [36, Theorem 1.3] implies
+that I(X) = ˜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 ⊆ βX be an invariant open subset. We say that X has partial
+Property A towards ∂βX \ U if G(X)∂βX\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 ⊆ βX be a countably generated invariant open subset. Then the following
+are equivalent:
+(1) X has partial Property A towards βX \ U;
+(2) ˜I(U) = I(U);
+(3) the ideal IG 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 βX directly. This has the
+advantage of playing within the given space rather than going to the mysterious
+Stone- ˇCech 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
+
+6
+QIN WANG AND JIAWEN ZHANG
+of geometric ideals (Definition A) studied in [12, 44], and we also discuss their
+minimality in the lattice of ideals IU 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 J´an ˇSpakula 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 ⊆ X, denote by χA
+the characteristic function of A, and set δx := χ{x} for x ∈ 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 Cb(X) the subset of bounded
+continuous functions on X. Recall that the support of a function f ∈ C(X) is the
+closure of {x ∈ X : f(x) � 0}, written as supp(f), and denote by Cc(X) the set
+of continuous functions with compact support. We also denote by C0(X) the set
+of continuous functions vanishing at infinity, which is the closure of Cc(X) with
+respect to the supremum norm ∥ f∥∞ := sup{| f(x)| : x ∈ X}.
+When X is discrete, denote ℓ∞(X) := Cb(X) and ℓ2(X) the Hilbert space of
+complex-valued square-summable functions on X. Denote by B(ℓ2(X)) the C∗-
+algebra of all bounded linear operators on ℓ2(X), and by K(ℓ2(X)) the C∗-subalgebra
+of all compact operators on ℓ2(X).
+For a discrete space X, denote by βX its Stone- ˇCech compactification and ∂βX :=
+βX \ X the Stone- ˇCech 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 BX(x, r) := {y ∈ X :
+d(x, y) ≤ r} for x ∈ X and r ≥ 0.
+For a subset A ⊆ X and r > 0, denote the
+r-neighbourhood of A in X by Nr(A) := {x ∈ X : dX(x, A) ≤ r}. For R > 0, denote the
+R-entourage by ER := {(x, y) ∈ X × X : d(x, y) ≤ R}.
+We saythat(X, d)hasbounded geometry ifforanyr > 0, the numbersupx∈X |BX(x, r)|
+is finite. Also say that (X, d) is strongly discrete if the set {d(x, y) : x, y ∈ X} is a dis-
+crete subset of 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.
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+7
+Now we recall the notion of Property A introduced by Yu (see, e.g., [45, Propo-
+sition 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 ε, R > 0
+there exist an S > 0 and a function f : X × X → [0, +∞) satisfying:
+(1) supp(f) ⊆ ES;
+(2) for any x ∈ X, we have �
+z∈X f(z, x) = 1;
+(3) for any x, y ∈ X with d(x, y) ≤ R, then �
+z∈X | f(z, x) − f(z, y)| ≤ ε.
+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 ε, R > 0 there exist an
+S > 0 and a function f : X × X → [0, +∞) satisfying:
+(1) supp(f) ⊆ ES;
+(2) for any x ∈ X, we have | �
+z∈X f(z, x) − 1| ≤ ε;
+(3) for any x, y ∈ X with d(x, y) ≤ R, then �
+z∈X | f(z, x) − f(z, y)| ≤ ε.
+We also need a characterisation for Property A using kernels. Recall that a kernel
+on X is a function k: X × X → R. We say that k is of positive type if for any n ∈ N,
+x1, . . . , xn ∈ X and λ1, . . . , λn ∈ R, we have:
+n
+�
+i,j=1
+λiλjk(xi, xj) ≥ 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 ε > 0, there
+exist S > 0 and a kernel k : X × X → R of positive type satisfying the following:
+(1) for x, y ∈ X, we have k(x, y) = k(y, x) and k(x, x) = 1;
+(2) for x, y ∈ X with d(x, y) ≥ S, we have k(x, y) = 0;
+(3) for x, y ∈ X with d(x, y) ≤ R, we have |1 − k(x, y)| ≤ ε.
+We also recall the notion of coarse embedding:
+Definition 2.4. Let (X, dX) and (Y, dY) be metric spaces and f : X → Y be a map.
+We say that f is a coarse embedding if there exist functions ρ± : [0, ∞) → [0, ∞) with
+limt→+∞ ρ±(t) = +∞ such that for any x, y ∈ X we have
+ρ−(dX(x, y)) ≤ dY(f(x), f(y)) ≤ ρ+(dX(x, y)).
+If additionally there exists C > 0 such that Y = NC(f(X)), then we say that f is a
+coarse equivalence and (X, dX), (Y, dY) are coarsely equivalent.
+2.3. Uniform Roe algebras. Let (X, d) be a discrete metric space. Each operator
+T ∈ B(ℓ2(X)) can be written in the matrix form T = (T(x, y))x,y∈X, where T(x, y) =
+⟨Tδy, δx⟩ ∈ C. We also regard T ∈ B(ℓ2(X)) as a bounded function on X × X, i.e.,
+an element in ℓ∞(X × X). Denote by ∥T∥ the operator norm of T in B(ℓ2(X)), and
+∥T∥∞ the supremum norm when regarding T as a function in ℓ∞(X × X). It is clear
+that ∥T∥∞ ≤ ∥T∥ for any T ∈ B(ℓ2(X)).
+Given an operator T ∈ B(ℓ2(X)), we define the support of T to be
+supp(T) := {(x, y) ∈ X × X : T(x, y) � 0},
+
+8
+QIN WANG AND JIAWEN ZHANG
+and the propagation of T to be
+prop(T) := sup{d(x, y) : (x, y) ∈ supp(T)}.
+Definition 2.5. Let (X, d) be a space.
+(1) The set of all finite propagation operators in B(ℓ2(X)) forms a ∗-algebra,
+called the algebraic uniform Roe algebra of X and denoted by Cu[X]. For each
+R ≥ 0, denote the subset
+CR
+u[X] := {T ∈ B(ℓ2(X)) : prop(T) ≤ R}.
+It is clear that Cu[X] = �
+R≥0 CR
+u[X].
+(2) The uniform Roe algebra of X is defined to be the operator norm closure of
+Cu[X] in B(ℓ2(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 ∈ C∗
+u(X) is called a ghost if T ∈ C0(X × X) when
+regarding T as a function in ℓ∞(X × X). In other words, for any ε there exists a
+finite subset F ⊆ X such that for any (x, y) � F × F, we have |T(x, y)| < ε.
+It is easy to see that all the ghost operators in C∗
+u(X) form an ideal in C∗
+u(X),
+denoted by IG. 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 termi-
+nology on groupoids. Details can be found in [28], or [38] in the ´etale case.
+Recall that a groupoid is a small category, in which every morphism is invertible.
+More precisely, a groupoid consists of a set G, a subset G(0) called the unit space, two
+maps s, r : G → G(0) called the source and range maps respectively, a composition
+law:
+G(2) := {(γ1, γ2) ∈ G × G : s(γ1) = r(γ2)} ∋ (γ1, γ2) �→ γ1γ2 ∈ G,
+and an inverse map γ �→ γ−1. These operationssatisfya couple ofaxioms, including
+associativity law and the fact that elements in G(0) act as units.
+For x ∈ G(0), denote Gx := r−1(x) and Gx := s−1(x). For Y ⊆ G(0), denote GY
+Y :=
+r−1(Y) ∩ s−1(Y). Note that GY
+Y is a subgroupoid of G (in the sense that it is stable
+under multiplication and inverse), called the reduction of G by Y. A subset Y is
+said to be invariant if r−1(Y) = s−1(Y), and we write GY instead of GY
+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 ´etale (also called r-discrete) if the range (hence the source) map is a local
+homeomorphism. Clearly in this case, each fibre Gx (and Gx) is discrete with the
+induced topology, and G(0) is clopen in G. The notion of ´etaleness 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×X as a set, whose unit space
+is {(x, x) ∈ X × X : x ∈ 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)· (y, z) = (x, z) for x, y, z ∈ X.
+When X is a discrete Hausdorff space, then X × X is a locally compact Hausdorff
+´etale groupoid.
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+9
+Now we introduce the algebras associated to groupoids. Here we only focus
+on the case of ´etaleness, and guide readers to [28] for the general case.
+Let G be a locally compact, Hausdorff and ´etale groupoid with unit space G(0).
+Note that the space Cc(G) is a ∗-involutive algebra with respect to the following
+operations: for f, g ∈ Cc(G),
+(f ∗ g)(γ)
+=
+�
+α∈Gs(γ)
+f(γα−1)g(α),
+f ∗(γ)
+=
+f(γ−1).
+Consider the following algebraic norm on Cc(G) defined by:
+∥ f∥I := max
+sup
+x∈G(0)
+�
+γ∈Gx
+| f(γ)|, sup
+x∈G(0)
+�
+γ∈Gx
+| f ∗(γ)|
+ .
+The completion of Cc(G) with respect to the norm ∥ · ∥I is denoted by L1(G).
+The maximal (full) groupoid C∗-algebra C∗
+max(G) is defined to be the completion of
+Cc(G) with respect to the norm:
+∥ f∥max := sup ∥π(f)∥,
+where the supremum is taken over all ∗-representations π of L1(G).
+In order to define the reduced counterpart, we recall that for each x ∈ G(0) the
+regular representation at x, denoted by λx : Cc(G) → B(ℓ2(Gx)), is defined as follows:
+(2.1)
+�
+λx(f)ξ
+�
+(γ) :=
+�
+α∈Gx
+f(γα−1)ξ(α),
+where f ∈ Cc(G) and ξ ∈ ℓ2(Gx).
+It is routine work to check that λx is a well-defined ∗-homomorphism. The reduced
+norm on Cc(G) is
+∥ f∥r := sup
+x∈G(0)
+∥λx(f)∥,
+and the reduced groupoid C∗-algebra C∗
+r(G) is defined to be the completion of the
+∗-algebra Cc(G) with respect to this norm. Clearly, each regular representation λx
+can be extended to a homomorphism λx : C∗
+r(G) → B(ℓ2(Gx)) automatically. It
+is also routine to check that there is a canonical surjective homomorphism from
+C∗
+max(G) to C∗
+r(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) :=
+�
+r>0
+Er
+β(X×X) ⊆ β(X × X).
+Recall from Example 2.7 that X × X is the pair groupoid with source and range
+maps s(x, y) = y and r(x, y) = x. These maps extend to maps G(X) → βX, still
+denoted by r and s.
+Now consider (r, s) : G(X) → βX × βX. It was shown in [39, Lemma 2.7] that the
+map (r, s) is injective, and hence G(X) can be endowed with a groupoid structure
+
+10
+QIN WANG AND JIAWEN ZHANG
+induced by the pair groupoid βX × βX, called the coarse groupoid of X. Therefore,
+G(X) can also be equivalently defined by
+G(X) :=
+�
+r>0
+Er
+βX×βX ⊆ βX × βX,
+with the weak topology. It was also shown in [39, Proposition 3.2] that the coarse
+groupoid G(X) is locally compact, Hausdorff, ´etale and principal. Clearly, the unit
+space of G(X) can be identified with βX.
+Given f ∈ Cc(G(X)), then f is a continuous function supported on Er for some
+r > 0; equivalently, we can interpret f as a bounded function on Er. Hence we
+define an operator θ(f) on ℓ2(X) by setting its matrix coefficients to be θ(f)(x, y) :=
+f(x, y) for x, y ∈ X. We have the following:
+Proposition 2.8 ([34, Proposition 10.29]). The map θ provides a ∗-isomorphism from
+Cc(G(X)) to Cu[X], and extends to a C∗-isomorphism Θ : C∗
+r(G(X)) → C∗
+u(X). Note that
+Θ maps the C∗-subalgebra C∗
+r(X × X) onto the compact operators K(ℓ2(X)).
+Recall from Section 2.3 that an operator T ∈ B(ℓ2(X)) can be regarded as an
+element in ℓ∞(X × X).
+Following the notation from [12], we denote by T the
+continuous extension of T on β(X × X) when regarding T ∈ ℓ∞(X × X). Then
+supp(T) = supp(T).
+Note that G(X) is open in β(X × X), hence C0(G(X)) is a subalgebra in C(β(X × X)).
+Restricting to G(X), we also regard T as a function on G(X) and hence we can talk
+of the value T(α, γ) for (α, γ) ∈ G(X). Moreover, we have the following:
+Lemma 2.9. For T ∈ Cu[X], we have T ∈ Cc(G(X)) and θ(T) = T. For T ∈ C∗
+u(X), we
+have T ∈ C0(G(X)) and Θ(T) = T.
+Proof. The first statement is a direct corollary of Proposition 2.8. Since ∥T∥∞ ≤ ∥T∥
+for any T ∈ B(ℓ2(X)), the second follows from the first.
+□
+2.6. Amenability and a-T-menability for groupoids. Amenable groupoids com-
+prise 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 ´etaleness, in which the notion of amenability behaves quite well.
+A standard reference is [1] and another reference for just ´etale groupoids is [9,
+Chapter 5.6].
+Definition 2.10 ([1]). A locally compact, Hausdorff and ´etale groupoid G is said
+to be (topologically) amenable if for any ε > 0 and compact K ⊆ G, there exists
+f ∈ Cc(G) with range in [0, 1] such that for any γ ∈ K we have
+�
+α∈Gr(γ)
+f(α) = 1
+and
+�
+α∈Gr(γ)
+| f(α) − f(αγ)| < ε.
+Similar to the case of Property A, we have the following by a standard normal-
+isation argument:
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+11
+Lemma 2.11. A locally compact, Hausdorff and ´etale groupoid G is amenable if and
+only if for any ε > 0 and compact K ⊆ G, there exists f ∈ Cc(G) with range in [0, +∞)
+such that for any γ ∈ K we have
+���
+�
+α∈Gr(γ)
+f(α) − 1
+��� < ε
+and
+�
+α∈Gr(γ)
+| f(α) − f(αγ)| < ε.
+Amenability for ´etale groupoids enjoy similar permanence properties as in the
+case of groups. For example, open or closed subgroupoids of amenable ´etale
+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 G be a locally compact, Hausdorff, ´etale
+and amenable groupoid. Then the natural quotient C∗
+max(G) → C∗
+r(G) is an isomorphism.
+Now we recall the notion of a-T-menability for groupoids introduced by Tu
+[40]. Let G be a locally compact, Hausdorff and ´etale groupoid. A continuous
+function f : G → R is said to be of negative type if
+(1) f|G(0) = 0;
+(2) for any γ ∈ G, f(γ) = f(γ−1);
+(3) Given γ1, · · · , γn ∈ G with the same range and λ1, · · · , λn ∈ R with �n
+i=1 λi =
+0, we have �
+i,j λiλj f(γ−1
+i γj) ≤ 0.
+A continuous function f : G → R is called locally proper if for any compact subset
+K ⊆ G(0), the restriction of f on GK
+K is proper.
