Coarse quotients of metric spaces and embeddings of uniform Roe algebras
aa r X i v : . [ m a t h . OA ] S e p COARSE QUOTIENTS OF METRIC SPACES ANDEMBEDDINGS OF UNIFORM ROE ALGEBRAS
BRUNO M. BRAGA
Abstract.
We study embeddings of uniform Roe algebras which have“large range” in their codomain and the relation of those with coarsequotients between metric spaces. Among other results, we show that if Y has property A and there is an embedding Φ : C ∗ u ( X ) → C ∗ u ( Y ) with“large range” and so that Φ( ℓ ∞ ( X )) is a Cartan subalgebra of C ∗ u ( Y ),then there is a bijective coarse quotient X → Y . This shows that thelarge scale geometry of Y is, in some sense, controlled by the one of X .For instance, if X has finite asymptotic dimension, so does Y . Introduction
Given a metric space X , one associates to it a C ∗ -algebra C ∗ u ( X ), calledthe uniform Roe algebra of X , which captures some of the large scale geomet-ric properties of X (see Section 2 for precise definitions regarding uniformRoe algebras and coarse geometry). This algebra was introduced by J. Roein the context of index theory of elliptical operators in noncompact manifolds[Roe88, Roe93], but its uses have spread way beyond this field. As a sampleapplication, their study entered the realm of mathematical physics in thecontext of topological materials and, in particular, of topological insulators.For instance, Y. Kubota has proposed the uniform Roe algebra of Z n as amodel for disordered topological materials while E. Ewert and R. Meyer haveused its non uniform version for the same purpose (see [Kub17, EM19]). Inthis context, in order to study various types of symmetries, it is importantto better understand embeddings/isomorphisms of (uniform) Roe algebra.This paper deals with preservation of the large scale geometry of uniformlylocally finite metric spaces under embeddings between their uniform Roealgebras — the study of such embeddings was initiated by I. Farah, A.Vignati and the current author in [BFV20a]. Firstly, notice that an injectivecoarse map X → Y induces a canonical embedding C ∗ u ( X ) → C ∗ u ( Y ): givensuch f : X → Y , the isometric embedding u f : ℓ ( X ) → ℓ ( Y ) determinedby u f δ x = δ f ( x ) gives the embedding Ad( u f ) : C ∗ u ( X ) → C ∗ u ( Y ) ([BFV20a,Theorem 1.2]). If f is, furthermore, a coarse embedding, then the image of Φ Date : September 16, 2020. Recall, a metric space (
X, d ) is uniformly locally fininte if given r > N ∈ N so that the balls of radius r in X contains at most N -many elements. Finitely generatedgroups and k -regular graphs are examples of such spaces. is a hereditary subalgebra of C ∗ u ( Y ) and, if f is a bijective coarse equivalence,then Φ is an isomorphism (see Figure (1)).Bijectivecoarse equivalence / / (cid:15) (cid:15) Injectivecoarse embedding / / (cid:15) (cid:15) Injectivecoarse map (cid:15) (cid:15)
Isomorphism ofuniform Roealgebras / / Embedding ofuniform Roe algebraswith hereditary range / / Embedding ofuniform Roealgebras
Figure 1.
Relation between coarse properties of maps be-tween metric spaces and embeddings of their uniform RoealgebrasRigidity questions for uniform Roe algebras deal with when the verticalarrows in Figure (1) can be reversed. The first two arrows are known tobe reversable if the codomain space has G. Yu’s property A (see [WW18,Theorem 6.13] and [BFV20a, Theorem 1.4]). As for the latter, the follow-ing holds: under some geometric conditions on Y , (1) a rank-preservingembedding C ∗ u ( X ) → C ∗ u ( Y ) gives a uniformly finite-to-one coarse map X → Y ([BFV20a, Theorem 5.4]), and (2) a compact-preserving embeddingC ∗ u ( X ) → C ∗ u ( Y ) gives a partition for X into finitely many pieces all ofwhich can be mapped into Y by an injective coarse map ([BFV20a, Theo-rem 1.2]). Although those results do not give injective coarse maps, theirpower comes from the fact that, for uniformly locally finite metric spaces,the existence of uniformly finite-to-one coarse maps X → Y often impliesthat X inherits large scale geometric properties of Y — among them, wehave property A, asymptotic dimension and finite decomposition complexity([BFV20a, Corollary 1.3]).The current paper focus on better understanding the form taken by em-beddings C ∗ u ( X ) → C ∗ u ( Y ) and on obtaining results on geometry preserva-tion which are “opposite” to the ones mentioned above. Precisely, we areinterested in when the existence of an embedding C ∗ u ( X ) → C ∗ u ( Y ) can make Y inherit geometric properties of X (instead of X inheriting properties of Y ).Obviously, we exclude the case in which the embedding Φ : C ∗ u ( X ) → C ∗ u ( Y )is an isomorphism from our range of interests. Indeed, if that is so, not onlyΦ − : C ∗ u ( Y ) → C ∗ u ( X ) is an embedding and the previous results apply,but one can also obtain a coarse equivalence between X and Y under somegeometric conditions (see [ˇSW13a, BF20] for precise statements). We arethen interested in non isomorphic embeddings C ∗ u ( X ) → C ∗ u ( Y ) which have“large enough range”, forcing then the geometry of Y is “controlled” by theone of X .Before giving the definition of an embedding with “large enough range”,we introduce the concept which motivated it. Recall, a surjective bounded Recall, a map f : X → Y is uniformly finite-to-one if sup y ∈ Y | f − ( n ) | < ∞ . OARSE QUOTIENTS AND UNIFORM ROE ALGEBRAS 3 linear map between Banach spaces is called a quotient map . By the openmapping theorem, the image of the unit ball under quotient maps contains aball with positive radius. With that in mind, S. Zhang extended in [Zha15]the concept of quotient maps to the (coarse) metric space category as follows:let X and Y be metric spaces and f : X → Y . A coarse map f is a coarsequotient map if there is K > ε > δ > B ( f ( x ) , ε ) ⊂ f ( B ( x, δ )) K for all x ∈ X , where f ( B ( x, δ )) K is the K -neighborhood of f ( B ( x, δ )). Coarse quotients have been studied further in [BZ16, Zha18] and a weakernotion, called weak coarse quotient map , was studied in [HW19].Notice that coarse quotients do not need to be coarse embeddings; noteven if the map is a bijection. We refer to Subsection 2.4 for examples ofcoarse quotient maps. For the time being, a quick spoiler: if G is a finitegroup acting on a metric space X by coarse equivalences, then the canonicalmap from X to the orbit space X/G is a coarse quotient map.As coarse quotients are objects in between coarse maps and coarse equiva-lences, there should be a diagram as the one in Figure 1 with coarse quotientsappearing in the middle column. More precisely, as it is automatic from thedefinition that a coarse quotient f : X → Y is K -dense in Y (for K asabove), there is not much loss in generality to look at bijective coarse quo-tients. The next definition introduces the terminology needed. Recall, thepropagation of a = [ a xy ] ∈ B ( ℓ ( X )) is defined asprop( a ) = sup { d ( x, y ) | a xy = 0 } . Definition 1.1.
Let X be a metric space and A ⊂ C ∗ u ( X ) be a C ∗ -subalgebra.1. Let ε, k, ℓ >
0. An element b ∈ C ∗ u ( X ) is ( ε, k, ℓ ) -cobounded if thereare a , . . . , a ℓ ∈ A , with k a i k ≤ ℓ , and c , . . . , c ℓ ∈ C ∗ u ( X ), withprop( c i ) ≤ k and k c i k ≤ ℓ , so that (cid:13)(cid:13)(cid:13) b − ℓ X i =1 c i a i (cid:13)(cid:13)(cid:13) ≤ ε ;and b is ( ε, k ) -cobounded if it is ( ε, k, ℓ )-cobounded for some ℓ .2. The subalgebra A is cobounded in C ∗ u ( X ) if there is k > b ∈ C ∗ u ( X ) is ( ε, k )-cobounded for all ε > A is strongly-cobounded in C ∗ u ( X ) if there is k > b ∈ C ∗ u ( X ) is ( ε, k, k )-cobounded for all ε > ∗ u ( X ) → C ∗ u ( Y )whose images are strongly-cobounded in C ∗ u ( Y ) and, in particular, cobounded(see Remark 4.3). When extending quotient maps to metric spaces, the absense of linearity gives riseto two distinct approaches: extend the concept taking into account the (1) large scalegeometry of the metric spaces or (2) the small scale geometry (i.e., its uniform structure).We deal with the former in this paper. For uniform quotient maps , see [BJL + B. M. BRAGA
We now proceed to describe the main results of this paper. We start givinga characterization of the existence of bijective coarse quotients X → Y interms of uniform Roe algebras. Notice that this characterization does notdepend on extra geometric conditions on either X or Y . Theorem 1.2.
Let X and Y be uniformly locally finite metric spaces. Thefollowing are equivalent:1. There is a bijective coarse quotient map X → Y .2. There is an embedding Φ : C ∗ u ( X ) → C ∗ u ( Y ) so that Φ( ℓ ∞ ( X )) = ℓ ∞ ( Y ) and Φ(C ∗ u ( X )) is cobounded in C ∗ u ( Y ) . Theorem 1.2 should be compared with two known results: (1) X and Y are bijectively coarsely equivalent if and only if there is an isomorphismΦ : C ∗ u ( X ) → C ∗ u ( Y ) so that Φ( ℓ ∞ ( X )) = ℓ ∞ ( Y ) ([BF20, Theorem 8.1]);and (2) there is an injective coarse map X → Y if and only if there is anembedding Φ : C ∗ u ( X ) → C ∗ u ( Y ) so that Φ( ℓ ∞ ( X )) ⊂ ℓ ∞ ( Y ) (although notexplicitly, this follows easily from [BFV20a, Theorem 4.3 and Lemma 5.3]).While Theorem 1.2 gives a complete characterization of the existence ofbijective coarse maps, its hypothesis are too rigid. Indeed, the demand thatΦ( ℓ ∞ ( X )) = ℓ ∞ ( Y ) uses too much of the structure given by a choice of basisof ℓ ( Y ) (namely, its canonical orthonormal basis). Hence, it is interestingto obtain results under milder conditions on the embeddings.We have the following in the presence of G. Yu’s property A: Theorem 1.3.
