A Gelfand-type duality for coarse metric spaces with property A
aa r X i v : . [ m a t h . OA ] J u l A GELFAND-TYPE DUALITY FOR COARSE METRICSPACES WITH PROPERTY A
BRUNO M. BRAGA AND ALESSANDRO VIGNATI
Abstract.
We prove the following two results for a given uniformlylocally finite metric space with Yu’s property A:1. The group of outer automorphisms of its uniform Roe algebra isisomorphic to its group of bijective coarse equivalences modulocloseness.2. The group of outer automorphisms of its Roe algebra is isomorphicto its group of coarse equivalences modulo closeness.The main difficulty lies in the latter. To prove that, we obtain severaluniform approximability results for maps between Roe algebras and usethem to obtain a theorem about the ‘uniqueness’ of Cartan masas of Roealgebras. We finish the paper with several applications of the resultsabove to concrete metric spaces. Introduction
Given the class of metric spaces, consider the following three kinds ofmorphisms: (1) homeomorphisms — maps preserving the topological struc-ture —, (2) coarse equivalences — maps preserving the large-scale geometry— and (3) bijective coarse equivalences. In short, coarse equivalences uni-formly send close points to close points, far points to far points, and havelarge image in their codomain. Although, the set of homeomorphisms ofa metric space forms a group under composition, this is not the case forcoarse equivalences. Indeed, coarse equivalences need to be neither injectivenor surjective. However, the set of coarse equivalences on a metric space(
X, d ) becomes a group after identifing coarse equivalences which are close to each other (see Definition 2.2). We denote by Coa( X ) the group of allcoarse equivalences of X modulo the closeness relation and by BijCoa( X )the group of all bijective coarse equivalences of X modulo closeness (we referthe reader to § X is locally compact, homeomorphisms correspond, thanks toGelfand’s transform, to automorphisms of the C ∗ -algebra of continuous func-tions on X vanishing at infinity, C ( X ). The goal of this paper is to give anoperator algebraic characterisation of the groups Coa( X ) and BijCoa( X ),at least when dealing with metric spaces with certain regularity properties.The objects apt to this coarse Gelfand-type correspondence are Roe-type Date : July 29, 2020.2010
Mathematics Subject Classification. C ∗ -algebras. These C ∗ -algebras were introduced by Roe in [16] for their con-nections to (higher) index theory and the associated applications to manifoldtopology and geometry ([17]). The Roe algebra and its uniform version werestudied precisely to detect C ∗ -algebraically the large-scale geometry of met-ric spaces. Their study was boosted due to their intrinsic relation with thecoarse Baum-Connes conjecture and consequently with the coarse Novikovconjecture ([25]). Recently, Roe-type algebras and their K -theory have beenused as a framework in mathematical physics to study the classification oftopological phases and the topology of quantum systems ([10, 12]).We now describe our main results. Given a metric space ( X, d ) and aHilbert space H , ℓ ( X, H ) denotes the Hilbert space of square summable H -valued functions on X . Operators in B ( ℓ ( X, H )) can be seen as X × X -matrices whose entries are in B ( H ). Given an operator a = ( a xy ) x,y ∈ X in B ( ℓ ( X, H )), we define its propagation as the quantityprop( a ) = sup { d ( x, y ) | a xy = 0 } . If H is separable and infinite-dimensional, the C ∗ -algebra of band-dominatedoperators of ( X, d ), denoted by BD( X ), is the norm closure of the ∗ -algebraof finite propagation operators. If in addition we demand each entry a xy tobe compact, we obtain the Roe algebra of X , C ∗ ( X ). Finally, if H = C , the uniform Roe algebra of X , C ∗ u ( X ), is defined once again as the norm closureof the ∗ -algebra of finite propagation operators on ℓ ( X ) = ℓ ( X, C ) (see § X ). A bijective coarse equiv-alence of X induces an automorphism of C ∗ u ( X ) in a canonical way, andtwo bijective coarse equivalences are close if and only if the associated iso-morphisms are unitarily equivalent in C ∗ u ( X ) (see § X ) into Out(C ∗ u ( X )),the latter being the group of outer automorphisms of C ∗ u ( X ), i.e.,Out(C ∗ u ( X )) = Aut(C ∗ u ( X )) / Inn(C ∗ u ( X )) . Problem A (Gelfand-type duality for bijective coarse equivalences) . Let X be a uniformly locally finite metric space. Is the canonical homomorphismBijCoa( X ) → Out(C ∗ u ( X ))a group isomorphism?The work of White and Willett on uniqueness of Cartan masas in uniformRoe algebras in presence of Yu’s property A can be used to give a positiveanswer to Problem A (again in the presence of property A). The following,proven as Theorem 2.4, is a consequence of [23, Theorem E]: Theorem A.