+Definition 2.13 ([40, Section 3.3]). A locallycompact, Hausdorff and ´etale groupoid
+G is said to be a-T-menable if there exists a continuous locally proper function
+f : G → R of negative type on G.
+Analogous to the case of groups, Tu [40] proved that a locally compact, σ-
+compact, Hausdorff and ´etale groupoid G is a-T-menable if and only if there exists
+a continuous field of Hilbert spaces over G(0) with a proper affine action of G. We
+also need the following significant result by Tu:
+Proposition 2.14 ([40, Th´eor`eme 0.1]). Let G be a locally compact, σ-compact, Haus-
+dorff, ´etale and a-T-menable groupoid. Then G is K-amenable, i.e., the quotient map
+induces an isomorphism K∗(C∗
+max(G)) → K∗(C∗
+r(G)) for ∗ = 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) X has Property A if and only if G(X) is amenable;
+(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 ˇSpakula and Willett in [43], which becomes an important
+tool for later use.
+
+12
+QIN WANG AND JIAWEN ZHANG
+Throughout the section, we always assume that (X, d) is a space (see “Conven-
+tion” 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 → R with D, R ⊆ X is called a
+partial translation if t is a bijection from D to R, and supx∈X d(x, t(x)) is finite. The
+graph of t is {(t(x), x) : x ∈ D}, denoted by 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 ω ∈ βX. A partial
+translation t : D → R on X is compatible with ω if ω(D) = 1. In this case, regarding
+t as a function from D to βX, we define the following thanks to Lemma A.2:
+t(ω) := lim
+ω t ∈ βX.
+In other words, consider the extension t : D → R then t(ω) = t(ω).
+An ultrafilter α ∈ βX is compatible with ω if there exists a partial translation t
+compatible with ω and t(ω) = α. Denote by X(ω) the collection of all ultrafilters on
+X compatible with ω. A compatible family for ω is a collection of partial translations
+{tα}α∈X(ω) such that each tα is compatible with ω and tα(ω) = α.
+Fix an ultrafilter ω on X, and a compatible family {tα}α∈X(ω). Define a function
+dω : X(ω) × X(ω) → [0, ∞) by
+dω(α, β) := lim
+x→ω d(tα(x), tβ(x)).
+It is shown in [43, Proposition 3.7] that dω is a uniformly discrete metric of bounded
+geometry on X(ω) which does not depend on the choice of {tα}.
+This leads to the following:
+Definition 3.2 ([43, Definition 3.8]). For each non-principal ultrafilter ω on X, the
+metric space (X(ω), dω) is called the limit space of X at ω, which is a space in the
+sense of “Convention” in Section 2.2.
+It is shown in [43, Proposition 3.9] that for any α ∈ X(ω), we have X(α) = X(ω)
+as metric spaces. Also note that when ω is principal, i.e., ω ∈ X, then it is clear
+that (X(ω), dω) = (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 ω be a non-principal ultrafilter on X, and
+{tα : Dα → Rα} a compatible family for ω. Then for each finite F ⊆ X(ω), there exists a
+subset Y ⊆ X with ω(Y) = 1 such that for each y ∈ Y, there is a finite subset G(y) ⊆ X
+such that the map
+fy : F → G(y), α �→ tα(y)
+is a surjective isometry. Such a collection {fy}y∈Y is called a local coordinate systerm
+for F, and the maps fy are called local coordinates.
+Furthermore, if F is a metric ball B(ω, r), then there exist Y ⊆ X with ω(Y) = 1 and a
+local coordinate system {fy : F → G(y)}y∈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):
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+13
+Lemma 3.4 ([43, Lemma C.3]). Given a non-principal ultrafilter ω ∈ βX, the map
+F : X(ω) → G(X)ω,
+α �→ (α, ω)
+is a bijection. Hence X(ω) is the smallest invariant subset of βX containing ω. Here we
+consider G(X) as a subset of βX × βX, as explained in Section 2.5.
+Consequently, we obtain the following:
+Corollary 3.5. As a set, we have
+G(X) =
+�
+X × X
+�
+⊔
+�
+ω∈∂βX
+�
+X(ω) × X(ω)
+�
+.
+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 → R be a partial translation on X. Then we have:
+gr(t)
+βX×βX = gr(t) ⊔
+�
+ω∈∂βX
+�
+(α, ω) : ω(D) = 1 and α = t(ω)
+�
+.
+In general, we have the following:
+Lemma 3.7. For any S ≥ 0, we have:
+ES
+βX×βX = ES ⊔
+�
+ω∈∂βX
+�
+(α, γ) ∈ X(ω) × X(ω) : dω(α, γ) ≤ S
+�
+.
+Proof. As explained at the beginning of this subsection, we can decompose
+ES = gr(t1) ⊔ · · · ⊔ gr(tN)
+where each ti : Di → Ri is a partial translation. Hence we have
+ES = gr(t1) ∪ · · · ∪ gr(tN).
+Applying Lemma 3.6, we obtain that ES is contained in the right hand side in the
+statement.
+On the other hand, given ω ∈ ∂βX and α, γ ∈ X(ω) with dω(α, γ) ≤ S, then we
+have (X(ω), dω) = (X(γ), dγ). Take a partial translation t : D → R such that γ(D) = 1
+and α = t(γ). Note that
+dγ(α, γ) = lim
+x→γ d(t(x), x) ≤ S.
+Hence for D′ := {x ∈ D : d(t(x), x) ≤ S}, we have γ(D′) = 1. Consider the restriction
+of t on D′, denoted by t′. Then t′ is also a partial translation and α = t′(γ). By
+Lemma 3.6, we obtain that (α, γ) ∈ gr(t′), which is contained in ES as desired.
+□
+Now we compute concrete examples of limit spaces. First we recall the case
+of groups from [43, Appendix B]. Let Γ be a countable discrete group, equipped
+with a left-invariant bounded geometry and strongly discrete metric d. For each
+g ∈ Γ, denote
+ρg : Γ → Γ, h �→ hg
+the right translation map. Each ρg is a partial translation with full domain, and
+hence is compatible with every ω ∈ βΓ. Moreover, we have the following:
+
+14
+QIN WANG AND JIAWEN ZHANG
+Lemma 3.8 ([43, Lemma B.1]). For each non-principal ultrafilter ω ∈ βΓ, the map
+bω : Γ −→ Γ(ω), g �→ ρg(ω)
+is an isometric bijection.
+Inspired by Lemma 3.8, we provide the following general method:
+Lemma 3.9. Let {tλ : Dλ → Rλ}λ∈Λ be a family of partial translations on X satisfying
+the following: for each S > 0, there exists a finite subset ΛS ⊆ Λ such that gr(tλ) ∩ gr(tµ)
+is finite for λ � µ in ΛS and ES \
+� �
+λ∈ΛS gr(tλ)
+�
+is finite. Then for any non-principal
+ultrafilter ω on X and α ∈ X(ω), there exists λ ∈ Λ such that ω(Dλ) = 1 and α = tλ(ω).
+Proof. By definition, we assume that α = t(ω) for some partial translation t : D → R
+with ω(D) = 1. For λ ∈ Λ, set �Dλ := {x ∈ D ∩ Dλ : t(x) = tλ(x)}. Choose S > 0 such
+that gr(t) ⊆ ES, and hence there exists a finite subset F ⊆ ES such that
+gr(t) ⊆
+� �
+λ∈ΛS
+gr(tλ)
+�
+⊔ F.
+This implies that D ⊆
+� �
+λ∈ΛS �Dλ
+�
+⊔ F′ for some finite F′ ⊆ X. Since ω is non-
+principal, ΛS is finite and �Dλ ∩ �Dµ is finite for any λ, µ ∈ ΛS, there exists a unique
+λ ∈ ΛS such that ω(�Dλ) = 1. This implies that ω(Dλ) = 1 and α = tλ(ω), which
+concludes the proof.
+□
+Back to the case of the group Γ, the set {ρg : Γ → Γ}g∈Γ satisfies the condition in
+Lemma 3.9, and hence the map bω in Lemma 3.8 is surjective. It is straightforward
+to check that bω is isometric, which recovers the proof for Lemma 3.8.
+Example 3.10. Consider X = N with the usual metric. For each k ∈ Z with k ≥ 0,
+define a partial translation
+ρk : N −→ N,
+n �→ n + k.
+For k ∈ Z with k < 0, define a partial translation
+ρk : [−k, ∞) ∩ N −→ N,
+n �→ n + k.
+Then it is clear that the set {ρk}k∈Z satisfies the condition in Lemma 3.9. Now for a
+non-principal ultrafilter ω on N, consider the map
+bω : Z −→ N(ω),
+k �→ ρk(ω).
+Note that for k, l ∈ Z, we have
+dω(ρk(ω), ρl(ω)) = lim
+n→ω |(k + n) − (l + n)| = |k − l|,
+which implies that bω is isometric. Moreover, Lemma 3.9 shows that bω is surjec-
+tive. Therefore, every limit space of N is isometric to 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.
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+15
+3.2. Limit operators. Now we recall the notion of limit operators for metric spaces
+introduced by ˇSpakula and Willett:
+Definition 3.11 ([43, Definition 4.4]). For a non-principal ultrafilter ω on X, fix a
+compatible family {tα}α∈X(ω) for ω and let T ∈ C∗
+u(X). The limit operator of T at ω,
+denoted by Φω(T), is an X(ω)-by-X(ω) indexed matrix defined by
+Φω(T)αγ := lim
+x→ω Ttα(x)tγ(x)
+for
+α, γ ∈ X(ω).
+It was studied in [43, Chapter 4] that the above definition does not depend on
+the choice of the compatible family {tα}α∈X(ω) for ω. Furthermore, the limit operator
+Φω(T) is indeed a bounded operator on ℓ2(X(ω)), and belongs to the uniform Roe
+algebra C∗
+u(X(ω)).
+Recall from Lemma 2.9 that for an operator T ∈ C∗
+u(X), the continuous extension
+T ∈ C0(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 ω on X and T ∈ C∗
+u(X), we have
+Φω(T)αγ = T(α, γ)
+for
+α, γ ∈ X(ω).
+Proof. Choose partial translations tα, tγ compatible with α, γ such that tα(ω) = α,
+tγ(ω) = γ. By definition, we have
+Φω(T)αγ = lim
+x→ω Ttα(x)tγ(x) = lim
+x→ω T(tα(x), tγ(x)) = T(α, γ)
+where the last equality comes from the discussion before Lemma 2.9.
+□
+Since the limit operator Φω(T) contains the information of the asymptotic be-
+haviour of T “in the ω-direction”, we introduce the following:
+Definition 3.13. For an ω ∈ ∂βX, we say that an operator T ∈ C∗
+u(X) is locally
+invisible (or vanishes) in the ω-direction if Φω(T) = 0. For a subset V ⊆ ∂βX, we say
+that T is locally invisible (or vanishes) in the V-direction if Φω(T) = 0 for any ω ∈ V.
+Finally, we recall from Proposition 2.8 that there is a C∗-isomorphism Θ :
+C∗
+r(G(X)) → C∗
+u(X). This allows us to relate limit operators to left regular rep-
+resentations of C∗
+r(G(X)):
+Lemma 3.14 ([43, Lemma C.3]). For a non-principal ultrafilter ω on X, let Wω :
+ℓ2(G(X)ω) → ℓ2(X(ω)) be the unitary representation induced by F in Lemma 3.4. Then
+we have the following commutative diagram:
+C∗
+r(G(X))
+λω
+�
+Θ �
+�
+B(ℓ2(G(X)ω))
+AdWω
+�
+�
+C∗
+u(X)
+Φω
+� B(ℓ2(X(ω))),
+where λω is the left regular representation from (2.1).
+
+16
+QIN WANG AND JIAWEN ZHANG
+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 ∈ C∗
+u(X) and ε > 0, the
+ε-support of T is defined to be
+suppε(T) := {(x, y) ∈ X × X : |T(x, y)| ≥ ε}.
+Also define the ε-truncation of T to be
+Tε(x, y) :=
+� T(x, y),
+if |T(x, y)| ≥ ε;
+0,
+otherwise.
+It is clear that supp(Tε) = suppε(T). We also record the following elementary
+result for later use. The proof is straightforward, hence omitted.
+Lemma 4.2. Given T ∈ C∗
+u(X) and ε > 0, we have
+supp(Tε) ⊆ { ˜ω ∈ β(X × X) : |T( ˜ω)| ≥ ε} ⊆ supp(Tε/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 ∈ I and ε > 0, we have Tε ∈ I ∩ Cu[X]. Moreover, we have
+I ∩ Cu[X] = {Tε : T ∈ I, ε > 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 ∩ Cu[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) = βX. 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 βX, then X ⊆ U.
+Proof. Since X is dense in βX and U is open and non-empty, we obtain that
+U ∩ X � ∅. Take an x ∈ U ∩ X, then the pair (x, y) ∈ G(X) for any y ∈ X. Thanks to
+the invariance of U, we obtain that y ∈ U. This implies that X ⊆ U.
+□
+Given an invariant open subset U ⊆ βX, denote G(X)U := G(X) ∩ s−1(U). Fol-
+lowing [12], we define
+Ic(U) : = {f ∈ Cc(G(X)) : f( ˜ω) = 0 for any ˜ω � G(X)U}
+= {T ∈ Cu[X] : T( ˜ω) = 0 for any ˜ω � G(X)U}.
+Obviously, Ic(U) is a two-sided ideal in Cc(G(X)). Denote its closure in C∗
+r(G(X))
+by I(U), which is a geometric ideal in C∗
+r(G(X)) � C∗
+u(X) from the definition (see
+also [12, Lemma 5.1]). This leads to the following:
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+17
+Definition 4.6. For an invariant open subset U ⊆ βX, 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 βX. 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 Cc(G(X)U) in C∗
+r(G(X)). Note that
+Cc(G(X)U) = {T ∈ Cu[X] : supp(T) ⊆ G(X)U} ⊆ Ic(U).
+On the other hand, for T ∈ Ic(U) we have T = limε→0 Tε since T has finite propaga-
+tion. Note that supp(Tε) ⊆ G(X)U, which implies Tε ∈ Cc(G(X)U). Hence we obtain
+that Cc(G(X)U) and Ic(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 × X.
+Hence combining Proposition 2.8 and Lemma 4.7, we obtain that the geometric
+ideal associated to X is I(X) = K(ℓ2(X)). On the other hand, for U = βX it is clear
+that I(βX) = C∗
+u(X).
+Conversely, following [12, Section 4 and 5] we can associate an invariant open
+subset of βX 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) :=
+�
+T∈I,ε>0
+r(suppε(T)) =
+�
+T∈I∩Cu[X],ε>0
+r(suppε(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 βX. 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 �→ 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
+βX, with the inverse map given by U �→ I(U).