Let X and Y be uniformly locally finite metric spaces, andassume that Y has property A. If there is an embedding Φ : C ∗ u ( X ) → C ∗ u ( Y ) so that Φ( ℓ ∞ ( X )) is a Cartan subalgebra of C ∗ u ( Y ) and Φ(C ∗ u ( X )) is strongly-cobounded in C ∗ u ( Y ) , then there is a bijective coarse quotient X → Y . Roughly speaking, property A is needed in Theorem 1.3 for three reasons:(1) selecting a map X → Y , (2) assuring the map can be taken to be abijection, and (3) to guarantee that (C ∗ u ( Y ) , Φ( ℓ ∞ ( X ))) is a Roe Cartan pair .The former can actually be obtained under milder geometric assumptionson Y (see Theorem 1.4). As for the latter, recall that Roe Cartan pairs wereintroduced by S. White and R. Willett in [WW18] as follows: a subalgebra B ⊂ C ∗ u ( Y ) is called coseparable if there is a countable S ⊂ C ∗ u ( Y ) sothat C ∗ u ( Y ) = C ∗ ( B, S ). Then (C ∗ u ( Y ) , B ) is a Roe Cartan pair if B is acoseparable Cartan subalgebra of C ∗ u ( Y ) which is isomorphic to ℓ ∞ ( N ) (seeSubsection 2.2). It is open if a Cartan subalgebra of C ∗ u ( Y ) isomorphic to ℓ ∞ ( N ) is automatically coseparable. However, it has been recently shownthat this is indeed the case if Y has property A ([BFV20b, Corollary 6.3]).We now describe a version of Theorem 1.3 which holds under weakergeometric conditions. A metric space ( X, d ) is sparse if there is a partition X = F n X n into finite subsets so that d ( X n , X m ) → ∞ as n + m → ∞ . Also,we say that X yields only compact ghost projections if all ghost projections OARSE QUOTIENTS AND UNIFORM ROE ALGEBRAS 5 in C ∗ u ( X ) are compact. The weaker geometric property which we look at isthe one of all sparse subspaces of X yielding only compact ghost projections .This is a fairly broad property: indeed, it is not only implied by property A,but also by coarse embeddability into ℓ and, more generally, by the validityof the coarse Baum-Connes conjecture with coefficients (see [BF20, Lemma7.3] and [BCL20, Theorem 5.3]).The following is a version of Theorem 1.3 outside the scope of propertyA. Since property A is not assumed, we loose bijectivity and coseparabilityof the range is needed in the hypothesis. Theorem 1.4.
Let X and Y be uniformly locally finite metric spaces, andassume that all sparse subspaces of Y yield only compact ghost projections. Ifthere is an embedding Φ : C ∗ u ( X ) → C ∗ u ( Y ) so that Φ( ℓ ∞ ( X )) is a coseparableCartan subalgebra of C ∗ u ( Y ) and Φ(C ∗ u ( X )) is strongly-cobounded in C ∗ u ( Y ) ,then there is a uniformly finite-to-one coarse quotient X → Y . Theorem 1.4 can be applied to obtain restrictions to the geometry of Y .Without getting into details, we mention that the next result is a corollaryof Theorem 1.4 and results on the “coarse version” of a function X → Y being n -to-1. Precisely, we show that a uniformly finite-to-one coarse quo-tient X → Y between uniformly locally finite metric spaces is automatically coarsely n -to-1 for some n ∈ N (see Definition 3.2 and Proposition 3.3). Corollary 1.5.
Let X and Y be uniformly locally finite metric spaces, andassume that all sparse subspaces of Y yield only compact ghost projections.Suppose there is an embedding Φ : C ∗ u ( X ) → C ∗ u ( Y ) so that Φ( ℓ ∞ ( X )) is acoseparable Cartan subalgebra of C ∗ u ( Y ) and Φ(C ∗ u ( X )) is strongly-coboundedin C ∗ u ( Y ) . Then:1. If X has finite asymptotic dimension, so does Y .2. If X has property A, so does Y .3. If X has asymptotic property C, so does Y .4. If X has straight finite decomposition complexity, so does Y . In Section 5, we provide formulas for strongly continuous embeddingsbetween uniform Roe algebras (see Theorems 5.3 and 5.5). Precisely, allisomorphisms Φ : C ∗ u ( X ) → C ∗ u ( Y ) between uniform Roe algebras arespacially implemented, i.e., there is a unitary u : ℓ ( X ) → ℓ ( Y ) so thatΦ = Ad( u ) ([ˇSW13a, Lemma 3.1]). This was later generalized for embed-dings Φ : C ∗ u ( X ) → C ∗ u ( Y ) onto a hereditary subalgebra of C ∗ u ( Y ) ([BFV20a,Lemma 6.1]). Notice that spacially implemented embeddings are automat-ically strongly continuous and rank-preserving. We generalize the two re-sults above by showing that an embedding C ∗ u ( X ) → C ∗ u ( Y ) is spaciallyimplemented if and only if it is strongly continuous and rank-preserving,see Theorem 5.3 (two other characterizations of such embeddings in terms Recall, an operator a = [ a xy ] ∈ B ( ℓ ( X )) is called a ghost if for all ε > A ⊂ X so that | a xy | < ε for all x, y A (see Subsection 2.1 for details). B. M. BRAGA of compact operators are also given). Moreover, we prove an equivalent re-sult for strongly continuous embeddings C ∗ u ( X ) → C ∗ u ( Y ) which send rank1 operators to rank n operators, where n ∈ N ∪ {∞} , see Theorem 5.5.We finish the paper stating some natural problems which are left open;see Section 6. 2. Preliminaries
Uniform Roe algebras.
Given a Hilbert space H , B ( H ) denotes theC ∗ -algebra of bounded operators on H and K ( H ) denotes its ideal of com-pact operators. Given a set X , ( δ x ) x ∈ X denotes the standard unit basisof the Hilbert space ℓ ( X ) and 1 X denotes the identity on ℓ ( X ). Given x, y ∈ X , e xy denotes the partial isometry in B ( ℓ ( X )) so that e xy δ x = δ y and e xy δ z = 0 for all z = x . Given A ⊂ X , we write χ A = SOT- P x ∈ X e xx ;so 1 X = χ X .If ( X, d ) is a metric space, a ∈ B ( ℓ ( X )) and r >
0, we say that the propagation of a is at most r , and write prop( a ) ≤ r , if d ( x, y ) > r implies h aδ x , δ y i = 0 , for all x, y ∈ X. The support of a is defined bysupp( a ) = n ( x, y ) ∈ X × X | h aδ x , δ y i 6 = 0 o . Given a metric space (
X, d ), the subset of all operators in B ( ℓ ( X )) withfinite propagation is a ∗ -algebra called the algebraic uniform Roe algebraof X , denoted by C ∗ u [ X ]. The closure of C ∗ u [ X ] is a C ∗ -algebra called the uniform Roe algebra of X , denoted by C ∗ u ( X ).Instead of presenting the original definition of property A, we give a def-inition in terms of ghost operators which is better suited for our goals: Definition 2.1.
Let X be a uniformly locally finite metric space.1. An operator a ∈ B ( ℓ ( X )) is called a ghost if for all ε > A ⊂ X so that |h aδ x , δ y i| ≤ ε for all x, y ∈ X \ A.
2. The metric space X has property A if all ghost operators in C ∗ u ( X )are compact (see [RW14, Theorem 1.3]).We recall the notions of coarse-likeness introduced in [BF20]. Definition 2.2.
Let X and Y be a metric space.1. Given ε > r >
0, an operator a ∈ B ( ℓ ( X )) can be ε - r -approximated (equivalently, a is ε - r -approximable ) if there is b ∈ C ∗ u ( X ) with prop( b ) ≤ r so that k a − b k ≤ ε .2. A map Φ : C ∗ u ( X ) → C ∗ u ( Y ) is coarse-like if for all ε, s > r > a ) can be ε - r -approximated for all contractions a ∈ C ∗ u ( X ) with prop( a ) ≤ s . OARSE QUOTIENTS AND UNIFORM ROE ALGEBRAS 7
The following is a simple consequence of [BF20, Lemma 4.9] (see [BFV20a,Proposition 3.3] for a precise proof; cf. [BF20, Theorem 4.4]). Theorem 2.3.
Let X and Y be uniformly locally finite metrics spaces and Φ : C ∗ u ( X ) → C ∗ u ( Y ) be a strongly continuous linear operator. Then Φ iscoarse-like. Roe Cartan pairs.