Let (
X, d ) be a uniformly locally finite metric space withproperty A. The canonical homomorphismBijCoa( X ) → Out(C ∗ u ( X ))is a group isomorphism. GELFAND-TYPE DUALITY FOR COARSE METRIC SPACES 3
Theorem A gives an alternative way to compute the outer automorphismgroup of a uniform Roe algebra. As a simple application, it can be used toshow that all automorphisms of C ∗ u ( N ) are inner and that Out(C ∗ u ( Z )) ∼ = Z (see Corollary 6.3).Let us now focus on the case of coarse equivalences. Although coarseequivalences do not induce uniform Roe algebra isomorphisms (for instance,all finite metric spaces are coarsely equivalent, but, if X and Y are finite,C ∗ u ( X ) and C ∗ u ( Y ) are isomorphic if and only if | X | = | Y | ), they do induceisomorphisms between Roe algebras. If X is a uniformly locally finite metricspace, assigning an element of Aut(C ∗ ( X )) to a coarse equivalence of X ishighly non-canonical (see § X ) to Aut(C ∗ ( X )) modulo Inn(BD( X )) —notice that there are a couple of hidden claims in here: (1) we are allowed tomod out our maps and (2) Inn(BD( X )) is a normal subgroup of Aut(C ∗ ( X ))(the latter follows since we prove that BD( X ) is the multiplier algebra ofC ∗ ( X ), see Theorem 4.1). We form the outer automorphism group of C ∗ ( X )by letting Out(C ∗ ( X )) = Aut(C ∗ ( X )) / Inn(BD( X )) . Problem B (Gelfand-type duality for coarse equivalences) . Let X be auniformly locally finite metric space. Is the canonical homomorphismCoa( X ) → Out(C ∗ ( X ))a group isomorphism?We give a positive answer to Problem B above in the case of property A.This is proven as Theorem 5.13. Theorem B.
Let (
X, d ) be a uniformly locally finite metric space withproperty A. The canonical homomorphismCoa( X ) → Out(C ∗ ( X ))is a group isomorphism.Computing Coa( X ) is in general a very difficult task, even for a simplespace such as Z . However, using results present in the literature, TheoremB gives us some interesting applications. For instance, Out(C ∗ ( Z )) containsisomorphic copies of Thompson’s group F and of the free group of rankcontinuum (see Corollary 6.6). Using results of Eskin, Fisher and Whyte([9]), and of Farb and Mosher ([11]), we obtain a complete computationof Out(C ∗ ( X )) for solvable Baumslag-Solitar groups, and for lamplightergraphs F ≀ Z , where F is a finite group (see Corollaries 6.8 and 6.9).For our main results (Theorem A and B), we assume Yu’s property A. Thisis one of the best known regularity properties in the setting of coarse spaces.It is equivalent to many algebraic and geometric properties such as thenon-existence of noncompact ghost operators in C ∗ u ( X ) ([18, Theorem 1.3]),nuclearity of C ∗ u ( X ) ([6, Theorem 5.5.7]), and the operator norm localisationproperty, ONL ([19, Theorem 4.1]) — the latter is in fact the formulation of B. M. BRAGA AND A. VIGNATI property A we use in our proofs. Property A is fairly broad: for example,all finitely generated exact groups have property A (more precisely, theirCayley graphs, when endowed with the shortest path metric, have propertyA). In