+Proposition 4.9 shows that geometric ideals in C∗
+u(X) can be fully determined
+by invariant open subsets of βX. In contrast, general ideals in C∗
+u(X) cannot be
+characterised merely by the associated subsets of βX. For example, direct calcu-
+lations show that the associated invariant open subsets for the ideal IG defined in
+Section 2.3 and for the ideal of compact operators in B(ℓ2(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)
+IU = {I is an ideal in C∗
+u(X) : U(I) = U}
+in (1.1) for each invariant open subset U ⊆ βX. The following result busts the
+smallest element in IU:
+
+18
+QIN WANG AND JIAWEN ZHANG
+Proposition 4.10. Let U be an invariant open subset of βX. Then the geometric ideal
+I(U) is the smallest element in the lattice IU 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)) = I ∩ Cu[X],
+where the closure is taken in C∗
+u(X). Hence we have I(U(I)) ⊆ I.
+Proof. Denoting ˚I := I ∩ Cu[X], it is clear that ˚I ∩ Cu[X] = I ∩ Cu[X]. This implies
+that ˚I is a geometric ideal in C∗
+u(X), and hence ˚I = I(U(˚I)) by Proposition 4.9. By
+definition, we have
+U(˚I) =
+�
+T∈˚I∩Cu[X],ε>0
+r(suppε(T)) =
+�
+T∈I∩Cu[X],ε>0
+r(suppε(T)) = U(I).
+Therefore, we obtain that I(U(I)) = I(U(˚I)) = ˚I = I ∩ Cu[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- ˇCech
+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 L of
+subsets of X satisfying the following:
+(1) if Y ∈ L and Z ⊆ Y, then Z ∈ L;
+(2) if R ≥ 0 and Y ∈ L, then NR(Y) ∈ L;
+(3) if Y, Z ∈ L, then Y ∪ Z ∈ L.
+For an ideal L in X, we define
+U(L) :=
+�
+Y∈L
+Y
+βX.
+Conversely, given an invariant open subset U of βX, we define
+L(U) := {Y ⊆ X : Y
+βX ⊆ U}.
+As a special case of [12, Theorem 6.3], we have the following:
+Proposition 4.13. For a space (X, d), the map L �→ U(L) provides an isomorphism
+between the lattice of all ideals in X and the lattice of all invariant open subsets of βX,
+with the inverse map given by U �→ 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(L)) = {T ∈ Cu[X] : supp(T) ⊆ Y × Y for some Y ∈ 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 ⊆ X, recall from [23, Section 5] that there is an associated
+ideal IA in C∗
+u(X) whose K-theory is isomorphic to that of the uniform Roe algebra
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+19
+C∗(A). More precisely, recall that an operator T ∈ B(ℓ2(X)) is near A if there exists
+R > 0 such that supp(T) ⊆ NR(A) × NR(A), and the ideal IA is defined to be the
+operator norm closure of all operators in Cu[X] near A.
+The ideal IA 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 LA := {Z ⊆ NR(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)
+UA := U(LA) =
+�
+R>0
+NR(A)
+is an invariant open subset of βX. Moreover, if U is an invariant open subset of βX
+containing A, then UA ⊆ U.
+Consequently, combining with (4.2) we reach the following:
+Corollary 4.15. Let A be a subset of X, then I(UA) = IA. Hence the spatial ideal IA is
+geometric.
+For later use, we record the following result concerning the set UA defined in
+(4.3). Recall from Corollary A.8 that for a subset Z ⊆ X, the closure Z in βX is
+homeomorphic to βZ.
+Lemma 4.16. Let A be a subset of X. Then we have:
+G(X)UA =
+�
+R>0
+G(NR(A)).
+Proof. By definition, we have G(X)UA = �
+R>0 G(X)NR(A). Note that for each R > 0
+and (α, ω) ∈ G(X)NR(A), we have ω ∈ NR(A) and there exists S > 0 such that
+(α, ω) ∈ ES due to Lemma 3.7. Hence the pair (α, ω) in NR+S(A) × NR+S(A) belongs
+to G(NR+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 = ⟨T⟩ be the principal ideal in C∗
+u(X) generated by T ∈ C∗
+u(X). Denote
+U :=
+�
+ε>0,R>0
+NR(r(suppε(T))).
+Then U is an invariant open subset of βX, and we have U(I) = U.
+Proof. By Lemma 4.14, it is clear that U is an invariant open subset of βX. By (4.1),
+U(I) contains r(suppε(T)) for any ε > 0. Since U(I) is invariant, we obtain that U(I)
+contains U again by Lemma 4.14.
+
+20
+QIN WANG AND JIAWEN ZHANG
+On the other hand, we consider S = �n
+i=1 aiTbi where ai, bi are non-zero with
+supports being partial translations contained in ER for some R > 0. Hence for any
+ε > 0, we have
+r
+�
+suppε(S)
+�
+⊆
+n
+�
+i=1
+r
+�
+supp ε
+n(aiTbi)
+�
+⊆
+n
+�
+i=1
+NR
+�
+r
+�
+supp
+ε
+n∥ai∥·∥bi∥(T)
+��
+,
+which implies that r
+�
+suppε(S)
+�
+⊆ U. Note that operators of the form �n
+i=1 aiTbi as
+above are dense in I. Hence for a general element ˜S ∈ I and ε > 0, there exists
+S = �n
+i=1 aiTbi where ai, bi are non-zero with supports being partial translations
+such that ∥ ˜S − S∥ < ε/2.
+Hence suppε( ˜S) ⊆ suppε/2(S), which concludes the
+proof.
+□
+5. Ghostly ideals
+In the previous section, we observe that for an invariant open subset U of βX,
+the associated geometric ideal I(U) is the smallest element in the lattice
+IU = {I is an ideal in C∗
+u(X) : U(I) = U}.
+In this section, we would like to explore the largest element in IU. As revealed in
+Section 1, a natural idea is to include all operators T ∈ 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 βX, denote
+˜I(U) := {T ∈ C∗
+u(X) : r(suppε(T)) ⊆ U for any ε > 0}.
+We call ˜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 ˜I(U) is indeed an ideal in C∗
+u(X).
+Lemma 5.2. For an invariant open subset U of βX, ˜I(U) is an ideal in C∗
+u(X).
+Proof. For T, S ∈ C∗
+u(X) and ε > 0, we have suppε(T + S) ⊆ suppε/2(T) ∪ suppε/2(S).
+It follows that ˜I(U) is a linear space in C∗
+u(X). Given T ∈ ˜I(U) and ε > 0, note that
+r(suppε(T∗)) = s(suppε(T)) and hence T∗ ∈ ˜I(U) since U is invariant. Now given a
+sequence {Tn}n in ˜I(U) converging to T ∈ C∗
+u(X) and an ε > 0, there exists n ∈ N
+such that suppε(T) ⊆ suppε/2(Tn). Hence we obtain that T ∈ ˜I(U).
+Finally, given T ∈ ˜I(U) and S ∈ C∗
+u(X) we need show that TS and ST belong to
+˜I(U). Note that ST = (T∗S∗)∗ and ˜I(U) is closed under taking the ∗-operation, hence
+it suffices to show that TS ∈ ˜I(U). Since ˜I(U) is closed in C∗
+u(X) with respect to
+the operator norm, it suffices to consider the case that S ∈ Cu[X]. We can further
+assume that S is a partial translation since ˜I(U) is closed under taking addition.
+In this case, we have r(suppε(TS)) ⊆ r(suppε/∥S∥(T)) for any ε > 0. Hence we
+conclude the proof.
+□
+Using the language of ideals in X, we record that for an ideal L in X and
+T ∈ C∗
+u(X), then T ∈ ˜I(U(L)) if and only if for any ε > 0 there exist R > 0 and Y ∈ L
+such that for any (x, y) � ER ∩ (Y × Y) then |T(x, y)| < ε.
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+21
+The following result shows that the ghostly ideal ˜I(U) is indeed the largest
+element in the lattice IU:
+Lemma 5.3. Given an invariant open subset U of βX, we have the following:
+(1) for an ideal I in C∗
+u(X) with U(I) = U, then I ⊆ ˜I(U).
+(2) U(˜I(U)) = U.
+Proof. (1). Given T ∈ I, the condition U(I) = U implies that r(suppε(T)) ⊆ U for
+each ε > 0. Hence by definition, we have T ∈ ˜I(U).
+(2). Note that
+U(˜I(U)) =
+�
+T∈˜I(U),ε>0
+r(suppε(T)) ⊆ U.
+On the other hand, Proposition 4.9 shows that U(I(U)) = U, which implies that
+˜I(U) ⊇ I(U) thanks to (1).
+Hence we have U(˜I(U)) ⊇ 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
+βX. Then any ideal I in C∗
+u(X) with U(I) = U sits between I(U) and ˜I(U). More precisely,
+the geometric ideal I(U) is the smallest element while the ghostly ideal ˜I(U) is the largest
+element in the lattice IU in (1.1).
+As mentioned in Section 1, Theorem 5.4 draws the border of the lattice IU.
+Therefore, once we can bust every ideal between I(U) and ˜I(U) for each invariant
+open subset U ⊆ βX, 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 ˜I(X) is the ideal IG defined in Section 2.3. Indeed,
+T ∈ ˜I(X) if and only if for any ε > 0, r(suppε(T)) is finite. This is equivalent to that
+T ∈ C0(X × X) since T ∈ C∗
+u(X). On the other hand, it is clear that ˜I(βX) = C∗
+u(X).
+More generally, we have the following result. Note that for any invariant open
+subset U of βX, G(X)U is an open subset of β(X × X). Hence both C0(G(X)U) and
+Ic(U) can be regarded as subalgebras in C(β(X × X)) � ℓ∞(X × X).
+Proposition 5.6. For an invariant open subset U ⊆ βX and T ∈ C∗
+u(X), the following
+are equivalent:
+(1) T ∈ ˜I(U);
+(2) T ∈ C0(G(X)U);
+(3) T ∈ Ic(U)
+∥·∥∞;
+(4) T vanishes in the (βX \ U)-direction, i.e., Φω(T) = 0 for any ω ∈ βX \ U.
+Proof. “(1) ⇒ (2)”: By definition, for any ε > 0 we have
+r(supp(Tε)) = r(supp(Tε)) = r(suppε(T)) ⊆ U.
+
+22
+QIN WANG AND JIAWEN ZHANG
+Consider the compact set K := { ˜ω ∈ β(X × X) : |T( ˜ω)| ≥ 2ε}. By Lemma 4.2, we
+have r(K) ⊆ r(supp(Tε)) ⊆ U, which implies that K ⊆ G(X)U. Moreover, we have
+|T( ˜ω)| < 2ε for any ˜ω ∈ β(X × X) \ K, which concludes that T ∈ C0(G(X)U).
+“(2) ⇒ (1)”: Assume that T ∈ C0(G(X)U) ⊆ C(β(X × X)). Then for any ε > 0 there
+exists a compact subset K ⊆ G(X)U such that for any ˜ω ∈ β(X × X) \ K, we have
+|T( ˜ω)| < ε. This implies that { ˜ω ∈ β(X × X) : |T( ˜ω)| ≥ ε} ⊆ K. Using Lemma 4.2, we
+obtain that supp(Tε) ⊆ K, which implies that r(suppε(T)) ⊆ U. Hence T ∈ ˜I(U).
+“(2) ⇔ (3)”: This is due to the fact that
+Ic(U)
+∥·∥∞ = {f ∈ C0(G(X)) : f( ˜ω) = 0 for ˜ω � G(X)U}.
+“(3) ⇔ (4)”: By Lemma 3.12, (4) holds if and only if T(α, γ) = 0 for any α, γ ∈ X(ω)
+and ω ∈ βX \ U. Applying Lemma 2.9 and Lemma 3.4, this holds if and only if
+T( ˜ω) = 0 whenever ˜ω � G(X)U, which describes elements in Ic(U)
+∥·∥∞. Hence we
+conclude the proof.
+□
+Note that G(X)X = X × X. Hence as a direct corollary, we recover the following
+characterisation for ghost operators:
+Corollary 5.7 ([43, Proposition 8.2]). An operator T ∈ C∗
+u(X) is a ghost if and only if
+Φω(T) = 0 for any non-principal ultrafilter ω 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. Thissuggestsustoconsideroperatorsin ˜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 ⊆ βX, notice that Uc = βX\U is also invariant.
+Denote by G(X)Uc := G(X) ∩ s−1(Uc) and clearly, we have a decomposition:
+G(X) = G(X)U ⊔ G(X)Uc.
+Note that G(X)U is open in G(X), hence the above induces the following short
+exact sequence of ∗-algebras:
+(5.1)
+0 −→ Cc(G(X)U) −→ Cc(G(X)) −→ Cc(G(X)Uc) −→ 0
+where the map Cc(G(X)U) −→ Cc(G(X)) is the inclusion and the map Cc(G(X)) −→
+Cc(G(X)Uc) 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 −→ C∗
+max(G(X)U) −→ C∗
+max(G(X)) −→ C∗
+max(G(X)Uc) −→ 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 −→ C∗
+r(G(X)U)
+iU
+−→ C∗
+r(G(X))
+qU
+−→ C∗
+r(G(X)Uc) −→ 0.
+By construction, iU is injective, qU is surjective and qU ◦ iU = 0.
+Also recall
+from Lemma 4.7 that iU(C∗
+r(G(X)U)) = I(U), the geometric ideal associated to U.
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+23
+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 Ker(qX) = IG, 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) = ˜I(X). More generally, we have the following:
+Proposition 5.9. Given an invariant open subset U ⊆ βX, the kernel of qU : C∗
+r(G(X)) →
+C∗
+r(G(X)Uc) coincides with the ghostly ideal ˜I(U). Hence the sequence (5.3) is exact if and
+only if I(U) = ˜I(U).
+Proof. It is easy to see that an operator T ∈ C∗
+u(X) � C∗
+r(G(X)) belongs to the
+kernel of qU if and only if λω(T) = 0 for any ω ∈ βX \ U, where λω is the left
+regular representation from (2.1). Now Lemma 3.14 implies that Φω(T) = 0 for
+any ω ∈ βX \ 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 {Xi}i∈N be a sequence of expander graphs or pertubed expander
+graphs (see [44] for the precise definition). Let Yi,j = Xi for all j ∈ N and set
+Y := �
+i,j Yi,j. We endow Y with a metric d such that it is the graph metric on each
+Yi,j and satisfies d(Yi,j, Yk,l) → ∞ as i + j + k + l → ∞.
+Let Pi,j ∈ B(ℓ2(Yi,j)) be the orthogonal projection onto constant functions on Yi,j,
+and we set P to be the direct sum of Pi,j in the strong operator topology. By the
+assumption on the expansion of {Xi}i∈N, it is clear that P ∈ 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 � ˜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 βY → βX × βN
+induced by the bijection of Y with X×N and the universal property of βY. Define:
+f : βY −→ βX × βN −→ βX
+where the second map is just the projection onto the first coordinate. Denote
+U = f −1(X), which is open in βY. Note that U = �
+i f −1(Xi), where each f −1(Xi) is
+homeomorphic to Xi × βN. On the other hand, note that
+U =
+�
+i
+f −1(Xi) =
+�
+i
+�
+j
+Yi,j =
+�
+ε>0
+r(suppε(P)) =
+�
+ε>0,R>0
+NR(r(suppε(P))).