Recall, given a C ∗ -algebra A , a C ∗ -subagebra B ⊂ A is called a Cartan subalgebra of A if1. B is a maximal abelian self-adjoint subalgebra of A ,2. B contains an approximate unit for A ,3. the normalizer of B in A , i.e., { a ∈ A | aBa ∗ , ⊂ a ∗ Ba ⊂ B } , generates A as a C ∗ -algebra, and4. there is a faithful condition expectation A → B .If X is a u.l.f. metric space, then ℓ ∞ ( X ) is a Cartan subalgebra of C ∗ u ( X )[WW18, Proposition 4.10].Let A be a unital C ∗ -algebra and B ⊂ A be a Cartan subalgebra of A .We say that ( A, B ) is
Roe Cartan pair if1. A contains the algebra of compact operators on a infinite dimensionalHilbert space as an essential ideal,2. B is isomorphic to the C ∗ -algebra ℓ ∞ ( N ), and3. B is co-separable in A , i.e., there is a countable S ⊂ A so that A =C ∗ ( B, S ).Roe Cartan pairs were recently introduced to the literature in [WW18].Notice that, as ℓ ∞ ( X ) is a Cartan subalgebra of C ∗ u ( X ), it is clear that(C ∗ u ( X ) , ℓ ∞ ( X )) is a Roe Cartan pair for any u.l.f. metric space X . More-over, we notice that it is not known whether co-separability is a necessaryproperty. Precisely, if (C ∗ u ( X ) , B ) satisfies (2), it is not known whether B isautomatically co-separable. By [BFV20b, Corollary 6.3], this is indeed thecase if X has property A.The importance of Roe Cartan pairs to our goals lies in the followingtheorem: Theorem 2.4 (Theorem B of [WW18]) . Let ( A, B ) be a Roe Cartan pair.Then there is a u.l.f. metric space X and an isomorphism Φ : A → C ∗ u ( X ) so that Φ( B ) = ℓ ∞ ( X ) . Coarse geometry.
Given a metric space (
X, d ), x ∈ X and ε > B ( x, ε ) the closed unit ball centered at x of radius ε . Given asubset A ⊂ X and K >
0, we write A K = n x ∈ X | d ( x, A ) ≤ K o . The hypothesis of [BFV20a, Proposition 3.33] actually demand Φ to be compact-preserving. However, this is so as its proof used an earlier version of [BF20, Lemma 4.9]which required compactness. The (newer) published version of [BF20, Lemma 4.9] doesnot do so, hence the proof of [BFV20a, Proposition 3.33] holds for noncompact preservingΦ’s.
B. M. BRAGA
A metric space (
X, d ) is called uniformly locally finite , u.l.f. for short, ifsup x ∈ X | B ( x, r ) | < ∞ for all r >
0, where | B ( x, r ) | denotes the cardinalityof B ( x, r ).Let ( X, d ) and (
Y, ∂ ) be metric spaces and f : X → Y be a map. The modulus of uniform continuity of f is defined by ω f ( t ) = n ∂ ( f ( x ) , f ( y )) | d ( x, y ) ≤ t o for t ≥
0. Then f is called coarse if ω f ( t ) < ∞ for all t ≥
0. Given anothermap g : X → Y , we say that f is close to g , and write f ∼ g , ifsup x ∈ X ∂ ( f ( x ) , g ( x )) < ∞ . The map f is a coarse equivalence if f is coarse and there is a coarse map h : Y → X so that f ◦ h ∼ Id Y and h ◦ f ∼ Id X .Given K >
0, we say that f is K -co-coarse if for all ε > δ > B ( f ( x ) , ε ) ⊂ f (cid:0) B ( x, δ ) (cid:1) K for all x ∈ X . The map f is co-coarse if it is K -co-coarse for some K > f is both coarse and co-coarse, f is a coarse quotient . Proposition 2.5.
Let ( X, d ) and ( Y, ∂ ) be metric spaces and f, g : X → Y be close maps. If f is a coarse quotient, so is g .Proof. Coarseness is well-known to be preserved under closeness. Moreover,if m = sup x ∈ X ∂ ( f ( x ) , g ( x )) and K > f is K -co-coarse, then itis straightforward to check that g is ( K + m )-co-coarse. (cid:3) From now on, we assume throughout the paper that d and ∂ are themetrics on X and Y , respectively.2.4. Examples of coarse quotients.
Firstly, notice that every coarseequivalence is a coarse quotient map ([Zha15, Proposition 2.5]). But coarseequivalences are far from the only examples of coarse quotient maps. Forinstance, given n, m ∈ N with n < m , the standard projection q : Z m → Z n is clearly a coarse quotient map (and clearly not a coarse equivalence). Themap f : Z → N given by f ( n ) = 2 n for n ∈ N and f ( n ) = − n + 1 for all n ∈ Z \ N is also a coarse quotient. In this case, f is a bijective coarse quo-tient which is not a coarse embedding/equivalence. Similar constructionsgive us bijective coarse quotients Z n → N n for all n ∈ N .More generally, group actions give us natural examples of coarse quotientmaps. Precisely, let ( X, d ) be a discrete metric space and G be a groupacting on X . We say that G acts on X uniformly by coarse equivalences if each g ∈ G acts on X by a coarse equivalence and if there is a map ω : [0 , ∞ ) → [0 , ∞ ) so that d ( g · x, g · y ) ≤ ω ( d ( x, y )) OARSE QUOTIENTS AND UNIFORM ROE ALGEBRAS 9 for all x, y ∈ X and all g ∈ G . If G is a finite group which acts on X bycoarse equivalences, then it is automatic that G acts on X uniformly bycoarse equivalences.Given G y X , denote the orbit space of this action by X/G , i.e., definean equivalence ∼ G on X by letting x ∼ G y if there is g ∈ G with g · y = x and X/G is the set of ∼ G -equivalence classes. The coarse structure of X induces a coarse structure on X/G . Precisely, we can endow
X/G with themetric ∂ ([ x ] , [ y ]) = min n sup x ′ ∈ [ x ] inf y ′ ∈ [ y ] d ( x ′ , y ′ ) , sup y ′ ∈ [ y ] inf x ′ ∈ [ x ] d ( x ′ , y ′ ) o for all [ x ] , [ y ] ∈ X/G . As X is discrete, ∂ is clearly a metric on X/G . Let q : X → X/G be the quotient map. If G acts on X uniformly by coarseequivalences, then it is straightforward to check that q is a coarse quotientmap. The coarse geometry of those spaces was studied in [HW19].3. Coarse quotients and geometry preservation
Coarse properties are those mathematical properties of metric spaceswhich are preserved under coarse equivalence. Some large scale proper-ties are also stable under coarse embeddings, i.e., if X coarsely embeds into Y and Y has such property, then so does X . Moreover, for u.l.f. metricspaces, it is also known that some large scale properties are stable under theexistence of uniformly finite-to-one coarse maps — for instance, this holdsfor property A, asymptotic dimension and finite dimension decomposition,see [BFV20a, Proof of Corollary 1.3, Proposition 2.5, and Corollary 5.8].In this section, we show that the “opposite” phenomena can happen foruniformly finite-to-one coarse quotient maps X → Y . Precisely, we showthat the existence of such maps is often enough so that large scale geometricproperties of X passes to Y . This is the case for finite asymptotic dimen-sion, straight finite decomposition complexity, asymptotic property C, andproperty A (Corollary 3.7).We start noticing that compositions of coarse quotients are coarse quo-tients. This is essentially done in [Zha15, Proposition 2.5], but we include ashort proof for the reader’s convenience. Proposition 3.1.
Let X , Y and Z be metric spaces, and f : X → Y and g : Y → Z be a coarse quotient maps. Then g ◦ f is a coarse quotient. Inparticular, g ◦ f is a coarse quotient map given that f is a coarse quotientmap and g is a coarse equivalence.Proof. If both f and g are coarse quotient maps, both are coarse and so istheir composition. We are left to show that g ◦ f is co-coarse. For that,fix K > f and g are K -co-coarse. Given ε >
0, let δ > B ( g ( y ) , ε ) ⊂ g ( B ( y, δ )) K for all y ∈ Y , and let γ > B ( f ( x ) , δ ) ⊂ f ( B ( x, γ )) K for all x ∈ X . Then, for L = K + ω g ( K ), we have B ( g ◦ f ( x ) , ε ) ⊂ g ( B ( f ( x ) , δ )) K ⊂ g ( f ( B ( y, γ )) K ) K ⊂ g ( f ( B ( y, γ ))) L for all x ∈ X . Hence the assignment ε γ witness that g ◦ f is L -co-coarse. (cid:3) Our main tool in order to obtain coarse geometry preservation, is basedon the next concept. This was introduced in [MV13, Subsection 3.2] asproperty “ B n ” and it is the “coarse version” of a function being n -to-1. Definition 3.2.
Let X and Y be metric spaces. Given f : X → Y and n ∈ N , we say that f is coarsely n -to-1 if for each s > r > B ⊂ Y , with diam( B ) ≤ s , there are A , . . . , A n ⊂ X , withdiam( A i ) ≤ r for all i ∈ { , . . . , n } , so that f − ( B ) ⊂ S ni =1 A i . The map f : X → Y is called uniformly coarsely finite-to-one if f is coarsely n -to-1for some n ∈ N .The next proposition is the main result of this section. Proposition 3.3.
Let X be a metric space and Y be a u.l.f. metric space.Any uniformly finite-to-one coarse quotient map X → Y is uniformly coarselyfinite-to-one. We need a couple of preliminary results before proving Proposition 3.3.We start by noticing that “coarsely n -to-1” is a coarse property for mapsbetween metric spaces. Proposition 3.4.
Let X and Y be metric spaces and f, g : X → Y be closemaps. Given n ∈ N , if f is coarsely n -to-1, so is g .Proof. As f and g are close, let k = sup x ∈ X ∂ ( f ( x ) , g ( x )) < ∞ . Given s >
0, let r > s + 2 k , f and n . Fix B ⊂ Y withdiam( B ) ≤ s . As diam( B k ) ≤ s + 2 k , our choice of r gives A , . . . , A n ⊂ X ,with diam( A i ) ≤ r for all i ∈ { , . . . , n } , so that f − ( B k ) ⊂ S ni =1 A i . Then g − ( B ) ⊂ S ni =1 A i . (cid:3) Lemma 3.5.