+Hence it follows from Lemma 4.17 that U is invariant (comparing with [20, Lemma
+5.2]), and U(⟨P⟩) = U. (Note that P ∈ ˜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 (∂βY \ U)-
+direction. In particular, fixing an index j0 ∈ N and taking a sequence {xi ∈ Yi,j0}i∈N,
+we choose a cluster point ω ∈ {xi}i. It is clear that ω � U. Intuitively speaking, this
+means that P vanishes “in the i-direction”.
+
+24
+QIN WANG AND JIAWEN ZHANG
+We remark that the first-named author proved in [44, Section 3] that the prin-
+cipal ideal ⟨P⟩ cannot be decomposed into I(U) + (IG ∩ ⟨P⟩), 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 ⟨P⟩ 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 βX.
+Hence we
+introduce the following:
+Definition 6.1. An invariant open subset U ⊆ βX is called maximal if U � βX and U
+is not properly contained in any proper invariant open subset of βX. Similarly, an
+invariant closed subset K ⊆ βX is called minimal if K � ∅ and K does not properly
+contain any non-empty invariant closed subset of βX.
+Proposition 6.2. For any maximal invariant open subset U ⊂ βX, the ghostly ideal ˜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 = ˜I(U(I)).
+Proof. For any ideal J in C∗
+u(X) containing ˜I(U), we have U(J) ⊇ U(˜I(U)) = U by
+Lemma 5.3(2). Since U is maximal, then either U(J) = βX or U(J) = U. If U(J) = βX,
+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 ⊆ ˜I(U(J)) = ˜I(U),
+which implies that J = ˜I(U). This concludes that ˜I(U) is maximal.
+Conversely for any maximal ideal I in C∗
+u(X), we have U(I) � βX. For any open
+invariant subset V � βX containing U, we have I ⊆ ˜I(U) ⊆ ˜I(V) by Theorem 5.4,
+and ˜I(V) � C∗
+u(X). Hence due to the maximality of I, we obtain that I = ˜I(U) = ˜I(V).
+This implies that U = V and also I = ˜I(U) as required.
+□
+Taking complements, we obtain the following:
+Corollary 6.3. For any minimal invariant closed subset K ⊆ βX, the ghostly ideal
+˜I(βX \ 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 βX. Recall from
+Lemma 3.4 that for each ω ∈ ∂βX, the limit space X(ω) is the smallest invariant
+subset of βX containing ω. However, note that X(ω) might not be closed in general.
+Definition 6.4. A point ω ∈ ∂βX is called minimal if the closure of the limit space
+X(ω) in βX is minimal in the sense of Definition 6.1.
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+25
+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 ⊆ βX, there exists a minimal point
+ω ∈ ∂βX such that K = X(ω). Conversely for any minimal point ω ∈ ∂βX, the set X(ω)
+is minimal.
+One might wonder whether every ω ∈ ∂βX 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 = Z. Firstly, we notice the following:
+Lemma 6.6. Let K be an invariant closed subset of βX. Then K contains a minimal point.
+In particular, for any ω ∈ ∂βX there exists a minimal point ω′ such that ω′ ∈ X(ω).
+Proof. This follows directly from the Zorn’s lemma together with the fact that βX
+is compact. Details are left to readers.
+□
+Consequently, we obtain:
+Corollary 6.7. For ω ∈ ∂βX, ω is minimal if and only if for any α ∈ X(ω), we have
+X(α) = X(ω). Writing X(ω) = �
+λ∈Λ X(ωλ) for certain ωλ ∈ ∂βX, then ω is minimal if
+and only if X(ωλ) = X(ω) for any λ ∈ Λ.
+As promised, now we study the case of X = 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 Z with the usual metric, there exist non-minimal
+points in the boundary ∂βZ. More precisely, for any sequence {hn}n∈N in Z tending to
+infinity such that |hn − hm| → +∞ when n + m → ∞ and n � m, and any ω ∈ ∂βZ with
+ω({hn}n∈N) = 1, then ω 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 Γ equipped with
+a left-invariant word length metric.
+Lemma 6.9. For ω, α ∈ ∂βΓ, then α ∈ Γ(ω) if and only if for any S ⊆ Γ with α(S) = 1,
+there exists gS ∈ Γ such that ω(S · g−1
+S ) = 1.
+Proof. Recall from Lemma 3.8 that Γ(ω) = {ρg(ω) : g ∈ Γ}, and from Appendix A
+that {S : S ⊆ X} forms a basis for βX. Hence by definition, we obtain that α ∈ X(ω)
+if and only if for any S ⊆ X with α ∈ S, there exists gS ∈ Γ such that ρgS(ω) ∈ S.
+Equivalently, this means that for any S ⊆ X with α(S) = 1, there exists gS ∈ Γ such
+that ρgS(ω)(S) = ω(S · g−1
+S ) = 1, which concludes the proof.
+□
+Now we return to the case of Γ = Z and prove Theorem 6.8:
+Proof of Theorem 6.8. Fix a subset H = {hn}n∈N ⊆ Z tending to infinity such that
+|hn − hm| → +∞ when n + m → ∞ and n � m. For any non-zero g ∈ Z, note that
+h ∈ (g + H) ∩ H if and only if there exists h′ ∈ H such that h − h′ = g. Since g is
+fixed and distances between different points in H tend to infinity, we obtain that
+(g + H) ∩ H is finite. Hence for any g1 � g2 in Z, (g1 + H) ∩ (g2 + H) is finite.
+
+26
+QIN WANG AND JIAWEN ZHANG
+Fixing a non-principal ultrafilter ω ∈ ∂βZ with ω(H) = 1, we denote
+U := {B ⊆ H : ω(B) = 1}.
+We claim that for each n ∈ N, there exists gn ∈ Z and Bn ∈ U such that {Bn + gn}n∈N
+are mutually disjoint.
+Indeed, we take g0 = 0 and B0 = H.
+Set g1 = 1 and
+B1 := H \ (H − g1). Since H ∩ (H − g1) is finite by the previous paragraph, then
+ω(B1) = 1, i.e., B1 ∈ U. Similarly for each n ∈ N, we take gn = n and Bn :=
+H \
+�
+(H − g1) ∪ (H − g2) ∪ · · · ∪ (H − gn)
+�
+, which concludes the claim.
+Consider �
+H := �
+n∈N(Bn + gn), and denote Un := {B ⊆ Bn : ω(B) = 1} for each
+n ∈ N. By Lemma A.4, Un is an ultrafilter on Bn. Choose a non-principal ultrafilter
+ω0 on N, and we consider:
+�
+U :=
+� �
+n∈N
+(An+gn) ⊆
+�
+n∈N
+(Bn+gn) = �
+H : ∃ J ⊆ N with ω0(J) = 1 s.t. ∀n ∈ J, An ∈ Un
+�
+.
+Lemma A.6 implies that �
+U is an ultrafilter on �
+H. We define a function α : P(Z) →
+{0, 1} by setting α(S) = 1 if and only if S ∩ �
+H ∈ �
+U, which is indeed an ultrafilter on
+Z thanks to Lemma A.5. Also note that α is non-principal and α(�
+H) = 1.
+Now we show that α ∈ Z(ω) while ω � Z(α), and hence conclude the proof. To
+see that α ∈ Z(ω), we will consult Lemma 6.9. For any S ⊆ Z with α(S) = 1, by
+definition we have S ∩ �
+H ∈ �
+U. Writing S ∩ �
+H = �
+n∈N(An + gn) with An ⊆ Bn, then
+there exists J ∈ ω0 such that for any n ∈ J we have An ∈ Un. Hence for any n ∈ J,
+we have S ⊇ S ∩ �
+H ⊇ An + gn and ω(An) = 1, which implies that ω(S − gn) = 1.
+Applying Lemma 6.9, we conclude that α ∈ Z(ω).
+Finally, it remains to check that ω � Z(α). Assume the contrary, then Lemma
+6.9 implies that there exists g ∈ Z and �B ⊆ �
+H with α(�B) = 1 such that H ⊇ �B − g.
+Writing �B = �
+n∈N(An + gn) with An ⊆ Bn, then there exists J ∈ ω0 such that for any
+n ∈ J, ω(An) = 1. This implies that
+H + g ⊇ �B ⊇
+�
+n∈J
+(An + gn) ⊇ An0 + gn0
+for some n0 ∈ J with gn0 � g (this can be achieved since J is infinite). However,
+it follows from the first paragraph that (H + g) ∩ (H + gn0) is finite. While this
+intersection contains An0 + gn0, which is infinite since ω(An0) = 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- ˇCech bound-
+ary 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 view-
+point, 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 ˜I(βX \ X(ω)) for some boundary point ω ∈ ∂βX. Moreover
+according to Proposition 5.9, ˜I(βX \ X(ω)) is the kernel of the following surjective
+homomorphism:
+qβX\X(ω) : C∗
+r(G(X)) −→ C∗
+r(G(X)X(ω)).
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+27
+Hence we obtain the following:
+Corollary 6.10. A point ω ∈ ∂βX is minimal if and only if C∗
+r(G(X)X(ω)) is simple.
+Example 6.11. Consider the case of a countable discrete group Γ.
+For a point
+ω ∈ ∂βΓ, it follows from Lemma 3.8 that the limit space Γ(ω) is identical to Γω,
+and hence C∗
+r(G(Γ)Γ(ω)) is C∗-isomorphic to the reduced crossed product C(Γω) ⋊ Γ.
+Thanks to Corollary 6.10, we obtain that ω is minimal if and only if C(Γω) ⋊ Γ
+is simple.
+Moreover, note that the action of Γ on βΓ is free (which is also a
+consequence of Lemma 3.8). Hence it follows from [29, Corollary 4.6] that C(Γω)⋊Γ
+is simple if and only if the action of Γ on Γω is minimal. In conclusion, we reach
+the following:
+Corollary 6.12. In the case of a countable discrete group Γ, a point ω ∈ ∂βΓ is minimal if
+and only if the action of Γ on Γω 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 X(ω)-direction (see
+Proposition 5.6). Now we show that this can be simplified by merely considering
+the ω-direction:
+Lemma 6.13. For ω ∈ ∂βX, an operator T ∈ C∗
+u(X) belongs to the ideal ˜I(βX \ X(ω)) if
+and only if T vanishes in the ω-direction, i.e., Φω(T) = 0.
+Proof. We assume that Φω(T) = 0, and it suffices to show that Φα(T) = 0 for any
+α ∈ X(ω). Fixing such an α, we take a net {ωλ}λ∈Λ in X(ω) such that ωλ → α and
+it follows that Φωλ(T) = 0. For any γ1, γ2 ∈ X(α), Lemma 3.4 and the fact that the
+coarse groupoid G(X) is ´etale imply that there exist γ1,λ and γ2,λ in X(ωλ) for each
+λ ∈ Λ such that γ1,λ → γ1 and γ2,λ → γ2. Now Lemma 3.12 implies that
+Φα(T)γ1γ2 = T((γ1, γ2)) = lim
+λ∈Λ T((γ1,λ, γ2,λ)) = lim
+λ∈Λ Φωλ(T)γ1,λγ2,λ = 0,
+which concludes the proof.
+□
+Hence for ω ∈ ∂βX, Lemma 6.13 implies that the associated ideal ˜I(β \ X(ω))
+coincides with the kernel of the following limit operator homomorphism (see also
+[43, Theorem 4.10]):
+(6.1)
+Φω : C∗
+u(X) −→ C∗
+u(X(ω)),
+T �→ Φω(T).
+Consequently, we reach the following:
+Corollary 6.14. A point ω ∈ ∂βX is minimal if and only if the image Im(Φω) is simple.
+Hence when X(ω) is infinite and Φω is surjective, the point ω 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(ω)), 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:
+
+28
+QIN WANG AND JIAWEN ZHANG
+Proposition 6.15. Let {hn}n∈N be a sequence in ZN tending to infinity such that
+∥hn − hk∥∞ ≥ 3k
+for any
+k > n.
+Then for any ω ∈ ∂βZN with ω({hn}n∈N) = 1, the map Φω : C∗
+u(ZN) −→ C∗
+u(ZN(ω)) is
+surjective.
+Note that the limit space ZN(ω) is bijective to ZN 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 {hn}n∈N in ZN tending to infinity such that ∥hn−hk∥∞ ≥
+3k for any k > n, and any ω ∈ ∂βZN with ω({hn}n∈N) = 1, then ω is not a minimal point.
+Remark 6.17. We notice from the discussion above that for ω ∈ ∂βX, there is a
+C∗-monomorphism:
+(6.2)
+C∗
+r(G(X)X(ω)) � C∗
+u(X)/˜I(βX \ X(ω)) −→ C∗
+u(X(ω))
+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(ω) = X(ω) × X(ω) and hence G(X)X(ω) =
+X(ω) × X(ω)
+G(X). Now we have
+G(X)X(ω) =
+�
+S>0
+�
+ES
+G(X) ∩ X(ω) × X(ω)
+G(X)�
+=
+�
+S>0
+ES
+G(X) ∩ (X(ω) × X(ω))
+G(X)
+,
+where the last inequality is due to the fact that ES
+G(X) is clopen in G(X). By Lemma
+3.7, we have
+ES
+G(X) ∩ (X(ω) × X(ω)) = {(α, γ) ∈ X(ω) × X(ω) : dω(α, γ) ≤ S} =: ES(X(ω), dω)
+i.e., the S-entourage in the limit space (X(ω), dω). Therefore, we conclude that
+G(X)X(ω) =
+�
+S>0
+ES(X(ω), dω)
+G(X).
+On the other hand, we have (by definition) that
+G(X(ω)) =
+�
+S>0
+ES(X(ω), dω)
+β(X(ω)×X(ω)).
+By the universal property of the Stone- ˇCech compactification, there is a surjective
+continuous map ES(X(ω), dω)
+β(X(ω)×X(ω)) −→ ES(X(ω), dω)
+G(X), which induces an in-
+jective map Cc(G(X)X(ω)) −→ Cc(G(X(ω))). Moreover, it is routine to check that it
+induces a C∗-monomorphism
+C∗
+r(G(X)X(ω)) −→ C∗
+r(G(X(ω))) � C∗
+u(X(ω)),
+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 = Γ. Fixing a point
+ω ∈ ∂βΓ, we mentioned in Example 6.11 that C∗
+r(G(Γ)Γ(ω)) � C(Γω)⋊Γ. On the other
+hand, we know from Lemma 3.8 that C∗
+u(Γ(ω)) � ℓ∞(Γω) ⋊ Γ. In this case, one can
+check that the map (6.2) is induced by the natural embedding C(Γω) ֒→ ℓ∞(Γω).
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+29
+7. Geometric ideals vs ghostly ideals
+In this section, we return to the lattice IU in (1.1). Recall from Theorem 5.4 that
+for an invariant open subset U ⊆ βX, the geometric ideal I(U) and the ghostly
+ideal ˜I(U) are the smallest and the largest elements in IU. Also as noticed before,
+generally IU consists of more than one element.
+Now we would like to study when I(U) = ˜I(U) or equivalently, when IU consists
+of a single element. Moreover, we will discuss their K-theories and provide a
+sufficient condition to ensure K∗(I(U)) = K∗(˜I(U)) for ∗ = 0, 1.
+First recall from Proposition 5.9 that we already have a characterisation for
+I(U) = ˜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) = ˜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) X has Property A;
+(2) G(X) is amenable;
+(3) G(X)∂βX is amenable;
+(4) I(U) = ˜I(U) for any invariant open subset U ⊆ βX;
+(5) I(X) = ˜I(X).
+We remark that “(1) ⇒ (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) ⇔ (2)” was proved in [39, Theorem 5.3]. “(2) ⇔ (3)” is
+due to the permanence properties of amenability (see Section 2.6) together with
+the fact that G(X)X � X × X is always amenable.
+“(2) ⇒ (4)”: Let U ⊆ βX be an invariant open subset. As open/closed sub-
+groupoids, both G(X)U and G(X)Uc are amenable as well. Consider the following
+commutative diagram coming from (5.2) and (5.3):
+0
+� C∗
+max(G(X)U)
+�
+�
+C∗
+max(G(X))
+�
+�
+C∗
+max(G(X)Uc)
+�
+�
+0
+0
+� C∗
+r(G(X)U)
+� C∗
+r(G(X))
+� C∗
+r(G(X)Uc)
+� 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) ⇒ (5)” holds trivially, and “(5) ⇒ (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) = ˜I(X)
+using Property A. However, we notice that assuming Property A is often too
+strong to ensure that I(U) = ˜I(U) for merely a specific invariant open subset U ⊆ βX.
+A trivial example is that I(βX) = ˜I(βX) holds for any space X. This suggests us to
+explore a weaker criterion for I(U) = ˜I(U), and we reach the following:
+
+30
+QIN WANG AND JIAWEN ZHANG
+Proposition 7.2. Let (X, d) be a space and U be an invariant open subset of βX. If the
+canonical quotient map C∗
+max(G(X)Uc) → C∗
+r(G(X)Uc) is an isomorphism, then I(U) =
+˜I(U). In particular, if the groupoid G(X)Uc is amenable then I(U) = ˜I(U).
+Proof. We consider the following commutative diagram:
+(7.1)
+0
+� C∗
+max(G(X)U)
+�
+πU
+�
+C∗
+max(G(X))
+�
+�
+C∗
+max(G(X)Uc)
+�
+�
+0
+0
+� ˜I(U)
+� C∗
+r(G(X))
+� C∗
+r(G(X)Uc)
+� 0.
+Here the map πU is the composition:
+(7.2)
+C∗
+max(G(X)U) → C∗
+r(G(X)U) � I(U) ֒→ ˜I(U),
+where the middle isomorphism comes from Lemma 4.7. Note that the top hor-
+izontal 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 chas-
+ing argument, we obtain that the left vertical map is surjective. This concludes
+that I(U) = ˜I(U) thanks to (7.2).
+□
+Remark 7.3. When U = X, Proposition 7.2 recovers “(3) ⇒ (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 S of subsets of X, denote L(S) the smallest ideal in X
+containing S, and we say that L(S) is generated by S. An ideal L in X is called
+countably generated if there exists a countable set S such that L = L(S).
+An invariant open subset U ⊆ βX is called countably generated if the associated
+ideal L(U) is countably generated.
+Example 7.5. For a space X and a subspace A ⊆ X, it follows from Lemma 4.14
+that the spatial ideal IA is countably generated. On the other hand, it follows from
+Lemma 4.17 that principal ideals are always countably generated. In particular,
+the ideal ⟨P⟩ 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 L be a countably generated ideal in X. Then there exists a countable
+subset {Y1, Y2, · · · , Yn, · · · } in L such that
+L = {Z ⊆ X : ∃ n ∈ N such that Z ⊆ Yn}.
+Consequently for any countably generated open invariant subset U of βX, the subgroupoid
+G(X)U is σ-compact.
+To prove Lemma 7.6, we need an auxiliary result on the structure of L(S):
+Lemma 7.7. For a set S of subsets of X, denote S(1) := {A1 ∪ · · · ∪ An : Ai ∈ S, n ∈ N}
+and S(2) := {Nk( ˜A) : ˜A ∈ S(1), k ∈ N}. Then we have:
+L(S) = {Z : ∃ Y ∈ S(2) such that Z ⊆ Y}.
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+31
+Proof. Denote S(3) := {Z : ∃ Y ∈ S(2) such that Z ⊆ Y}. It is clear that S(3) ⊆ L(S),
+and any ideal containing S must contain S(3). Hence it remains to check that S(3)
+is an ideal.
+For Z ∈ S(3) and W ⊆ Z, there exists Y ∈ S(2) such that Z ⊆ Y. Hence W ⊆ Y,
+which implies that W ∈ S(3). Also note that Y ∈ S(2) implies that there exists
+˜A ∈ S(1) and k′ ∈ N such that Y = Nk′( ˜A).
+Hence for any k ∈ N, we have
+Nk(Z) ⊆ Nk(Y) ⊆ Nk+k′( ˜A), which implies that Nk(Z) ∈ S(3).
+Finally for Z1, Z2 ∈ S(3), there exist ˜A1, ˜A2 ∈ S(1) and k1, k2 ∈ N such that
+Zi ⊆ Nki( ˜Ai) for i = 1, 2. Hence
+Z1 ∪ Z2 ⊆ Nk1( ˜A1) ∪ Nk2( ˜A2) ⊆ Nk1+k2( ˜A1 ∪ ˜A2) ∈ S(2),
+which implies that Z1 ∪ Z2 ∈ S(3). Therefore, we conclude the proof.
+□
+Proof of Lemma 7.6. By assumption, there exists a countable S such that L = L(S).
+Using the notation of Lemma 7.7, the set S(2) is countable as well. Hence the first
+statement follows directly from Lemma 7.7.
+For the second, note that U = �
+n∈N Yn and hence G(X)U = �
+n∈N G(X)Yn. Since
+each Yn is closed, we obtain that G(X)U is σ-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 ⊆ βX, we have an
+isomorphism
+(ιU)∗ : K∗(I(U)) −→ K∗(˜I(U))
+for ∗ = 0, 1, where ιU is the inclusion map. Therefore for any ideal I in C∗
+u(X) with U(I)
+countably generated, we have an injective homomorphism
+(ιI)∗ : K∗(I ∩ Cu[X]) −→ K∗(I)
+for ∗ = 0, 1, where ιI is the inclusion map.
+Proof. Fixing such a U ⊆ βX, 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:
+· · ·
+� K∗(C∗
+max(G(X)U))
+�
+(πU)∗
+�
+K∗(C∗
+max(G(X)))
+�
+�
+K∗(C∗
+max(G(X)Uc))
+�
+�
+K∗+1(C∗
+max(G(X)U))
+�
+(πU)∗+1
+�
+· · ·
+· · ·
+� K∗(˜I(U))
+� K∗(C∗
+r(G(X)))
+� K∗(C∗
+r(G(X)Uc))
+� K∗+1(˜I(U))
+� · · ·
+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)Uc are a-T-menable.
+Moreover, it is clear that G(X) and G(X)Uc are σ-compact by definition. Hence
+Proposition 2.14 implies that G(X) and G(X)Uc are K-amenable. This shows that
+the middle two vertical maps in the above diagram are isomorphisms, which
+implies that (πU)∗ is an isomorphism by the Five Lemma. Therefore from (7.2),
+we obtain that the composition
+K∗(C∗
+max(G(X)U)) −→ K∗(C∗
+r(G(X)U)) � K∗(I(U))
+(ιU)∗
+−→ K∗(˜I(U))
+
+32
+QIN WANG AND JIAWEN ZHANG
+is an isomorphism for ∗ = 0, 1.
+On the other hand, it follows from Lemma 7.6 that G(X)U is σ-compact, which
+implies that G(X)U is K-amenable again by Proposition 2.14. Hence we conclude
+that the map (ιU)∗ is an isomorphism for ∗ = 0, 1.
+For the last statement, we assume that U = U(I). By Lemma 4.11, we have that
+I ∩ Cu[X] = I(U). Also Lemma 5.3(1) shows that I ⊆ ˜I(U). Hence the inclusion
+map ιU can be decomposed as follows:
+I(U) = I ∩ Cu[X]
+ιI֒→ I ֒→ ˜I(U).
+Therefore, (ιU)∗ being an isomorphism implies that (ιI)∗ is injective for ∗ = 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 counterex-
+amples 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 K(ℓ2(X)) into IG 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 (ιI)∗. 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 = N × N, equipped with the metric induced from the Eu-
+clidean metric dE on R2. For each θ ∈ [0, π
+2] and k ∈ N, we define
+ℓθ := {(x, y) ∈ R × R : y = tan(θ)x}
+and
+Sθ,k := {(x, y) ∈ X : dE((x, y), ℓθ) ≤ k} = Nk(ℓθ) ∩ X.
+Consider S := {Sθ,k : θ ∈ [0, π
+2], k ∈ N} and set S(1) := {A1 ∪ · · · ∪ An : Ai ∈ S, n ∈ N}
+as in Lemma 7.7. For any R > 0 and A = A1 ∪ · · · ∪ An ∈ S(1) where Ai ∈ S, we
+have NR(A) = NR(A1) ∪ · · · ∪ NR(An), which is contained in some element in S.
+Hence applying Lemma 7.7, the ideal L(S) generated by S is:
+(7.3)
+L(S) = {Z : ∃ Y ∈ S(1) such that Z ⊆ Y}.
+We claim that L(S) is not countably generated. Otherwise, there exists a count-
+able subset S′ generating L(S). Moreover, according to (7.3) we can assume that
+S′ = {Yn : n ∈ N}
+where
+Yn = Sθn,1,kn ∪ · · · ∪ Sθn,pn,kn ∈ S(1).
+Choose θ ∈ [0, π
+2] \ {θn,i : i = 1, 2, · · · , pn; n ∈ N}, and consider Y = Sθ,1. Since S′
+generates L(S), it follows from Lemma 7.7 that there exist R > 0 and Ym1, · · · , Yml ∈
+S′ such that
+Sθ,1 = Y ⊆ NR(Ym1 ∪ · · · ∪ Yml).
+Note thatthe righthand side iscontained in a finite union ofsome R′-neighbourhood
+of lines (in R2 crossing the origin) with slopes in the set
+�
+tan(θn,i) : i = 1, 2, · · · , pn; n ∈ N
+�
+.
+This leads to a contradiction due to the choice of θ, which concludes that L(S)
+cannot be countably generated.
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+33
+8. Partial Property A and partial operator norm localisation property
+In the previous section, we find a sufficient condition (Proposition 7.2) to ensure
+˜I(U) = I(U) for a given invariant open subset U ⊆ βX. 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
+∂βX, 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)∂βX is amenable, which characterises I(X) = ˜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 ⊆ βX be an invariant open subset. We
+say that X has partial Property A towards ∂βX \ U if G(X)∂βX\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 ∂βX. On the other hand, it follows from
+Proposition 7.2 that if X has partial Property A towards ∂βX \ U, then we have
+I(U) = ˜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 geomet-
+ric 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 ⊆ βX an invariant open subset. Then X
+has partial Property A towards Uc = ∂βX \ U if and only if for any ε, R > 0, there exist
+S > 0, a subset D ⊆ X with D ⊇ Uc and a function f : X × X → [0, 1] satisfying:
+(1) supp(f) ⊆ ES;
+(2) for any x ∈ D, we have �
+z∈X f(z, x) = 1;
+(3) for any x, y ∈ D with d(x, y) ≤ R, we have �
+z∈X | f(z, x) − f(z, y)| ≤ ε.
+Comparing Proposition 8.2 with Definition 2.1, it is clear that Property A implies
+partial Property A towards any invariant closed subset of ∂βX.
+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 Uc if
+and only if for any ε, R > 0, there exist S > 0 and a function f : X × X → [0, 1]
+satisfying:
+(1) supp(f) ⊆ ES;
+(2) for any ω ∈ Uc, we have �
+α∈X(ω) f(α, ω) = 1;
+(3) for ω ∈ Uc and α ∈ X(ω) with dω(α, ω) ≤ R, then �
+γ∈X(ω) | f(γ, α) − f(γ, ω)| ≤ ε,
+where f ∈ C0(G(X)) is the continuous extension from Lemma 2.9.
+Proof. By definition, X has partial Property A towards Uc if and only if for any
+ε > 0 and compact K ⊆ G(X)Uc, there exists g ∈ Cc(G(X)Uc) with range in [0, 1] such
+
+34
+QIN WANG AND JIAWEN ZHANG
+that for any γ ∈ K we have
+�
+α∈Gr(γ)
+g(α) = 1
+and
+�
+α∈Gr(γ)
+|g(α) − g(αγ)| < ε.
+Recall from (5.1) that the restriction map Cc(G(X)) → Cc(G(X)Uc) is surjective, and
+hence g can be regarded as a function in Cc(G(X)). Taking f to be the restriction
+of g on X × X, then f ∈ ℓ∞(X × X) and there exists S > 0 such that supp(f) ⊆ ES
+for some S > 0. Using the notation from Lemma 2.9, we have g = f. Note that
+G(X)Uc = �
+R>0(ER ∩ G(X)Uc), and hence compact subsets of G(X)Uc are always
+contained in those of the form ER ∩ G(X)Uc. Furthermore, Lemma 3.7 implies
+ER ∩ G(X)Uc =
+�
+ω∈Uc
+{(α, γ) ∈ X(ω) × X(ω) : dω(α, γ) ≤ 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 Uc. Then the family of
+metric spaces {(X(ω), dω)}ω∈Uc 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 X(ω) might contain points outside X(ω) as discussed in Theorem 6.8, hence we
+do not know whether functions on X(ω) × X(ω) can be glued together to provide
+a continuous function on G(X)Uc.
+Proof of Proposition 8.2. Sufficiency: For any ε, R > 0, choose S > 0, D ⊆ X and a
+function g : X × X → [0, 1] satisfying the conditions (1)-(3) for ε and 3R. Take a
+map p : NR(D) → D such that the restriction of p on D is the identity map and
+d(p(x), x) ≤ R. Now we define:
+f(x, y) =
+� g(x, p(y)),
+y ∈ NR(D);
+g(x, y),
+otherwise.
+It is clear that supp(f) ⊆ ER+S. Moreover for any y1, y2 ∈ NR(D) with d(y1, y2) ≤ R,
+we have d(p(y1), p(y2)) ≤ 3R and hence
+�
+x∈X
+| f(x, y1) − f(x, y2)| =
+�
+x∈X
+|g(x, p(y1)) − g(x, p(y2))| ≤ ε.