Let X be a metric space and Y be a u.l.f. metric space. Anyinjective coarse quotient X → Y is uniformly coarsely finite-to-one.Proof. Let f : X → Y be an injective coarse quotient map. Without loss ofgenerality, assume that f is surjective. Fix K > f is K -co-coarseand let m = sup y ∈ Y | B ( y, K ) | . Let us show that f is coarsely m -to-1. Fix ε >
0. By our choice of K , there is δ = δ ( ε ) > B ( f ( x ) , ε ) ⊂ f (cid:0) B ( x, δ ) (cid:1) K for all x ∈ X .Fix y ∈ Y . We construct a finite sequence ( x i ) i of elements of X byinduction as follows. Pick x ∈ X so that f ( x ) ∈ B ( y, ε ). Let ℓ ∈ N andassume that x , . . . , x ℓ ∈ X have been chosen. If B ( y, ε ) ⊂ ℓ [ i =1 f (cid:0) B ( x i , δ ) (cid:1) , OARSE QUOTIENTS AND UNIFORM ROE ALGEBRAS 11 we stop the procedure and ( x i ) ℓi =1 is the outcome of it. If not, then pick x ℓ +1 ∈ X so that f ( x ℓ +1 ) ∈ B ( y, ε ) \ ℓ [ i =1 f (cid:0) B ( x i , δ ) (cid:1) . This completes the induction. As B ( y, ε ) contains finitely many elements,this procedure is finite.Let ( x i ) ℓi =1 be the finite sequence obtained by the procedure above. Let usshow that ℓ ≤ m . Assume by contradiction that ℓ > m and let z = f ( x m +1 ).As each f ( x i ) belongs to B ( y, ε ), we have that ∂ ( z, f ( x i )) ≤ ε for all i ∈ { , . . . , ℓ } . Hence, by our choice of δ , there is a finite sequence ( w i ) mi =1 in X so that d ( x i , w i ) ≤ δ and ∂ ( z, f ( w i )) ≤ K for all i ∈ { , . . . , m } .By the construction of ( x i ) ℓi =1 , f ( x i ) f ( B ( x j , δ )) for j < i . Hence, d ( x i , x j ) > δ for all i = j , which implies that ( w i ) mi =1 is a distinct sequence.As f is injective, ( f ( w i )) mi =1 is a distinct sequence. As z S mi =1 f ( B ( x i , δ )), { z, f ( w ) , . . . , f ( w m ) } is a subset of B ( z, K ) with m + 1 elements. Thiscontradicts our choice of m .As ℓ = m , B ( y, ε ) ⊂ S mi =1 f ( B ( x i , δ )). As f is injective, this implies that f − ( B ( y, ε )) ⊂ m [ i =1 B ( x i , δ ) . As y was arbitrary, the assignment ε δ ( ε ) witness that f is a coarsely m -to-1 map. (cid:3) Lemma 3.6.
Let X and Y be metric spaces, and f : X → Y be a uniformlyfinite-to-one map. If Y is u.l.f., then there is a u.l.f. metric space Z , with Y ⊂ Z , and an injective map g : X → Z which is close to f . Moreover, theinclusion Y ֒ → Z is a coarse equivalence.Proof. Let n = sup y ∈ Y | f − ( y ) | . Let Z = Y × { , . . . , n } and define a metric ∂ Z on Z by letting ∂ Z (( y, i ) , ( z, j )) = ∂ ( y, z ) + 1for all distinct ( y, i ) , ( z, j ) ∈ Z . As ( Y, ∂ ) is u.l.f., so is (
Z, ∂ Z ). Foreach y ∈ Y , enumerate f − ( y ), say f − ( y ) = { x y , . . . , x yi ( y ) } . As X = F y ∈ Y { x y , . . . , x yi ( y ) } , we can define a map g : X → Z by letting g ( x yj ) = ( y, j )for all y ∈ Y and all j ∈ { , . . . , i ( y ) } . It is clear that f is close to g andthat Y × { } ֒ → Z is a coarse equivalence. By identifying Y with Y × { } ,we can assume that Y ⊂ Z . (cid:3) Proof of Proposition 3.3.
Let f : X → Y be a uniformly finite-to-one coarsequotient map. Let Z and g be given Lemma 3.6 applied to f : X → Y .As the inclusion i : Y ֒ → Z is a coarse equivalence, f = i ◦ f : X → Z isa coarse quotient map (Proposition 3.1). Hence, as f is close to g , g is acoarse quotient map (Proposition 2.5). As g is injective, Lemma 3.5 givesus that g is uniformly coarsely finite-to-one. Using that f and g are close to each other once again, this shows that f is uniformly coarsely finite-to-one(Proposition 3.4). (cid:3) We can now use Proposition 3.3 in order to obtain that some coarseproperties of X passes to Y in the presence of a uniformly finite-to-onecoarse quotient map X → Y . For brevity, we do not introduce the geometricproperties mentioned in the corollary below. Instead, we refer the reader to[NY12, Definitions 2.2.1 and 2.7.7] and [DZ14, Definition 2.2]. Corollary 3.7.
Let X and Y be metric spaces, assume that Y is u.l.f. andthat there is a uniformly finite-to-one coarse quotient map X → Y . Thefollowing holds:1. If X has finite asymptotic dimension, so does Y .2. If X has property A, so does Y .3. If X has asymptotic property C, do does Y .4. If X has straight finite decomposition complexity, so does Y .Proof. (1) This follows from Proposition 3.3 and [MV13, Theorem 1.4].(2) This follows from Proposition 3.3 and and [DV16, Corollary 7.5].(3) This follows from Proposition 3.3 and [DV16, Theorem 6.2].(4) This follows from Proposition 3.3 and [DV16, Theorem 8.4 and The-orem 8.7]. (cid:3) Cobounded embeddings between uniform Roe algebras
In this section, we study embeddings into uniform Roe algebras whoseimages are “large”. We obtain that such embeddings can often be enoughto guarantee the existence of coarse quotient maps between the spaces. The-orems 1.2, 1.3, 1.4, and Corollary 1.5 are proven in this section.The next proposition is well known and its proof can be found for instancein [BF20, Proposition 2.4]. Recall, given
K >
0, a subset A of a metric space X is called K -separated if d ( x, y ) ≥ K for all distinct x, y ∈ A . Proposition 4.1.
Let X be a u.l.f. metric space. Given any K > , thereis n ∈ N and a partition X = X ⊔ X ⊔ . . . ⊔ X n so that each X i is K -separated. Proposition 4.2.
Let X and Y be u.l.f. metric spaces, and f : X → Y be an injective map so that f : X → f ( X ) is a coarse quotient map. Thenthere is a spacially implemented embedding Φ : C ∗ u ( X ) → C ∗ u ( Y ) so that Φ( ℓ ∞ ( X )) = ℓ ∞ ( f [ X ]) and Φ(C ∗ u ( X )) is cobounded in C ∗ u ( f [ X ]) .Proof. Let u f : ℓ ( X ) → ℓ ( Y ) be the isometric embedding given by u f δ x = δ f ( x ) for all x ∈ X . Then it is easy to see that Φ = Ad( u f ) : C ∗ u ( X ) → C ∗ u ( Y )is an embedding (see [BFV20a, Theorem 1.2] for a detailed proof). Since itis clear that Φ( ℓ ∞ ( X )) = ℓ ∞ ( f [ X ]), we only need to notice that Φ(C ∗ u ( X ))is cobounded in C ∗ u ( f [ X ]). OARSE QUOTIENTS AND UNIFORM ROE ALGEBRAS 13
Fix
K > f is K -co-bounded and let Z = f [ X ]. Fix ε > a ∈ C ∗ u [ Z ] with prop( a ) ≤ ε . Without loss of generality, assume ε > K .By our choice of K , there is δ > B ( f ( x ) , ε ) ⊂ f (cid:0) B ( x, δ ) (cid:1) K for all x ∈ X . As Y is u.l.f., there is n = n ( ε, Y ) ∈ N and a partition Z = F ni =1 Z i so that each Z i is 3 ε -separated (Proposition 4.1). For each( i, j ) ∈ { , . . . , n } , let a ( i, j ) = χ Z i aχ Z j .Fix i, j ∈ { , . . . , n } . For simplicity, let b = a ( i, j ). Clearly, prop( b ) ≤ prop( a ) ≤ ε . Hence, as Z i and Z j are 3 ε -separated, there are 3 ε -separatedsequences ( y n ) n and ( y ′ n ) n in Z so that b = X n ∈ N b n e y n y ′ n where b n = h bδ y n , δ y ′ n i for all n ∈ N . In particular, ∂ ( y n , y ′ n ) ≤ ε for all n ∈ N . As f : X → Z is bijective, fix asequence ( x n ) n of distinct elements in X so that f ( x n ) = y n for all n ∈ N .By our choice of δ , for each n ∈ N , there are z n ∈ X so that d ( x n , z n ) ≤ δ and ∂ ( f ( z n ) , y ′ n ) ≤ K . As ( y ′ n ) n is 3 ε -separated and K < ε , it follows that( f ( z n )) n is a sequence of distinct elements and, as f is injective, so is ( z n ) n .In particular, c = SOT- P n ∈ N b n e x n z n is well defined and, as prop( c ) ≤ δ , c ∈ C ∗ u ( X ). Similarly, d = SOT- P n ∈ N e f ( z n ) y ′ n is also well defined and ithas propagation at most K .Notice that b = d Φ( c ). As ε > i, j ∈ { , . . . , n } were arbitrary, weare done. (cid:3) Remark . Notice that if f : Z → N is the bijective coarse quotient givenby f ( n ) = 2 n for n ∈ N and f ( n ) = − n + 1 for n ∈ Z \ N , then Φ = Ad( u f )is actually strongly-cobounded. Indeed, let I and P denote the odd andeven natural numbers, respectively. Then any a ∈ C ∗ u ( Y ) can be written as a = χ I aχ I + χ P aχ P + χ P aχ I + χ I aχ P . Clearly, χ I aχ I , χ P aχ P ∈ Φ(C ∗ u ( Z )). Let g : P → I be the bijectiongiven by g ( n ) = n − n ∈ P , and let c = SOT- P n ∈ P e ng ( n ) ; soprop( c ) = 1. Moreover, it is clear that cχ P aχ I , c ∗ χ I aχ P ∈ Φ(C ∗ u ( Z )). As χ P aχ I = c ∗ cχ P aχ I and χ I aχ P = cc ∗ χ P aχ I , it easily follows that Φ(C ∗ u ( Z ))is strongly-cobounded in C ∗ u ( N ).We do not know if an arbitrary bijective coarse quotient X → Y is enoughto produce an embedding Φ : C ∗ u ( X ) → C ∗ u ( Y ) with strongly-coboundedrange (see Problem 6.3).For the next technical lemma, we introduce a weakening of Definition 1.1. Definition 4.4.
Let X be a metric space and let A ⊂ A be C ∗ -subalgebrasof C ∗ u ( X ). Given ε >
0, the algebra A is called ε -almost cobounded in A if there is k > b ∈ A there are a , . . . , a ℓ ∈ A and c , . . . , c ℓ ∈ A , with prop( c i ) ≤ k , so that (cid:13)(cid:13)(cid:13) b − ℓ X i =1 c i a i (cid:13)(cid:13)(cid:13) ≤ ε. Lemma 4.5.
Let X and Y be u.l.f. metric spaces, Φ : C ∗ u ( X ) → C ∗ u ( Y ) be an embedding so that Φ( ℓ ∞ ( X )) ⊂ ℓ ∞ ( Y ) and Φ( e xx ) has rank 1 for all x ∈ X . Let Z ⊂ Y be a subset so that χ Z = SOT - X x ∈ X Φ( e xx ) . Assume that
Φ(C ∗ u ( X )) is ε -almost cobounded in χ Z C ∗ u ( Y ) χ Z for some ε ∈ (0 , . Then there exists a bijective coarse quotient map X → Z .Proof. As Φ is a ∗ -homomorphism, the hypothesis imply that each Φ( e xx )is a projection of rank 1 in ℓ ∞ ( Z ). Hence, for each x ∈ X there is y x ∈ Z sothat Φ( e xx ) = e y x y x . Define a map f : X → Y by letting f ( x ) = y x for each x ∈ X . Notice that Z = f ( X ). Also, the map f is clearly injective, and, by[BFV20a, Lemma 5.3], it is also coarse. We now show that f : X → Z is co-coarse. For that, fix ε ∈ (0 ,
1) and k ∈ N which witness that Φ(C ∗ u ( X )) is ε -almost cobounded in χ Z C ∗ u ( Y ) χ Z .Assume for a contradiction that f is not k -co-coarse. Hence, there exists γ > x n ) n and ( y n ) n in X so that1. ∂ ( f ( x n ) , f ( y n )) ≤ γ for all n ∈ N , and2. for all z ∈ X , ∂ ( f ( y n ) , f ( z )) ≤ k implies d ( x n , z ) ≥ n .Without loss of generality, by going to a subsequence, we can assume that f ( x n ) = f ( x m ) and f ( y n ) = f ( y m ) for all n = m . Indeed, if this is not thecase, then there is an infinite N ⊂ N so that either f ( x n ) = f ( x m ), for all n, m ∈ N , or f ( y n ) = f ( y m ), for all n, m ∈ N . Then (1) and u.l.f.-ness of Y imply that, going to a further infinite subset of N if necessary, we can assumethat f ( x n ) = f ( x m ) and f ( y n ) = f ( y m ), for all n, m ∈ N . As f is injective,there is x, y ∈ X so that x = x n and y = y n for all n ∈ N . However, (2)implies that d ( x, y ) = d ( x n , y n ) ≥ n for all n ∈ N ; contradiction.As f ( x n ) = f ( x m ) and f ( y n ) = f ( y m ) for all n = m in N , (1) aboveimplies that P n ∈ N e f ( x n ) f ( y n ) converges in the strong operator topology andit belongs to C ∗ u ( Y ). By our choice of k , there are a , . . . , a ℓ ∈ C ∗ u ( X ) and c , . . . , c ℓ ∈ χ Z C ∗ u ( Y ) χ Z , with prop( c i ) ≤ k , so that (cid:13)(cid:13)(cid:13) χ Z (cid:16) X n ∈ N e f ( x n ) f ( y n ) (cid:17) χ Z − ℓ X i =1 c i Φ( a i ) (cid:13)(cid:13)(cid:13) ≤ ε. Notice that, although not explicit in the statement, the only assumption needed inorder for [BFV20a, Lemma 5.3] to hold for Φ is that Φ( e xx ) has rank 1 for all x ∈ X . OARSE QUOTIENTS AND UNIFORM ROE ALGEBRAS 15
Hence, we have that, for all n ∈ N , (cid:13)(cid:13)(cid:13) e f ( y n ) f ( y n ) (cid:16) ℓ X i =1 c i Φ( a i ) (cid:17) e f ( x n ) f ( x n ) (cid:13)(cid:13)(cid:13) ≥ (cid:13)(cid:13)(cid:13) e f ( y n ) f ( y n ) χ Z (cid:16) X i ∈ N e f ( x i ) f ( y i ) (cid:17) χ Z e f ( x n ) f ( x n ) (cid:13)(cid:13)(cid:13) − ε = (cid:13)(cid:13)(cid:13) e f ( x n ) f ( y n ) k − ε = 1 − ε. For each n ∈ N , let B n = B ( f ( y n ) , K ) ∩ Z and let A n = f − ( B n ).As each c i has propagation at most k and supp( c i ) ⊂ Z × Z , we have that e f ( y n ) f ( y n ) c i = e f ( y n ) f ( y n ) c i χ B n for all n ∈ N and all i ∈ { , . . . , ℓ } . Therefore,as Φ( χ A n ) = χ B n , we have that (cid:13)(cid:13)(cid:13) e f ( y n ) f ( y n ) (cid:16) ℓ X i =1 c i Φ( a i ) (cid:17) e f ( x n ) f ( x n ) (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) ℓ X i =1 e f ( y n ) f ( y n ) c i χ B n Φ( a i ) e f ( x n ) f ( x n ) (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) ℓ X i =1 e f ( y n ) f ( y n ) c i Φ( χ A n )Φ( a i )Φ( e x n x n ) (cid:13)(cid:13)(cid:13) ≤ max i k c i k ℓ X i =1 k χ A n a i e x n x n k for all n ∈ N . By going to a subsequence, a simple pigeonhole argumentallow us to assume that, for some i ∈ { , . . . , ℓ } , we haveinf n ∈ N k χ A n a i e x n x n k > . By the definition of ( A n ) n and (2) above, we have that lim n d ( x n , A n ) = ∞ .This, together with the previous inequality, contradicts the fact that a i ∈ C ∗ u ( X ). (cid:3) Theorem 4.6.
Let X and Y be uniformly locally finite metric. The follow-ing are equivalent:1. There is a bijective coarse quotient map X → Y .2. There is an embedding Φ : C ∗ u ( X ) → C ∗ u ( Y ) so that Φ( ℓ ∞ ( X )) = ℓ ∞ ( Y ) and Φ(C ∗ u ( X )) is cobounded in C ∗ u ( Y ) .3. There is ε ∈ (0 , and an embedding Φ : C ∗ u ( X ) → C ∗ u ( Y ) so that Φ( ℓ ∞ ( X )) = ℓ ∞ ( Y ) and Φ(C ∗ u ( X )) is ε -almost cobounded in C ∗ u ( Y ) .Proof. (1) ⇒ (2) This follows from Proposition 4.2.(2) ⇒ (3) This is straightforward. (3) ⇒ (1) As Φ is a ∗ -homomorphism, each Φ( e xx ) is a projection. Hence,as the restriction Φ ↾ ℓ ∞ ( X ) : ℓ ∞ ( X ) → ℓ ∞ ( Y ) is an isomorphism, Φ( e xx )has rank 1 for each x ∈ X . Moreover, this isomorphism also implies that1 Y = SOT- P x ∈ X Φ( e xx ). So, the result now follows from Lemma 4.5 (cid:3) Proof of Theorem 1.2.
This follows from Theorem 4.6. (cid:3)
We are ready to prove Theorem 1.3 and Theorem 1.4. Since our proofsgive us slightly stronger results than the ones stated in Section 1, we needthe following more general version of Definition 1.1(3).
Definition 4.7.
Let X be a metric space and let A ⊂ A be a C ∗ -subalgebra of C ∗ u ( X ). We say that A is strongly-cobounded in A if thereis k ∈ N so that for all ε > b ∈ A , there are a , . . . , a k ∈ A , with k a i k ≤ k , and c , . . . , c k ∈ A , with prop( c i ) ≤ k and k c i k ≤ k , so that (cid:13)(cid:13)(cid:13) b − k X i =1 c i a i (cid:13)(cid:13)(cid:13) ≤ ε. We now prove the main theorem of this section. Theorem 1.3 and Theo-rem 1.4 will follow from it.