+Therefore (enlarging S to S + R) we obtain that for any ε, R > 0, there exist S > 0,
+a subset D ⊆ X with D ⊇ Uc and a function f : X × X → [0, 1] satisfying:
+(1) supp(f) ⊆ ES;
+(2) for any x ∈ D, we have �
+z∈X f(z, x) = 1;
+(3) for any x ∈ D and y ∈ X with d(x, y) ≤ R, we have �
+z∈X | f(z, x) − f(z, y)| ≤ ε.
+Now we fix ε, R > 0 and take such S, D and function f.
+Given ω ∈ Uc, we have ω(D) = 1. Choose {tα : Dα → Rα} to be a compatible
+family for ω. Applying Proposition 3.3, there exists Yω ⊆ X with ω(Y) = 1 and a
+local coordinate system {hy : B(ω, R + S) → B(y, R + S)}y∈Yω such that the map
+hy : B(ω, R + S) → B(y, R + S),
+α �→ tα(y)
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+35
+is a surjective isometry for each y ∈ Yω. Replacing Y by Yω ∩ D, we assume that
+Yω ⊆ D. Note that supp(f) ⊆ ES, and hence applying Lemma 3.7 we have
+�
+α∈X(ω)
+f(α, ω) =
+�
+α∈B(ω,S)
+f(α, ω) =
+�
+α∈B(ω,S)
+lim
+x→ω f(tα(x), x) =
+lim
+x→ω,x∈Yω
+�
+α∈B(ω,S)
+f(tα(x), x)
+=
+lim
+x→ω,x∈Yω
+�
+z∈B(x,S)
+f(z, x) =
+lim
+x→ω,x∈Yω
+�
+z∈X
+f(z, x),
+(8.1)
+where the last item equals 1 thanks to the assumption.
+On the other hand, for any α ∈ B(ω, R) we have limx→ω d(tα(x), x) ≤ R. Hence
+shrinking Yω if necessary, we can assume that d(tα(x), x) ≤ R for any x ∈ Yω.
+Therefore, we have
+�
+γ∈X(ω)
+| f(γ, α) − f(γ, ω)| =
+�
+γ∈B(ω,R+S)
+| f(γ, α) − f(γ, ω)|
+(8.2)
+=
+lim
+x→ω,x∈Yω
+�
+γ∈B(ω,R+S)
+| f(tγ(x), tα(x)) − f(tγ(x), x)|
+=
+lim
+x→ω,x∈Yω
+�
+z∈B(x,R+S)
+| f(z, tα(x)) − f(z, x)|
+=
+lim
+x→ω,x∈Yω
+�
+z∈X
+| f(z, tα(x)) − f(z, x)|,
+where the last item is no more than ε by assumption. Therefore applying Lemma
+8.3, we conclude the sufficiency.
+Necessity: Given ε, R > 0, Lemma 8.3 provides S > 0 and a function f : X × X →
+[0, 1] satisfying the conditions (1)-(3) therein.
+Fix an ω ∈ Uc and we choose
+{tα : Dα → Rα} to be a compatible family for ω. Applying Proposition 3.3, there
+exists Yω ⊆ X with ω(Y) = 1 and a local coordinate system {hy : B(ω, R + 2S) →
+B(y, R + S)}y∈Yω such that the map
+hy : B(ω, R + S) → B(y, R + S),
+α �→ tα(y)
+is a surjective isometry for each y ∈ Yω. By the calculations in (8.1), we obtain that
+lim
+x→ω,x∈Yω
+�
+z∈X
+f(z, x) = 1.
+Hence for the given ε, there exists Y′
+ω ⊆ Yω with ω(Y′
+ω) = 1 such that for any x ∈ Y′
+ω
+we have
+�
+z∈X
+f(z, x) ∈ (1 − ε, 1 + ε).
+On the other hand, for any α ∈ B(ω, R) we apply the calculations in (8.2) and
+obtain:
+lim
+x→ω,x∈Y′ω
+�
+z∈X
+| f(z, tα(x)) − f(z, x)| =
+�
+γ∈X(ω)
+| f(γ, α) − f(γ, ω)| ≤ ε.
+Note that for x ∈ Y′
+ω, Proposition 3.3 implies that {tα(x) : α ∈ B(ω, R)} = B(x, R).
+Hence there exists Y′′
+ω ⊆ Y′
+ω with ω(Y′′
+ω) = 1 such that for any x ∈ Y′′
+ω and y ∈ B(x, R),
+we have
+�
+z∈X
+| f(z, y) − f(z, x)| < 2ε.
+
+36
+QIN WANG AND JIAWEN ZHANG
+Taking D := �
+ω∈Uc Y′′
+ω, then it is clear that D ⊇ Uc. Moreover, the analysis above
+shows that:
+• for any x ∈ D we have �
+z∈X f(z, x) ∈ (1 − ε, 1 + ε);
+• for any x ∈ D and y ∈ B(x, R), we have �
+z∈X | f(z, y) − f(z, x)| < 2ε.
+Finallyusinga standard normalisation argument(orequivalently, applyingLemma
+2.11 and modifying Lemma 8.3 accordingly), we conclude the proof.
+□
+Setting ξy(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 ⊆ βX be an invariant open subset. Then
+X has partial Property A towards Uc if and only if for any ε, R > 0, there exist S > 0, a
+subset D ⊆ X with D ⊇ Uc and a function ξ : D → ℓ1(X)1,+, x �→ ξx satisfying:
+(1) supp(ξx) ⊆ B(x, S) for any x ∈ D;
+(2) for any x, y ∈ D with d(x, y) ≤ R, we have ∥ξx − ξy∥1 ≤ ε.
+Remark 8.6. We remark that the function ξ in Proposition 8.2′ can be made such
+that ξx ∈ ℓ1(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 ℓ1(D)1,+ by ℓ2(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 L in X, we denote
+U(L) := �
+Y∈L Y. We need the following auxiliary lemma:
+Lemma 8.7. Let L be an ideal in X and D ⊆ X. Then D ⊇ U(L)c if and only if there
+exists Y ∈ L such that D ⊇ Yc.
+Proof. Assume Y ∈ L such that D ⊇ Yc. Note that Y ∩ Yc = ∅ and Y ∪ Yc = βX.
+Hence we have D ⊇ Yc = βX \ Y ⊇ βX \ U(L) = U(L)c.
+Conversely, assume that D ⊇ U(L)c. Then (D)c ⊆ U(L) = �
+Y∈L Y. Since D is
+clopen in the compact space βX, the set (D)c is compact as well. Hence there exists
+Y1, · · · , Yn ∈ L such that (D)c ⊆ Y1 ∪· · ·∪Yn = Y1 ∪ · · · ∪ Yn. Since L is an ideal, the
+set Y := Y1∪· · ·∪Yn ∈ L. Then we have (D)c ⊆ Y, which implies that D ⊇ (Y)c = Yc.
+Finally we obtain D = D ∩ X ⊇ Yc ∩ X = Yc, 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 ⊆ βX an invariant open subset and L = L(U)
+the associated ideal in X. Then X has partial Property A towards Uc if and only if for any
+ε, R > 0, there exist S > 0, a subset Y ∈ L(U) and a function ξ : Yc → ℓ2(Yc)1,+, x �→ ξx
+satisfying:
+(1) supp(ξx) ⊆ B(x, S) for any x ∈ Yc;
+(2) for any x, y ∈ Yc with d(x, y) ≤ R, we have ∥ξx − ξy∥1 ≤ ε.
+Similar to the proof in the case of Property A, we also have the following:
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+37
+Corollary 8.8. Let (X, d) be a space, U ⊆ βX an invariant open subset and L = L(U) the
+associated ideal in X. Then X has partial Property A towards Uc if and only if for any
+R > 0 and ε > 0, there exist S > 0, a subset Y ∈ L(U) and a kernel k : Yc × Yc → R of
+positive type satisfying:
+(1) for x, y ∈ Yc, we have k(x, y) = k(y, x) and k(x, x) = 1;
+(2) for x, y ∈ Yc with d(x, y) ≥ S, we have k(x, y) = 0;
+(3) for x, y ∈ Yc with d(x, y) ≤ R, we have |1 − k(x, y)| ≤ ε.
+8.2. Partial operator norm localisation property. Recall that the notion of oper-
+ator norm localisation property (ONL) was introduced by Chen, Tessera, Wang
+and Yu in [10], and proved by Sako in [37] that ONL is equivalent to Property A.
+Here we introduce a partial version of ONL, parallel to Definition 8.1.
+Let ν be a positive locally finite Borel measure on X and H be a separable
+infinite-dimensional Hilbert space. For an operator T ∈ B(L2(X, ν) ⊗ H), we can
+also define its propagation as in Section 2.3. We introduce the following:
+Definition 8.9. Let (X, d) be a space and U ⊆ βX an invariant open subset. We
+say that X has partial operator norm localisation property (partial ONL) towards Uc =
+∂βX \ U if there exists c ∈ (0, 1] such that for any R > 0 there exist S > 0 and
+D ⊆ X with D ⊇ Uc satisfying the following: for any positive locally finite Borel
+measure ν on X with supp(ν) ⊆ D and any a ∈ B(L2(X, ν) ⊗ H) with propagation
+at most R, there exists a non-zero ζ ∈ L2(X, ν) ⊗ H with diam(supp(ζ)) ≤ S such
+that c∥a∥ · ∥ζ∥ ≤ ∥aζ∥.
+The aim of the rest of this subsection is to show that partial ONL is equivalent
+to partial Property A. We will follow the outline of [37].
+To simplify the statement, denote CR
+u[X; H] := {T ∈ B(ℓ2(X) ⊗ H) : prop(a) ≤ R}
+for R ≥ 0.
+For a subspace Y ⊆ X, it is clear that CR
+u[Y] � χYCR
+u[X]χY (resp.
+CR
+u[Y; H] � χYCR
+u[X; H]χY), and hence can be regarded as a subset of CR
+u[X] (resp.
+CR
+u[X; H]) with support in Y × Y. Similarly, C∗
+u(Y) � χYC∗
+u(X)χY can be regarded
+as a C∗-subalgebra in C∗
+u(X). Hence we will not tell the difference in the sequel.
+For S > 0, denote
+BY
+S :=
+�
+x∈X
+B(ℓ2(B(x, S) ∩ Y)),
+whose elements will be written as b = ([bx(y, z)]y,z∈B(x,S)∩Y)x∈X. We also consider the
+map
+ψY
+S : C∗
+u(Y) � χYC∗
+u(X)χY −→ BY
+S
+by
+a �→ ([a(y, z)]y,z∈B(x,S)∩Y)x∈X.
+Recall the notions of completely positive map and completely bounded map:
+• A self-adjoint closed subspace F of a unital C∗-algebra B such that 1B ∈ F is
+called an operator system.
+• A linear map φ from F to a C∗-algebra C is said to be completely positive if
+the map φ(n) = φ ⊗ id : F ⊗ Mn(C) → C ⊗ Mn(C) is positive for every n.
+• A linear map θ : F → C is said to be completely bounded if the sequence
+{∥θ(n) : F⊗Mn(C) → C⊗Mn(C)∥} is bounded. Denote ∥θ∥cb := supn∈N ∥θ(n)∥.
+We have the following characterisation for partial ONL, which is analogous to
+[37, Proposition 3.1]. The proof is almost identical, hence omitted.
+
+38
+QIN WANG AND JIAWEN ZHANG
+Lemma 8.10. Let (X, d) be a space and U ⊆ βX an invariant open subset. Then the
+following are equivalent:
+(1) X has partial ONL towards Uc;
+(2) there exists c ∈ (0, 1] such that for any R > 0 there exist S > 0 and D ⊆ X with
+D ⊇ Uc satisfying condition (α): for any a ∈ CR
+u[D; H] there exists a non-zero
+ζ ∈ ℓ2(X) ⊗ H with diam(supp(ζ)) ≤ S and c∥a∥ · ∥ζ∥ ≤ ∥aζ∥;
+(3) for any c ∈ (0, 1) and R > 0, there exist S > 0 and D ⊆ X with D ⊇ Uc satisfying
+condition (α);
+(4) for any c ∈ (0, 1) and R > 0, there exist S > 0 and D ⊆ X with D ⊇ Uc
+satisfying condition (β): for any a ∈ CR
+u[D] there exists a non-zero ξ ∈ ℓ2(X) with
+diam(supp(ξ)) ≤ S and c∥a∥ · ∥ξ∥ ≤ ∥aξ∥;
+(5) for any ε, R > 0 there exist S > R and D ⊆ X with D ⊇ Uc such that
+∥(ψD
+S |CR
+u[D])−1 : ψD
+S (CR
+u[D]) −→ CR
+u[D]∥ < 1 + ε;
+(6) for any ε, R > 0 there exist S > R and D ⊆ X with D ⊇ Uc such that
+∥(ψD
+S |CR
+u[D])−1 : ψD
+S (CR
+u[D]) −→ CR
+u[D]∥cb < 1 + ε;
+We record the following, which comes directly from Lemma 8.7 and 8.10.
+Lemma 8.11. Let (X, d) be a space, U ⊆ βX an invariant open subset and L = L(U) the
+associated ideal in X. Then X has partial ONL towards Uc if and only if for any c ∈ (0, 1)
+and R > 0 there exist S > 0 and Y ∈ L(U) satisfying the following: for any a ∈ CR
+u[Yc]
+there exists a non-zero ξ ∈ ℓ2(X) with diam(supp(ξ)) ≤ S and c∥a∥ · ∥ξ∥ ≤ ∥aξ∥.
+Finally, we can mimic the proof of [37, Theorem 4.1] using Proposition 8.2′,
+Remark 8.6, Corollary 8.8 and Lemma 8.10 instead, and reach the following. The
+proof is almost identical, and hence omitted.
+Proposition 8.12. Let (X, d) be a space and U ⊆ βX be an invariant open subset. Then
+X has partial Property A towards Uc if and only if X has partial ONL towards Uc.
+To end this subsection, we study a permanence property of partial ONL, which
+will help to prove the main result in the next subsection.
+Let X be a space and L an ideal in X. Assume that X can be decomposed into
+X = X1 ∪ X2. Consider
+(8.3)
+Li := {Y ∩ Xi : Y ∈ L}
+for
+i = 1, 2.
+Then it is routine to check that Li is an ideal in Xi for i = 1, 2, and
+L = {Y1 ∪ Y2 : Yi ∈ Li, i = 1, 2}.
+With respect to the decomposition above, we now show that partial ONL is
+preserved under finite unions.
+Proposition 8.13. With the notation as above, assume that Xi has partial ONL towards
+βXi \ U(Li) for i = 1, 2. Then X has partial ONL towards βX \ U(L).