Theorem 4.8.
Let X and Y be u.l.f. metric spaces, and assume that Y has property A. Suppose there is an embedding Φ : C ∗ u ( X ) → C ∗ u ( Y ) and ahereditary C ∗ -subalgebra A of C ∗ u ( Y ) so that ( A, Φ( ℓ ∞ ( X ))) is a Roe Cartanpair and Φ(C ∗ u ( X )) is strongly-cobounded in A . Then there is Z ⊂ Y and abijective coarse quotient X → Z .Proof. Fix Φ : C ∗ u ( X ) → C ∗ u ( Y ) and A ⊂ C ∗ u ( Y ) as above. By hypothesis,( A, Φ( ℓ ∞ ( X ))) is a Roe Cartan pair, so Theorem 2.4 implies that thereis a u.l.f. metric space Z and an isomorphism Ψ : C ∗ u ( Z ) → A so thatΨ( ℓ ∞ ( Z )) = Φ( ℓ ∞ ( X )). Let Θ = Ψ − ◦ Φ, so Θ : C ∗ u ( X ) → C ∗ u ( Z ) is anembedding so that Θ( ℓ ∞ ( X )) = ℓ ∞ ( Z ). Claim 4.9.
Given ε >
0, Θ(C ∗ u ( X )) is ε -almost cobounded in C ∗ u ( Z ). Proof.
Let ε >
0. Fix k ∈ N which witness that Φ(C ∗ u ( X )) is strongly-cobounded in A and let δ = ε/ ( k + 1). As A is a hereditary subalgebra ofC ∗ u ( Y ), [BFV20a, Lemma 6.1] gives an isometric embedding u : ℓ ( Z ) → ℓ ( Y ) so that Ψ = Ad( u ). As Ad( u ∗ ) : C ∗ u ( Y ) → C ∗ u ( Z ) is compact-preserving and strongly continuous, Theorem 2.3 implies that Ad( u ∗ ) iscoarse-like. Notice that Ψ − = Ad( u ∗ ) ↾ A . So, coarse-likeness gives R > − ( c ) is δ - R -approximable for all contractions c ∈ A withprop( c ) ≤ k .Let b ∈ C ∗ u ( Z ). By our choice of k , there are a , . . . , a k ∈ C ∗ u ( X ), with k a i k ≤ k , and c , . . . , c k ∈ A , with prop( c i ) ≤ k and k c i k ≤ k , and so that (cid:13)(cid:13)(cid:13) Ψ( b ) − k X i =1 c i Φ( a i ) (cid:13)(cid:13)(cid:13) ≤ δ. OARSE QUOTIENTS AND UNIFORM ROE ALGEBRAS 17
For each i ∈ { , . . . , k } , let d i ∈ C ∗ u ( Z ) be so that prop( d i ) ≤ R and (cid:13)(cid:13) Ψ − ( c i ) − k c i k d i (cid:13)(cid:13) ≤ δ k c i k . Then (cid:13)(cid:13)(cid:13) b − k X i =1 k c i k d i Θ( a i ) (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) b − k X i =1 Ψ − ( c i Φ( a i )) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) k X i =1 Ψ − ( c i )Θ( a i ) − k X i =1 k c i k d i Θ( a i ) (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) Ψ( b ) − k X i =1 c i Φ( a i ) (cid:13)(cid:13)(cid:13) + k X i =1 (cid:13)(cid:13) Ψ − ( c i ) − k c i k d i (cid:13)(cid:13) · k a i k≤ δ + δk . By our choice of δ , we are done. (cid:3) As Θ( ℓ ∞ ( X )) = ℓ ∞ ( Z ), the previous claim and Theorem 4.6 imply thatthere is a bijective coarse quotient map f : X → Z . As Y has property A and A is a hereditary subalgebra of C ∗ u ( Y ), [BFV20a, Theorem 1.4] gives us aninjective coarse embedding g : Z → Y . By Proposition 3.1, g ◦ f : X → g ( Z )is a coarse quotient. So we are done. (cid:3) Proof of Theorem 1.3. If Y has property A, then C ∗ u ( Z ) is isomorphic toC ∗ u ( Y ) if and only if Z and Y are bijectively coarsely equivalent [WW18,Corollary 6.13]. Therefore, the result follows by running the proof of The-orem 4.8 for A = C ∗ u ( Y ), and using [WW18, Corollary 6.13] instead of[BFV20a, Theorem 1.4] in the last paragraph of the proof. Indeed, the map g : Z → Y obtained at the end of the proof of Theorem 4.8 becomes abijection, and so does the coarse quotient g ◦ f : X → Y (cid:3) Proof of Theorem 1.4.
If the sparse subspaces of Y yield only compact ghostprohections, then an isomorphism between C ∗ u ( Z ) and C ∗ u ( Y ) implies that Y and Z are coarsely equivalent (this follows form the proof of [BFV20a,Theorem 1.4 and Corollary 1.5]; equivalently, this is given by [BCL20, The-orem 1.3]). The result then follows by running the proof of Theorem 4.8 for A = C ∗ u ( Y ) and the result above instead of [BFV20a, Theorem 1.4] in thelast paragraph of the proof. (cid:3) Proof of Corollary 1.5.
This follows straightforwardly from Theorem 1.4 andCorollary 3.7 (cid:3) Characterization of spacially implemented embeddings
The study of embeddings between uniform Roe algebras is highly depen-dent on the embeddings being spacially implemented or strongly continuousand rank-preserving. In this section, we prove Theorem 5.3 and show that those two kinds of embedding coincide. We also give two other characteri-zation of such embeddings and prove a version of it for non rank-preservingstrongly continuous embeddings (see Theorem 5.5).For that, we need a result of [BFV20a]. For that, recall that given ametric space X , an operator a ∈ B ( ℓ ( X )) is called quasi-local if for all ε > S > d ( A, B ) > S implies k χ A aχ B k ≤ ε for all A, B ⊂ X . The norm closure of all quasi-local operators forms a C ∗ -algebracalled the quasi-local algebra of X ; and this algebra is denoted by C ∗ ql ( X ).Clearly, C ∗ u ( X ) ⊂ C ∗ ql ( X ) and it remains open whether this inclusion is anequality; when this is so, the metric space X is called quasi-local . We knowhowever that this is so if X has property A ([ˇSZ18, Theorem 3.3]).Before stating [BFV20a, Theorem 4.3], notice that, if Φ : C ∗ u ( X ) → C ∗ u ( Y )is an embedding, then (Φ( χ F )) F ⊂ X, | F | < ∞ is an increasing net of projections.Therefore, p = SOT- P x ∈ X Φ( e xx ) is well defined. Theorem 5.1 (Theorem 4.3 of [BFV20a]) . Let X and Y be u.l.f. metricspaces, and Φ : C ∗ u ( X ) → C ∗ u ( Y ) be an embedding. Then, the projection p = SOT - P x ∈ X Φ( e xx ) belong to C ∗ ql ( Y ) and the map a ∈ C ∗ u ( X ) p Φ( a ) p ∈ C ∗ ql ( Y ) is a strongly continuous embedding. Moreover, if Φ is compact-preserving,then p ∈ C ∗ u ( Y ) ; so the map above is a strongly continuous embedding into C ∗ u ( Y ) . Theorem 5.1 allows us to obtain the following characterization of strongcontinuity of embeddings between uniform Roe algebras.
Corollary 5.2.
Let X and Y be u.l.f. metric spaces, and Φ : C ∗ u ( X ) → C ∗ u ( Y ) be an embedding. Let p = SOT - P x ∈ X Φ( e xx ) . The following areequivalent:1. Φ is strongly continuous, and2. p = Φ(1 X ) .Moreover, if either Y is quasi-local or Φ is compact-preserving, then theitems above are also equivalent to:3. Φ( K ( ℓ ( X ))) is an essential ideal of the hereditary subalgebra of C ∗ u ( Y ) generated by Φ(C ∗ u ( X )) .Proof. (1) ⇒ (2) If Φ is strongly continuous, p = SOT- P x ∈ X Φ( e xx ) = Φ(1 X ).(2) ⇒ (1) This follows from Theorem 5.1.(2) ⇒ (3) Let a belong to the hereditary subalgebra of C ∗ u ( Y ) generatedby Φ(C ∗ u ( X )). If a Φ( K ( ℓ ( X ))) = 0, then a Φ( P x ∈ F e xx ) = 0 for all finite F ⊂ X . So ap = a Φ(1 X ) = 0. As Φ(1 X ) is the unit of the hereditarysubalgebra of C ∗ u ( Y ) generated by Φ(C ∗ u ( X )), it follows that a = 0. Hence,Φ( K ( ℓ ( X ))) is an essential ideal of Φ(C ∗ u ( X )).(3) ⇒ (2) By Theorem 5.1, if either Y is quasi-local or Φ is compact-preserving, then p ∈ C ∗ u ( Y ). Let q = Φ(1 X ) − p , so q ∈ C ∗ u ( Y ) and OARSE QUOTIENTS AND UNIFORM ROE ALGEBRAS 19 q ≤ Φ(1 X ). Hence q belongs to the hereditary subalgebra of C ∗ u ( Y ) gen-erated by Φ(C ∗ u ( X )). If a ∈ K ( ℓ ( X )), then a = lim n χ X n aχ X n , where( X n ) n is an increasing sequence of finite subsets of X so that X = S n X n .Then q Φ( a ) = lim n (cid:16) (Φ(1 X ) − p )Φ( χ X n )Φ( aχ X n ) (cid:17) = 0 . Therefore, by the arbitrariness of a and the fact that Φ( K ( ℓ ( X ))) is anessential ideal of the hereditary subalgebra of C ∗ u ( Y ) generated by Φ(C ∗ u ( X )),this implies that q = 0, i.e., p = Φ(1 X ). (cid:3) We point out that (3) of Corollary 5.2 cannot be weakened to “Φ( K ( ℓ ( X )))is an essential ideal of Φ(C ∗ u ( X ))”. We refer to Remark 5.4 for a further dis-cussion on that.The next result shows that spacially implemented embeddings and em-beddings which are both strongly continuous and rank-preserving coincide. Theorem 5.3.
Let X and Y be u.l.f. metric spaces and Φ : C ∗ ( X ) → C ∗ ( Y ) be an embedding. The following are equivalent:1. Φ is spacially implemented by an isometric embedding ℓ ( X ) → ℓ ( Y ) ,2. Φ is rank-preserving and strongly continuous,3. Φ(1 X ) K ( ℓ ( Y ))Φ(1 X ) is contained in Φ(C ∗ u ( X )) , and4. there is a subspace H ′ ⊂ H so that K ( H ′ ) ⊂ Φ(C ∗ u ( X )) and K ( H ′ ) isan essential ideal of the hereditary subalgebra generated by Φ(C ∗ u ( X )) .Proof. The implication (1) ⇒ (2) is straightforward. Moreover, if Φ = Ad( u )for an isometric embedding u : ℓ ( X ) → ℓ ( Y ), then u K ( ℓ ( X )) u ∗ = K (Im( u )) = uu ∗ K ( ℓ ( Y )) uu ∗ = Φ(1 X ) K ( ℓ ( Y ))Φ(1 X ) . So the implication (1) ⇒ (3) follows. As Φ(C ∗ u ( X )) ⊂ B (Im( u )), K (Im( u )) isan essential ideal of Φ(C ∗ u ( X )). So the implication (1) ⇒ (4) follows.(2) ⇒ (1) As Φ is rank-preserving, Φ( e xx ) is a rank 1 projection for each x ∈ X . So, for each x ∈ X , pick a normalized ξ x ∈ ℓ ( X ) so that Φ( e xx )is the projection onto span { ξ x } . Strong continuity gives that Φ(1 X ) =SOT- P x ∈ X Φ( e xx ). Hence, (Φ( e xx )) n is a maximal set of rank 1 projec-tions for the Hilbert space H ′ = Im(Φ(1 X )), so ( ξ n ) n is an orthonormalbasis for H ′ . As e xy ∈ C ∗ u ( X ), it follows that p xy = h· , ξ x i ξ y ∈ Φ(C ∗ u ( X )) forall x, y ∈ X . Hence, Φ(C ∗ u ( X )) contains K ( H ′ ).This implies that Φ( K ( ℓ ( X ))) is a nontrivial ideal of K ( H ′ ). So, theequality Φ( K ( ℓ ( X ))) = K ( H ′ ) must hold; hence there is a unitary u : ℓ ( X ) → H ′ so that Φ( a ) = uau ∗ for all a ∈ K ( ℓ ( X )) (see [Mur90, Theorem2.4.8]). On can easily check that Φ = Ad( u ) (cf. [ˇSW13b, Lemma 3.1]).(3) ⇒ (2) Let us notice that Φ( e xx ) has rank 1 for all x ∈ X . Indeed, ifthis is not the case for some x ∈ X , there is a rank 1 projection q < Φ( e xx ).As q = Φ(1 X ) q Φ(1 X ) ∈ Φ(C ∗ u ( X )), there is a projection p ∈ C ∗ u ( X ) withΦ( p ) = q . Then 0 < p < e xx ; contradiction.As Φ( e xx ) has rank 1 for all x ∈ X , we obtain that Φ( e xy ) has rank 1for all x, y ∈ X ; Φ( e xy ) is actually a partial isometry taking Im(Φ( e xy )) onto Im(Φ( e yy )). Therefore, Φ ↾ B ( ℓ ( F )) is rank-preserving for all finite F ⊂ X . As S F ⊂ X, | F | < ∞ B ( ℓ ( F )) is dense in the finite rank operators,it easily follows that Φ is rank-preserving. In particular, Φ is compact-preserving.As Φ( χ F ) ≤ Φ(1 X ) for all F ⊂ X , we have that p ≤ Φ(1 X ). If this isa strict inequality, pick a rank 1 projection p ≤ Φ(1 X ) − p . Then p =Φ(1 X ) p Φ(1 X ) ∈ Φ(C ∗ u ( X )). Pick a nonzero projection p ∈ C ∗ u ( X ) withΦ( p ) = p . Then k e xx p k = k Φ( e xx )Φ( p ) k ≤ k pp k = 0for all x ∈ X . So p = 0; contradiction. Then p = Φ(1 X ) and we haveΦ = Ad( p ) ◦ Φ. Hence, as Φ is compact-preserving, Theorem 5.1 impliesthat Φ is strongly continuous.(4) ⇒ (3) Fix H ′ ⊂ H as in the hypothesis and denote by A the hereditaryC ∗ -algebra generated by Φ(C ∗ u ( X )). We claim that Φ(1 X ) = 1 H ′ . Indeed, as K ( H ′ ) ⊂ Φ(C ∗ u ( X )), all finite rank projections q ∈ K ( H ′ ) are bellow Φ(1 X ).Hence, 1 H ′ ≤ Φ(1 X ). Let q = Φ(1 X ) − H ′ , so q ∈ A . As K ( H ′ ) q = 0 and K ( H ′ ) is an essential ideal of A , q = 0.This shows thatΦ(1 X ) K ( ℓ ( Y ))Φ(1 X ) = 1 H ′ K ( ℓ ( Y ))1 H ′ = K ( H ′ ) ⊂ Φ(C ∗ u ( X )) , so we are done. (cid:3) Remark . As noticed in [BFV20a, Proposition 4.1], there are embeddingsbetween uniform Roe algebras which are not strongly continuous. More-over, the embedding can also be taken to be rank-preserving. We recall theexample: Let X = { n | n ∈ N } and let U be a nonprincipal ultrafilter on X . Let Φ : C ∗ u ( X ) → C ∗ u ( X ) be the identity and letΦ ( χ A ) = (cid:26) χ A , if A ∈ U , , if A
6∈ U . The map Φ extends to a ∗ -homomorphism Φ : C ∗ u ( X ) → C ∗ u ( X ) whichsends K ( ℓ ( X )) to zero. The mapΦ = Φ ⊕ Φ : a ∈ C ∗ u ( X ) Φ ( a ) ⊕ Φ ( a ) ∈ C ∗ u ( X ) ⊕ C ∗ u ( X )is a rank-preserving embedding which is not strongly continuous. Moreover,as C ∗ u ( X ) ⊕ C ∗ u ( X ) ⊂ C ∗ u ( X ⊔ X ) (where X ⊔ X is any metric space whosemetric restricted to both copies of X coincide with X ’s original metric), thismap can be considered as a map C ∗ u ( X ) → C ∗ u ( X ⊔ X ).Notice that K ( ℓ ( X )) ⊕ { } is an essential ideal of Φ(C ∗ u ( X )). Therefore,this example shows that (4) cannot be weakened to “there is a subspace H ′ ⊂ H so that K ( H ′ ) is an essential ideal of Φ(C ∗ u ( X ))”.Given n ∈ N , an operator between operator algebras is called 1 -to- n rank-preserving if it sends operators of rank 1 to operators of rank n . We nowlook at strongly continuous 1-to- n rank-preserving maps between uniformRoe algebras. OARSE QUOTIENTS AND UNIFORM ROE ALGEBRAS 21
Given a metric space X and n ∈ N ∪ {∞} , we have a canonical embedding I n = I X,n : C ∗ u ( X ) → C ∗ u ( X ) ⊗ B ( C n ). Precisely, I n : a = [ a xy ] SOT- lim F ⊂ X, | F | < ∞ X x,y ∈ F a xy e xy ⊗ Id n ∈ C ∗ u ( X ) ⊗ B ( C n )(if n = ∞ , C n is assume to be ℓ ). In other words, identifying C ∗ u ( X ) ⊗B ( C n )with a subalgebra of B ( ℓ ( X, C n )) in the standard way, and thinking aboutoperators on B ( ℓ ( X, C n )) as X -by- X matrices with entries in B ( C n ), wehave that I n ([ a xy ]) = [ a xy Id n ] for all a = [ a xy ] ∈ C ∗ u ( X ). The map I n isclearly strongly continuous.We now prove the n -version of Theorem 5.3. Theorem 5.5.