+One way to prove Proposition 8.13 is to follow the proof of [16, Lemma 3.3]
+with minor changes. Here we choose another approach using Proposition 8.12.
+Recall from Corollary A.8 that for a subset Z ⊆ X, the closure Z in βX is
+homeomorphic to βZ. Hence we will regard them as the same object in the sequel.
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+39
+Now we can easily transfer the restriction of ideals in (8.3) to that of invariant
+open subsets (the proof is straightforward, hence omitted):
+Lemma 8.14. Let X = X1 ∪ X2, L be an ideal in X and U = U(L) ⊆ βX be the associated
+invariant open subset of βX. Then U ∩ Xi is an invariant open subset of Xi = βXi, which
+corresponds to the ideal Li in (8.3) for i = 1, 2.
+Although U ∩ Xi is invariant in βXi, generally it is not invariant in βX. This
+coincides with the fact that χXiC∗
+u(X)χXi � C∗
+u(Xi) is a just subalgebra in C∗
+u(X)
+rather than an ideal. Instead, we consider the spatial ideal IXi recalled in Section
+4, and prove the following permanence property for partial Property A:
+Lemma 8.15. Let X = X1 ∪ X2, U ⊆ βX be an invariant open subset and Ui = U ∩ Xi
+for i = 1, 2. If Xi has partial Property A towards βXi \ Ui for i = 1, 2, then X has partial
+Property A towards βX \ U.
+Proof. For any R > 0 and i = 1, 2, set Ui(R) := �
+Y∈L(Ui) NR(Y).
+Clearly, Ui(R)
+is an invariant open subset of NR(Xi) = β(NR(Xi)). Moreover, it follows from
+Proposition 8.2 that NR(Xi) has partial Property A towards NR(Xi) \ Ui(R). By
+definition, this means that the groupoid G(NR(Xi))NR(Xi)\Ui(R) is amenable. Hence
+as a subgroupoid, G(NR(Xi))NR(Xi)\U is amenable, which further implies that the
+groupoid �
+R>0 G(NR(Xi))NR(Xi)\U is amenable.
+Note from Lemma 4.14 that �
+R NR(Xi) \ U is invariant in βX and Lemma 4.16
+implies that
+G(X)�
+R NR(Xi)\U =
+�
+R>0
+G(NR(Xi))NR(Xi)\U,
+which is hence amenable. Note that �
+R NR(X1) ∪ �
+R NR(X2) = βX, and hence due
+to the extension property we obtain that
+G(X)βX\U = G(X)�
+R NR(X1)\U ∪ G(X)�
+R NR(X2)\U
+is amenable as required.
+□
+Combining Proposition 8.12 and Lemma 8.15, we conclude Proposition 8.13.
+8.3. Characterisation for I(U) = ˜I(U). Having established all the necessary ingre-
+dients above, now we present the main result of this section:
+Theorem 8.16. Let (X, d) be a space as in Section 2.2 and U ⊆ βX be a countably
+generated invariant open subset. Then the following are equivalent:
+(1) X has partial Property A towards βX \ U;
+(2) ˜I(U) = I(U);
+(3) the ideal IG of all ghost operators is contained in I(U).
+Note that U = X is countably generated, and hence Theorem 8.16 recovers [36,
+Theorem 1.3] (see Example 4.8 and Example 5.5). Borrowing the language of
+[36], condition (3) in Theorem 8.16 says that all the ghosts can be busted in the
+geometric ideal I(U).
+We follow the outline of the proof for [36, Theorem 1.3]. Firstly, we need a
+modified version of [36, Lemma 4.2]:
+
+40
+QIN WANG AND JIAWEN ZHANG
+Lemma 8.17. Let (X, d) be a space, U ⊆ βX be a countably generated invariant open
+subset and L = L(U) the associated ideal in X. Assume that X does not have partial ONL
+towards Uc. Then there exist κ ∈ (0, 1), R > 0, a sequence (Tn) in Cu[X], a sequence (Bn)
+of finite subsets of X and a sequence (Sn) of positive real numbers such that:
+(a) (Sn) is an increasing sequence tending to infinity as n → ∞;
+(b) each Tn is positive and has norm 1;
+(c) for n � m, then Bn ∩ Bm = ∅;
+(d) each Tn is supported in Bn × Bn;
+(e) for each n and ξ ∈ ℓ2(X) with ∥ξ∥ = 1 and diam(suppξ) ≤ Sn, then ∥Tnξ∥ ≤ κ;
+(f) for each Y ∈ L(U), there exists n such that Bn ∩ Y = ∅.
+Proof. Fixing a basepoint x0 ∈ X, consider the decomposition X = X(1) ∪ X(2) with
+X(1) :=
+�
+m even
+{x ∈ X : m2 ≤ d(x, x0) < (m + 1)2}
+and
+X(2) :=
+�
+m odd
+{x ∈ X : m2 ≤ d(x, x0) < (m + 1)2}.
+Set Li := {Y ∩ X(i) : Y ∈ L} for i = 1, 2. By assumption, X does not have partial
+Property A towards Uc. Hence without loss of generality, we can assume that
+X(1) does not have partial Property A towards βX(1) \ U(L1) thanks to Proposition
+8.13. Note that this implies that U(L1) � βX(1). It is clear that L1 is also countably
+generated, and hence according to Lemma 7.6 there exists a countable subset
+{Y1, Y2, · · · , Yn, · · · } of L1 such that
+L1 = {Z ⊆ X(1) : ∃ n ∈ N such that Z ⊆ Yn}.
+In the sequel, we fix such a sequence {Y1, Y2, · · · , Yn, · · · }.
+Due to Lemma 8.11, we know that there exist c ∈ (0, 1) and R > 0 such that for
+any Y ∈ L1 and S > 0, there exists T ∈ CR
+u[X(1) \ Y] with ∥T∥ = 1 satisfying: for any
+ξ ∈ ℓ2(X(1)) with diam(suppξ) ≤ S and ∥ξ∥ = 1, then ∥Tξ∥ < c. We call such an
+operator (R, c, S, Y)-localised. Replacing T by T∗T (and R by 2R and c by √c), we
+see that there exist c ∈ (0, 1) and R > 0 such that for any Y ∈ L1 and S > 0, there
+exists a positive (R, c, S, Y)-localised operator of norm one. Let us fix such c and R
+in the rest of the proof, and set κ :=
+2c
+1+c < 1.
+Note that X(1) can be decomposed into:
+X(1) :=
+�
+m∈N
+Xm
+where each Xm is finite and d(Xm, Xn) > R for any n � m. Hence each T ∈ CR
+u[X(1)]
+splits as a block diagonal sum of finite rank operators T =
+�
+m T(m) where T(m) ∈
+B(ℓ2(Xm)), with respect to this decomposition.
+Take S1 = 1. By assumption, there exists a positive (R, c, S1, Y1)-localised op-
+erator T ∈ CR
+u[X(1) \ Y1] with norm 1.
+Note that ∥T∥ = supm ∥T(m)∥, and then
+there exists m1 ∈ N such that ∥T(m1)∥ > 1+c
+2 . We set T1 := T(m1)/∥T(m1)∥ and denote
+B1 := Xm1 ∩ (X(1) \ Y1), which is nonempty since T has support in B1 × B1 by as-
+sumption. Then for any ξ ∈ ℓ2(X(1)) with ∥ξ∥ = 1 and diam(supp(ξ)) ≤ S1, we
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+41
+have
+∥T1ξ∥ ≤
+2c
+1 + c = κ < 1.
+Now we take S2 > max
+�
+diam
+� �
+k≤m1 Xk
+�
+, 2
+�
+.
+By assumption, there exists a
+positive (R, c, S2, Y2)-localised operator T ∈ CR
+u[X(1) \ Y2] with norm 1. Again there
+exists m2 such that ∥T(m2)∥ > 1+c
+2 , which forces m2 > m1. We set T2 := T(m2)/∥T(m2)∥
+and denote B2 := Xm2 ∩(X(1)\Y2), which is nonempty since T has support in B2×B2
+by assumption. Similarly for any ξ ∈ ℓ2(X(1)) with ∥ξ∥ = 1 and diam(supp(ξ)) ≤ S2,
+we have ∥T2ξ∥ ≤ κ.
+Inductively, we can construct a sequence (Tn) in Cu[X(1)] ⊆ Cu[X], a sequence
+(Bn) of finite subsets of X(1) ⊆ X and a sequence (Sn) of positive real numbers
+satisfying condition (a)-(e) in the statement. Furthermore for each Z ∈ L(U), there
+exists Yn containing Z∩X(1) for some n. By construction, we know that Bn ∩Yn = ∅
+and Bn ⊆ X(1). Hence we have:
+Bn ∩ Z = Bn ∩ X(1) ∩ Z ⊆ Bn ∩ Yn = ∅,
+which provides condition (f) and concludes the proof.
+□
+Remark 8.18. Comparing Lemma 8.17 with [36, Lemma 4.2], we note that condition
+(f) is the only extra condition added in Lemma 8.17.
+It seems hard to write
+condition (f) in the language of the invariant open subset U instead of the ideal
+L(U), which indicates the importance of using the notion of ideals in spaces as
+mentioned in Section 1.
+Now we are in the position to prove Theorem 8.16.
+Proof of Theorem 8.16. “(1) ⇒ (2)” is contained in Proposition 7.2, and “(2) ⇒ (3)”
+holds trivially since IG = ˜I(X) ⊆ ˜I(U). Hence it suffices to show “(3) ⇒ (1)”, and
+we follow the outline of the proof for [36, Theorem 1.3].
+Assume that X does not have partial Property A towards Uc, then it follows
+from Proposition 8.12 that X does not have partial ONL towards Uc. Then from
+Lemma 8.17, there exist κ ∈ (0, 1), R > 0, a sequence (Tn) in Cu[X], a sequence
+(Bn) of finite subsets of X and a sequence (Sn) of positive real numbers satisfying
+condition (a)-(f) therein. Now we consider the operator
+T :=
+�
+n
+Tn,
+which is a positive operator in Cu[X] of norm one.
+Now we take a continuous function f : [0, 1] → [0, 1] such that supp f ⊆ [1+κ
+2 , 1]
+and f(1) = 1. Consider the operator f(T) ∈ C∗
+u(X), which is positive, norm one,
+and admits a decomposition
+f(T) =
+�
+n
+f(Tn),
+where each f(Tn) ∈ B(ℓ2(Bn)). We will show that f(T) ∈ ˜I(X) \ I(U), and hence
+conclude a contradiction.
+First we show that f(T) � I(U). Recall from (4.2) that
+I(U) = {T′ ∈ Cu[X] : supp(T′) ⊆ Y × Y for some Y ∈ L(U)},
+
+42
+QIN WANG AND JIAWEN ZHANG
+Now for any T′ ∈ Cu[X] with supp(T′) ⊆ Y × Y for some Y ∈ L(U), condition (f) in
+Lemma 8.17 implies that there exists n such that Bn ∩ Y = ∅. Hence we have:
+∥ f(T) − T′∥ ≥ ∥χBn f(T)χBn − χBnT′χBn∥ = ∥ f(Tn) − 0∥ = 1,
+which implies that f(T) � I(U).
+On the other hand, using the same argument as for [36, Theorem 1.3] (since
+Lemma 8.17 provides all the conditions required in [36, Lemma 4.2]), we obtain
+that f(T) is a ghost operator. Hence according to Example 5.5, we have f(T) ∈ ˜I(X).
+Therefore, we conclude the proof.
+□
+9. Open questions
+Here we collect several open questions around this topic.
+First recall from Theorem 5.4 that for a space (X, d), any ideal I in the uniform
+Roe algebra C∗
+u(X) must lie between I(U) and ˜I(U) for U = U(I). However, the
+structure of the lattice
+IU = {I is an ideal in C∗
+u(X) : U(I) = U}
+in (1.1) is still unclear. Note that for any invariant open subset V ⊇ U of βX, the
+ideal I(V)∩ ˜I(U) belongs to the lattice IU. Unfortunately, we do not know whether
+these ideals can bust every element in IU. Hence we pose the following:
+Question 9.1. Let (X, d) be a space and U ⊆ βX be an invariant open subset. Can we
+describe elements in the lattice IU = {I is an ideal in C∗
+u(X) : U(I) = U} in details? For
+I ∈ IU, can we find an invariant open subset V ⊇ U such that I = I(V) ∩ ˜I(U)?
+Note that an answer to the above question together with Theorem 5.4 will
+provide a full description for the ideal structure of the uniform Roe algebra.
+Our next question concerns minimal points discussed in Section 6.1. Recall that
+minimal points in the Stone- ˇCech boundary correspond to maximal ideals in the
+uniform Roe algebra. However as shown in Theorem 6.8, there exist a number of
+non-minimal points in the boundary. Hence it would be interesting to explore a
+practical approach to distinguish minimal points.
+Question 9.2. Given a space (X, d), can we find a practical approach to distinguish
+minimal points in the Stone- ˇCech boundary ∂βX?
+Note from Theorem 6.8 that the answer might not be easy even in the elementary
+case that X = Z.
+Our last questions concern the assumption of countably generatedness used
+in Section 7 and 8. Recall that in Proposition 7.8 we prove that the inclusion
+ιU : I(U) ֒→ ˜I(U) induces an isomorphism in K-theory when the space is coarsely
+embeddable and U is countably generated. We ask the following:
+Question 9.3. Does the inclusion ιU : I(U) ֒→ ˜I(U) induce an isomorphism in K-theory
+for coarsely embeddable X without the assumption that U is countably generated?
+Also note that even under the assumption of countable generatedness, we are
+merely able to show that (ιI)∗ : K∗(I ∩ Cu[X]) −→ K∗(I) is injective in Proposition
+7.8. Hence we also pose the following:
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+43
+Question 9.4. For an ideal I in C∗
+u(X) with U(I) countably generated and X coarsely
+embeddable, is the map (ιI)∗ : K∗(I ∩ Cu[X]) −→ K∗(I) surjective for ∗ = 0, 1?
+Our final question is designed for Theorem 8.16. Recall that the assumption of
+countably generatedness plays an important role in Lemma 8.17, which is in turn
+crucial in the proof of Theorem 8.16. Hence we ask the following:
+Question 9.5. Let (X, d) be a space and U ⊆ βX be an invariant open subset.
+If
+˜I(U) = I(U), can we deduce that X has partial Property A towards βX \ U?
+Appendix A. Ultrafilters
+Here we collect some basic knowledge on ultrafilters, which is used throughout
+the paper. The material should be fairly well-known (see, e.g., [9, Appendix A],
+[34, Chapter 7.4] or [43, Appendix A]). While some of the results might not be
+standard and we have not dug into the reference, we include the proofs for
+convenience to readers.
+Definition A.1. Let X be a set and P(X) be its power set. An ultrafilter on X is a
+family U ⊆ P(X) satisfying the following:
+(I.1) ∅ � U;
+(I.2) for A, B ∈ U, then A ∩ B ∈ U;
+(I.3) for A ∈ U and A ⊆ B, then B ∈ U;
+(I.4) for any A ⊆ X, either A ∈ U or X \ A ∈ U.