Let X and Y be u.l.f. metric spaces and Φ : C ∗ u ( X ) → C ∗ u ( Y ) be an embedding. Given n ∈ N ∪ {∞} , the following are equivalent.1. There is an isometric embedding u : ℓ ( X, C n ) → ℓ ( Y ) so that Φ =Ad( u ) ◦ I n .2. Φ is -to- n rank-preserving and strongly continuous.Proof. As I n is 1-to- n rank-preserving and Ad( u ) : B ( ℓ ( X, C n )) → B ( ℓ ( Y ))is rank-preserving, (1) ⇒ (2) follows.(2) ⇒ (1) Fix x ∈ X . As Im(Φ( e x x )) has dimension n , fix a surjetiveisometry v : C n → Im(Φ( e x x )). Notice that Φ( e x x ) is a partial isome-try which takes Im(Φ( e x x )) isometrically onto Im(Φ( e xx )), for all x ∈ X .Define an isometric embedding u : ℓ ( X, C n ) → ℓ ( Y ) by letting uδ x ⊗ ξ = Φ( e x x ) vξ for all x ∈ X and all ξ ∈ C n . So, u ↾ ℓ ( { x } , C n ) is a surjective isometry be-tween ℓ ( { x } , C n ) and Im(Φ( e xx )) for all x ∈ X . Moreover, as Φ is stronglycontinuous, u is an isometry onto H = span { Im(Φ( e xx )) | x ∈ X } = Im(Φ(1 X )) . In particular, u ∗ [ H ⊥ ] = 0.Let us notice that Φ = Ad( u ) ◦ I n . As Φ is strongly continuous, it is enoughto show that Φ( e xy ) = Ad( u ) ◦ I n ( e xy ) for all x, y ∈ X . Fix x, y ∈ X . As u ∗ [ H ⊥ ] = 0 and Φ( e xy ) ξ = 0 for all ξ ∈ H ⊥ , it is enough to show thatΦ( e xy ) ξ = Ad( u ) ◦ I n ( e xy ) ξ for all ξ ∈ H .Fix ξ ∈ H . As H = (cid:16) M x ∈ X Im(Φ( e xx )) (cid:17) ℓ , write ξ = ( ξ x ) x ∈ X where ξ x ∈ Im(Φ( e xx )) for each x ∈ X . If z = x ,Φ( e xx ) ξ z = 0 as Φ( e xx ) and Φ( e zz ) are orthogonal projections. As u ∗ ( ξ z ) ∈ ℓ ( { z } , C n ), we also have thatAd( u ) ◦ I n ( e xy ) ξ z = ue xy ⊗ Id n u ∗ ξ z = 0 . Hence, we only need to notice that Φ( e xy ) ξ x = Ad( u ) ◦ I n ( e xy ) ξ x . Thisfollows since the formula of u gives that ue xy ⊗ Id n u ∗ ξ z = Φ( e x y ) vv ∗ Φ( e xx ) ξ x = Φ( e x y )Φ( e xx ) ξ x = Φ( e xy ) ξ x , so we are done. (cid:3) Questions
We finish the paper with a selection of natural questions left unsolved.Firstly, when working with quotient maps, it is natural to look at sectionsof such maps. Recall, given a surjection f : X → Y , we say that a map g : Y → X is a section of f if f ◦ g = Id Y . In the coarse category, the notionof coarse-section is more natural: precisely, g : Y → X is a coarse-section of f if it is coarse and f ◦ g ∼ Id Y . The following is straightforward. Proposition 6.1.
Let f : X → Y be a coarse quotient between metricspaces. If g : Y → X is a section of f which is coarse, then g : Y → g ( Y ) is a bijective coarse equivalence. If g : Y → X is a coarse-section of f , then g : Y → g ( Y ) is a coarse equivalence. (cid:3) Finding conditions for the existence of a coarse-section of a given coarsequotient would be interesting. More generally:
Problem 6.2.
Let X and Y be u.l.f. metric spaces and assume that thereis a uniformy finite-to-one coarse quotient X → Y . Does Y coarsely embedinto X ? While Theorem 1.2 gives us an embedding with cobounded range, The-orem 1.3 assumes strong-coboundedness on the embedding. Without as-suming some further structure on the metric space X , we do not know if abijective coarse quotient is enough to obtain an embeddings with strongly-cobounded range (see Remark 4.3). Problem 6.3.
Let X and Y be u.l.f. metric spaces, and f : X → Y be abijective coarse quotient map. Is there an embedding Φ : C ∗ u ( X ) → C ∗ u ( Y ) with strongly-cobounded range so that Φ( ℓ ∞ ( X )) is a Cartan subalgebra of C ∗ u ( Y ) ? Although Corollary 3.7 shows that the existence of a uniformly finite-to-one coarse quotient X → Y gives us that Y has finite asymptotic dimensionprovided that the same holds for X , we do not have a precise bounded forasydim( Y ). Problem 6.4.
Say X and Y are u.l.f. metric spaces, and assume thatthere is a uniformly finite-to-one coarse map X → Y . Does it follows that asydim( Y ) ≤ asydim( X ) ? It is known that, for proper metric spaces X with asydim( X ) < ∞ , wehave that asydim( X ) = dim( νX ), where νX is the Higson corona of X (see[Dra00, Theorem 6.2] for this result and [Roe03, Subsection 2.3] for details OARSE QUOTIENTS AND UNIFORM ROE ALGEBRAS 23 on the Higson corona). Hence, in order to give a positive answer to Problem6.4, it would be enough to give a positive answer to the following problem:
Problem 6.5.
Let X and Y be u.l.f. metric spaces and assume that there isa bijective coarse quotient X → Y . Does it follow that dim( νX ) = dim( νY ) ? A positive answer to the next problem would be very useful in order toextend the current rigidity results in the literature to non rank-reservingembeddings C ∗ u ( X ) → C ∗ u ( Y ). Problem 6.6.
Can v : C n → Im(Φ( e x x )) in the proof of Theorem 5.5 bechosen so that Ad( u )(C ∗ u ( X ) ⊗ M n ( C )) ⊂ C ∗ u ( Y ) ? On a different direction, but still trying to better understand non rank-reserving embeddings C ∗ u ( X ) → C ∗ u ( Y ), a version of Theorem 5.1 would bevery interesting. Problem 6.7.
Let X and Y be u.l.f. metric spaces, and Φ : C ∗ u ( X ) → C ∗ u ( Y ) be an embedding. Is there a projection p ∈ C ∗ u ( Y ) so that the map a ∈ C ∗ u ( X ) p Φ( a ) p ∈ C ∗ ql ( Y ) is a rank-preserving embedding? References [BCL20] B. M. Braga, Y.-C. Chung, and K. Li,
Coarse Baum-Connes conjecture andrigidity for Roe algebras , Journal of Functional Analysis, Volume 279, Issue 9,15 November 2020. doi: https://doi.org/10.1016/j.jfa.2020.108728.[BF20] B. M. Braga and I. Farah,
On the rigidity of uniform Roe algebras over uni-formly locally finite coarse spaces , To appear in the Transactions of the Amer-ican Mathematical Society (2020). doi: https://doi.org/10.1090/tran/8180[BFV20a] B. M. Braga, I. Farah, and A. Vignati,
Embeddings of uniform roe algebras ,Commun. Math. Phys. (2020), 1853–1882.[BFV20b] ,
General uniform Roe algebra rigidity , arXiv e-prints (2020),arXiv:2001.10266.[BJL +
99] S. Bates, W. B. Johnson, J. Lindenstrauss, D. Preiss, and G. Schechtman,
Affine approximation of Lipschitz functions and nonlinear quotients , Geom.Funct. Anal. (1999), no. 6, 1092–1127. MR 1736929[BZ16] F. Baudier and S. Zhang, ( β ) -distortion of some infinite graphs , J. Lond. Math.Soc. (2) (2016), no. 2, 481–501. MR 3483124[Dra00] A. N. Dranishnikov, Asymptotic topology , Uspekhi Mat. Nauk (2000),no. 6(336), 71–116. MR 1840358[DV16] J. Dydak and ˇZ. Virk, Preserving coarse properties , Rev. Mat. Complut. (2016), no. 1, 191–206. MR 3438031[DZ14] A. Dranishnikov and M. Zarichnyi, Asymptotic dimension, decompositioncomplexity, and Haver’s property C , Topology Appl. (2014), 99–107.MR 3199862[EM19] E. Ewert and R. Meyer,
Coarse Geometry and Topological Phases , Comm.Math. Phys. (2019), no. 3, 1069–1098. MR 3927086[HW19] L. Higginbotham and Th. Weighill,
Coarse quotients by group actions and themaximal Roe algebra , J. Topol. Anal. (2019), no. 4, 875–907. MR 4040015[Kub17] Y. Kubota, Controlled topological phases and bulk-edge correspondence , Comm.Math. Phys. (2017), no. 2, 493–525. MR 3594362 [Mur90] G. Murphy, C ∗ -algebras and operator theory , Academic Press, Inc., Boston,MA, 1990. MR 1074574[MV13] T. Miyata and ˇZ. Virk, Dimension-raising maps in a large scale , Fund. Math. (2013), no. 1, 83–97. MR 3125134[NY12] P. Nowak and G. Yu,
Large scale geometry , EMS Textbooks in Mathematics,European Mathematical Society (EMS), Z¨urich, 2012. MR 2986138[Roe88] J. Roe,
An index theorem on open manifolds. I, II , J. Differential Geom. (1988), no. 1, 87–113, 115–136. MR 918459[Roe93] , Coarse cohomology and index theory on complete Riemannian mani-folds , Mem. Amer. Math. Soc. (1993), no. 497, x+90. MR 1147350[Roe03] ,
Lectures on coarse geometry , University Lecture Series, vol. 31, Amer-ican Mathematical Society, Providence, RI, 2003. MR 2007488[RW14] J. Roe and R. Willett,
Ghostbusting and property A , J. Funct. Anal. (2014),no. 3, 1674–1684. MR 3146831[ˇSW13a] J. ˇSpakula and R. Willett,
On rigidity of Roe algebras , Adv. Math. (2013),289–310. MR 3116573[ˇSW13b] ,
On rigidity of Roe algebras , Adv. Math. (2013), 289–310.MR 3116573[ˇSZ18] J. ˇSpakula and J. Zhang,
Quasi-Locality and Property A , arXiv:1809.00532(2018).[WW18] S. White and R. Willett,
Cartan subalgebras of uniform Roe algebras ,arXiv:1808.04410 (2018).[Zha15] S. Zhang,
Coarse quotient mappings between metric spaces , Israel J. Math. (2015), no. 2, 961–979. MR 3359724[Zha18] ,
Asymptotic properties of Banach spaces and coarse quotient maps ,Proc. Amer. Math. Soc. (2018), no. 11, 4723–4734. MR 3856140
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