+For an ultrafilter U on X, we can associate a function ω : P(X) → {0, 1} by
+setting ω(A) = 1 if and only if A ∈ U. It follows from (I.1)-(I.4) above that ω is a
+finitely additive {0, 1}-valued probability measure on P(X). Conversely for such
+a function ω on P(X), we can associate a family Uω := {A ⊆ X : ω(A) = 1}. It is
+clear that Uω is an ultrafilter on X, and these two procedures are inverse to each
+other. Therefore throughout the paper, we slide between these two notions freely
+without further explanation.
+For a ∈ X, it is clear that the family {A ∈ P(X) : a ∈ A} is an ultrafilter on
+X. Such an ultrafiler is called principal. An ultrafilter which is not principal is
+called non-principal. An argument using Zorn’s lemma shows that non-principal
+ultrafilters always exist whenever X is infinite.
+The following is well-known (see, e.g., [43, Lemma A.2]):
+Lemma A.2. Let ω be an ultrafilter on a set X, and D ⊆ X with ω(D) = 1. Let f : D → Y
+be a function from D to a compact Hausdorff topological space Y. Then there exists a
+unique point y ∈ Y such that for any open neighbourhood U of y, we have ω(f −1(U)) = 1.
+Definition A.3. The unique point in Lemma A.2 is called the ultralimit of f along
+ω or the ω-limit of f, denoted by limω f or lima→ω f(a).
+We record the following localisation result:
+Lemma A.4. Let U be an ultrafilter on a set X, and A ⊆ X with A ∈ U. Then we have:
+(1) {S ∩ A : S ∈ U} = {S ⊆ A : S ∈ U} is an ultrafilter on A, denoted by UA.
+(2) U = {S ⊆ X : S ∩ A ∈ UA} = {S ⊆ X : ∃ S′ ∈ UA such that S′ ⊆ S}.
+
+44
+QIN WANG AND JIAWEN ZHANG
+Proof. (1). It follows from Definition A.1 that {S ∩ A : S ∈ U} = {S ⊆ A : S ∈ U},
+and hence (I.1)-(I.3) hold for UA. Concerning (I.4): given B ⊆ A, if B ∈ U then
+B ∈ UA as well; if B � U then X \ B ∈ U, and hence A \ B = A ∩ (X \ B) ∈ UA.
+(2). This is straightforward, hence omitted.
+□
+We can also extend an ultrafilter on a subset to the whole space. The proof is
+straightforward, hence omitted.
+Lemma A.5. Let Y be a subset of a set X, and U0 an ultrafilter on Y. Define
+U := {S ⊆ X : S ∩ Y ∈ U0}.
+Then U is an ultrafilter on X.
+The following result provides an approach to combine a family of ultrafilters
+into a single one:
+Lemma A.6. Let {Xi}i∈I be a family of sets, and Ui be an ultrafilter on Xi for each i ∈ I.
+Let ω0 be an ultrafilter on I. Consider the set X := �
+i∈I Xi and define:
+U :=
+� �
+i∈I
+Ai ⊆
+�
+i∈I
+Xi : ∃ J ⊆ I with ω0(J) = 1 such that ∀i ∈ J, Ai ∈ Ui
+�
+.
+Then U is an ultrafilter on X.
+Proof. Firstly, it is clear that ∅ � U. Assume that �
+i∈I Ai and �
+i∈I Bi ∈ U, i.e., there
+exist JA, JB ⊆ I with ω0(JA) = ω0(JB) = 1 such that Ai ∈ Ui for any i ∈ JA and Bi ∈ Ui
+for any i ∈ JB. Consider (�
+i∈I Ai) ∩ (�
+i∈I Bi) = �
+i∈I(Ai ∩ Bi) and J = JA ∩ JB. Then
+ω0(J) = 1 and for each i ∈ J, Ai and Bi are in Ui. This implies that Ai ∩ Bi ∈ Ui,
+which concludes (I.2).
+It is clear that (I.3) holds for U and finally, we consider (I.4).
+Given A =
+�
+i∈I Ai ⊆ X, denote J := {i ∈ I : Ai ∈ Ui}. If ω0(J) = 1, then it follows that A ∈ U.
+Otherwise, assume that ω0(J) = 0. Then we consider X \ A = �
+i∈I(Xi \ Ai). Then
+I \ J = {i ∈ I : Xi \ Ai ∈ Ui} and ω0(I \ J) = 1, which implies that X \ A ∈ U and
+concludes the proof.
+□
+Recall that ultrafilters can also be characterised by the Stone- ˇCech compactifi-
+cation. More precisely, we have the following (see, e.g., [34, Chapter 7.4]):
+Lemma A.7. Let X be a set and βX be the Stone- ˇCech compactification of X.
+(1) Given ω ∈ βX, the family {A ⊆ X : ω ∈ A} is an ultrafilter on X.
+(2) Given an ultrafilter U on X, the intersection �
+A∈U A consists of a single point.
+The procedures above are inverse to each other, and hence βX can be characterised by
+ultrafilters on X. Moreover, points in ∂βX correspond to non-principal ultrafilters.
+Thanks to Lemma A.7, we will also use ultrafilters and points in the Stone- ˇCech
+compactification freely without further explanation throughout the paper.
+For convenience, we also record that for D ⊆ X, its closure D in βX satisfies:
+D = {ω ∈ βX : ω(D) = 1}
+and D is clopen in βX. The topology of βX is generated by {D : D ⊆ X}.
+
+GHOSTLY IDEALS IN UNIFORM ROE ALGEBRAS
+45
+Finally we recall the following, which can be proved either directly using the
+universal property of the Stone- ˇCech compactification or deduced directly from
+Lemma A.4, A.5 and A.7:
+Corollary A.8. For a subset Z ⊆ X, the closure Z in βX is homeomorphic to βZ.
+References
+[1] Claire Anantharaman-Delaroche and Jean Renault. Amenable groupoids, volume 36 of Mono-
+graphies de L’Enseignement Math´ematique. L’Enseignement Math´ematique, Geneva, 2000. With
+a foreword by Georges Skandalis and Appendix B by E. Germain.
+[2] Florent P. Baudier, Bruno M. Braga, Ilijas Farah, Ana Khukhro, Alessandro Vignati, and Rufus
+Willett. Uniform Roe algebras of uniformly locally finite metric spaces are rigid. Invent. Math.,
+230(3):1071–1100, 2022.
+[3] Paul Baum and Alain Connes. Geometric K-theory for Lie groups and foliations. Enseign.
+Math., 46(1/2):3–42, 2000 (firstly circulated in 1982).
+[4] Paul Baum, Alain Connes, and Nigel Higson. Classifying space for proper actions and K-
+theory of group C∗-algebras. Contemporary Mathematics, 167:241–241, 1994.
+[5] Bruno M. Braga, Yeong Chyuan Chung, and Kang Li. Coarse Baum-Connes conjecture and
+rigidity for Roe algebras. J. Funct. Anal., 279(9):108728, 21, 2020.
+[6] Bruno M. Braga and Ilijas Farah. On the rigidity of uniform Roe algebras over uniformly
+locally finite coarse spaces. Trans. Amer. Math. Soc., 374(2):1007–1040, 2021.
+[7] Bruno M. Braga, Ilijas Farah, and Alessandro Vignati. Embeddings of uniform Roe algebras.
+Comm. Math. Phys., 377(3):1853–1882, 2020.
+[8] Bruno M. Braga, Ilijas Farah, and Alessandro Vignati. General uniform Roe algebra rigidity.
+Ann. Inst. Fourier (Grenoble), 72(1):301–337, 2022.
+[9] Nathanial P. Brown and Narutaka Ozawa. C∗-algebras and finite-dimensional approximations,
+volume 88 of Graduate Studies in Mathematics. Amer. Math. Soc., Providence, RI, 2008.
+[10] Xiaoman Chen, Romain Tessera, Xianjin Wang, and Guoliang Yu. Metric sparsification and
+operator norm localization. Adv. Math., 218(5):1496–1511, 2008.
+[11] Xiaoman Chen and Qin Wang. Notes on ideals of Roe algebras. Q. J. Math., 52(4):437–446,
+2001.
+[12] Xiaoman Chen and Qin Wang. Ideal structure of uniform Roe algebras of coarse spaces. J.
+Funct. Anal., 216(1):191 – 211, 2004.
+[13] Xiaoman Chen and Qin Wang. Ideal structure of uniform Roe algebras over simple cores.
+Chinese Ann. Math. Ser. B, 25(2):225–232, 2004.
+[14] Xiaoman Chen and Qin Wang. Ghost ideals in uniform Roe algebras of coarse spaces. Arch.
+Math. (Basel), 84(6):519–526, 2005.
+[15] Xiaoman Chen and Qin Wang. Rank distributions of coarse spaces and ideal structure of Roe
+algebras. Bull. London Math. Soc., 38(5):847–856, 2006.
+[16] Xiaoman Chen, Qin Wang, and Xianjin Wang. Operator norm localization property of metric
+spaces under finite decomposition complexity. J. Funct. Anal., 257(9):2938–2950, 2009.
+[17] Xiaoman Chen, Qin Wang, and Guoliang Yu. The maximal coarse Baum–Connes conjecture
+for spaces which admit a fibred coarse embedding into Hilbert space. Adv. Math., 249:88–130,
+2013.
+[18] Eske Ellen Ewert and Ralf Meyer. Coarse geometry and topological phases. Comm. Math.
+Phys., 366(3):1069–1098, 2019.
+[19] Martin Finn-Sell. Fibred coarse embeddings, a-T-menability and the coarse analogue of the
+Novikov conjecture. J. Funct. Anal., 267(10):3758–3782, 2014.
+[20] Martin Finn-Sell and Nick Wright. Spaces of graphs, boundary groupoids and the coarse
+Baum-Connes conjecture. Adv. Math., 259:306–338, 2014.
+[21] Nigel Higson, Vincent Lafforgue, and Georges Skandalis. Counterexamples to the Baum-
+Connes conjecture. Geom. Funct. Anal., 12(2):330–354, 2002.
+[22] Nigel Higson and John Roe. On the coarse Baum-Connes conjecture. In Novikov conjectures,
+index theorems and rigidity, Vol. 2 (Oberwolfach, 1993), volume 227 of London Math. Soc. Lecture
+Note Ser., pages 227–254. Cambridge Univ. Press, Cambridge, 1995.
+
+46
+QIN WANG AND JIAWEN ZHANG
+[23] Nigel Higson, John Roe, and Guoliang Yu. A coarse Mayer-Vietoris principle. Math. Proc.
+Cambridge Philos. Soc., 114(1):85–97, 1993.
+[24] Gennadi Kasparov and Guoliang Yu. The coarse geometric Novikov conjecture and uniform
+convexity. Adv. Math., 206(1):1–56, 2006.
+[25] Kang Li, J´an ˇSpakula, and Jiawen Zhang. Measured asymptotic expanders and rigidity for
+roe algebras. arXiv:2010.10749, to appear in Int. Math. Res. Not., 2020.
+[26] Paul S. Muhly, Jean. N. Renault, and Dana P. Williams. Continuous-trace groupoid C∗-
+algebras. III. Trans. Amer. Math. Soc., 348:3621–3641, 1996.
+[27] Piotr W. Nowak and Guoliang Yu. Large scale geometry. EMS Textbooks in Mathematics.
+European Mathematical Society (EMS), Z¨urich, 2012.
+[28] Jean Renault. A Groupoid Approach to C∗-Algebras, volume 793 of Lecture Notes in Mathematics.
+Springer-Verlag, 1980.
+[29] Jean Renault. The ideal structure of groupoid crossed product C∗-algebras. J. Operator Theory,
+25(1):3–36, 1991. With an appendix by Georges Skandalis.
+[30] Steffen Roch. Ideals of band-dominated operators. Complex Var. Elliptic Equ., 67(3):701–715,
+2022.
+[31] John Roe. An index theorem on open manifolds. I, II. J. Differential Geom., 27(1):87–113, 115–
+136, 1988.
+[32] John Roe. Coarse cohomology and index theory on complete Riemannian manifolds, volume 497.
+Amer. Math. Soc., 1993.
+[33] John Roe. Index theory, coarse geometry, and topology of manifolds, volume 90. Amer. Math. Soc.,
+1996.
+[34] John Roe. Lectures on coarse geometry, volume 31 of University Lecture Series. Amer. Math. Soc.,
+Providence, RI, 2003.
+[35] John Roe. Band-dominated Fredholm operators on discrete groups. Integr. Equ. Oper. Theory,
+51(3):411–416, 2005.
+[36] John Roe and Rufus Willett. Ghostbusting and property A. J. Funct. Anal., 266(3):1674–1684,
+2014.
+[37] Hiroki Sako. Property A and the operator norm localization property for discrete metric
+spaces. J. Reine Angew. Math. (Crelle’s Journal), 2014(690):207–216, 2014.
+[38] Aidan Sims. ´Etale groupoids and their C∗-algebras. arxiv preprint arXiv:1710.10897, 2017.
+[39] Georges Skandalis, Jean-Louis Tu, and Guoliang Yu. The coarse Baum-Connes conjecture and
+groupoids. Topology, 41(4):807–834, 2002.
+[40] Jean-Louis Tu. La conjecture de Baum-Connes pour les feuilletages moyennables. K-Theory,
+17(3):215–264, 1999.
+[41] J´an ˇSpakula. Uniform K-homology theory. J. Funct. Anal., 257(1):88–121, 2009.
+[42] J´an ˇSpakula and Rufus Willett. On rigidity of Roe algebras. Adv. Math., 249:289–310, 2013.
+[43] J´an ˇSpakula and Rufus Willett. A metric approach to limit operators. Trans. Amer. Math. Soc.,
+369(1):263–308, 2017.
+[44] Qin Wang. Remarks on ghost projections and ideals in the Roe algebras of expander se-
+quences. Arch. Math. (Basel), 89(5):459–465, 2007.
+[45] Rufus Willett. Some notes on property A. In Limits of graphs in group theory and computer
+science, pages 191–281. EPFL Press, Lausanne, 2009.
+[46] Rufus Willett and Guoliang Yu. Higher index theory for certain expanders and Gromov
+monster groups, I. Adv. Math., 229(3):1380–1416, 2012.
+[47] Guoliang Yu. The coarse Baum-Connes conjecture for spaces which admit a uniform embed-
+ding into Hilbert space. Invent. Math., 139(1):201–240, 2000.
+(Q. Wang) Research Center for Operator Algebras, and Shanghai Key Laboratory of Pure
+Mathematics and Mathematical Practice, School of Mathematical Sciences, East China Nor-
+mal University, Shanghai, 200241, China.
+Email address: qwang@math.ecnu.edu.cn
+(J. Zhang) School of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai,
+200433, China.
+Email address: jiawenzhang@fudan.edu.cn
+