particular, this includes the classes of linear groups, groups with finiteasymptotic dimension and amenable groups.A key step in the proof of Theorem B is to show ‘uniqueness of Cartanmasas’ in Roe algebras, generalising [23, Theorem E] to Roe algebras. Inthe case of uniform Roe algebras, the canonical Cartan masa of C ∗ u ( X ) is ℓ ∞ ( X ). In C ∗ ( X ), there are many canonical Cartan masas which dependon the choice of an orthonormal basis of H . Precisely, if ¯ ξ = ( ξ n ) n is anorthonormal basis of H , we obtain a Cartan masa of C ∗ ( X ) by consideringall operators a ∈ B ( ℓ ( X, H )) such that for all x ∈ X there is ( λ n ) n ∈ c forwhich a ( δ x ⊗ ξ n ) = λ n ( δ x ⊗ ξ n ) for all n ∈ N . This masa is isomorphic to ℓ ∞ ( c ) and we denote it by ℓ ∞ ( X, ¯ ξ ). Theorem C.
Let X be a uniformly locally finite metric space with propertyA, and let ¯ ξ and ¯ ζ be orthonormal bases of the separable Hilbert space H .If Φ ∈ Aut(C ∗ ( X )), then there is a unitary v ∈ BD( X ) such thatAd( v ) ◦ Φ( ℓ ∞ ( X, ¯ ξ )) = ℓ ∞ ( X, ¯ ζ ) . Theorem C is proven below as Theorem 5.10. We point out that this resultis actually stronger: the unitary v ∈ BD( X ) is chosen so that Ad( v ) ◦ Φ :C ∗ ( X ) → C ∗ ( X ) respects the coarse geometry of X in a very strong sense(see Definition 3.1 and Theorem 5.10 for details).To prove Theorem C, and consequently Theorem B, we state and proveuniform approximability results for Roe algebras. These results ensure thatcertain maps between Roe algebras respect in some sense the coarse ge-ometry of the metric spaces. The concept of uniform approximability wasintroduced in [3], and then further developed in [5] and [4], for maps be-tween uniform Roe algebras. These tools were already applied for a betterunderstanding of uniform Roe algebras (e.g., [13, 23]), and we believe ourgeneralisations can be the key to a further development of the theory of Roealgebras.The paper is structured as follows: § §
3, and appliedin § §
5, where our main theorems are proved. § Notation and preliminaries If H is a Hilbert space, B ( H ) denotes the space of bounded operatorson H and K ( H ) its ideal of compact operators. We denote the identity of B ( H ) by 1 H . Given a set X , we denote by ℓ ( X, H ) the Hilbert space ofsquare-summable functions X → H . If x ∈ X and η ∈ H , δ x ⊗ η is thefunction that sends x to η and all other elements of X to 0. Elements in B ( ℓ ( X, H )) can be viewed as (bounded) X × X matrices where each entry GELFAND-TYPE DUALITY FOR COARSE METRIC SPACES 5 is an operator in B ( H ). With this identification, given a ∈ B ( ℓ ( X, H )), and x, y ∈ X , we denote by a xy the operator in B ( H ) determined by h a xy ξ, η i = h a ( δ x ⊗ ξ ) , δ y ⊗ η i , for all ξ, η ∈ H. Multiplication in B ( ℓ ( X, H )) corresponds to matrix multiplication, that is,( ab ) xy = X z ∈ X a xz b zy . We make the following abuse of notation throughout this article: given ametric space X and a Hilbert space H , we write χ C to denote both1. the projection in ℓ ∞ ( X ) ⊂ B ( ℓ ( X )) defined by χ C δ x = δ x if x ∈ C and χ C δ x = 0 if x C , and2. the projection in ℓ ∞ ( X, B ( H )) defined by χ C δ x ⊗ ξ = δ x ⊗ ξ if x ∈ C and χ C δ x ⊗ ξ = 0 if x C .If χ C denotes a projection in ℓ ∞ ( X ), when considering elements of B ( ℓ ( X, H )), χ C will always be accompanied by an operator on H — for instance: χ C ⊗ p ,where p ∈ B ( H ). Given a ∈ B ( ℓ ( X, H )) and x, y ∈ X , the operator a xy can be identified with the standard restriction of χ { x } aχ { y } to an operatorin B ( H ).Let ( X, d ) be a metric space. For a ∈ B ( ℓ ( X, H )), the support of a isdefined by supp( a ) = { ( x, y ) ∈ X | a xy = 0 } and its propagation byprop( a ) = sup { d ( x, y ) | ( x, y ) ∈ supp( a ) } . If H is infinite-dimensional and separable, we construct the band-dominatedalgebra of X , denoted BD( X ), by lettingBD( X ) = { a ∈ B ( ℓ ( X, H )) | prop( a ) is finite } . The
Roe algebra is the ideal of BD( X ) given by locally compact operatorsi.e., a xy is compact for all x, y ∈ X . SoC ∗ ( X ) = { a ∈ BD( X ) | ∀ x, y ∈ X, a xy ∈ K ( H ) } . If H = C , all band-dominated operators are locally compact, as K ( H ) = B ( H ). In this case, the algebra of band-dominated operators is called the uniform Roe algebra of X and denoted by C ∗ u ( X ).The algebra C ∗ ( X ) is not unital, but it has a well-behaved approximateidentity if X is uniformly locally finite. A metric space ( X, d ) is uniformlylocally finite ( u.l.f. ) if sup x ∈ X | B r ( x ) | < ∞ for all r > , where | B r ( x ) | denotes the cardinality of the closed ball of radius r centered at x . Let ( p j ) j be a sequence of finite rank projections on H which converges tothe identity 1 H in the strong operator topology. Given a function f : X → N ,let q f = SOT- X x ∈ X χ { x } ⊗ p f ( x ) . B. M. BRAGA AND A. VIGNATI
Each q f is in C ∗ ( X ), and q f ≤ q g if and only if for all x ∈ X we have f ( x ) ≤ g ( x ). Proposition 2.1.
Let X be u.l.f. metric space. The net { q f | f : X → N } is an approximate identity for C ∗ ( X ) consisting of projections.Proof. Pick a ∈ C ∗ ( X ) with prop( a ) ≤ r , and let ε >
0. Since X is u.l.f.and r is fixed, we can find a and a in C ∗ ( X ) with propagation at most r and finite sets A n ⊆ X , for n ∈ N , such that • a = a + a , • supp( a ) ∩ supp( a ) = ∅ , • a = P n χ A n a χ A n and a = P n χ A n +1 a χ A n +1 , and • if | i − j | ≥ A i ∩ A j = ∅ .Since each A i is finite and each entry of χ A i a is compact, there is a sequenceof natural numbers ( n ( i )) i such that (cid:13)(cid:13) ( χ A i ⊗ p n ( i ) ) a − χ A i a (cid:13)(cid:13) < ε for all i ∈ N . Define f : X → N by letting f ( x ) = n ( i ) if x ∈ A i , and f ( x ) = 0 elsewhere. Then k q f a − a k < ε . Notice that if f ≤ f , then q f q f = g f , hence k q f a − a k ≤ k q f a − q f q f a k + k q f q f a − a k ≤ ε. One can analogously define f : X → X such that k q f a − a k < ε . Then k q g a − a k ≤ ε for all g : X → N with g ≥ max { f , f } . Since ε wasarbitrary, we are done. (cid:3) By its definition, the Roe algebra C ∗ ( X ) is an essential ideal in BD( X ),hence BD( X ) ⊆ M (C ∗ ( X )) (see e.g., [1, II.7.3.5]). We will prove that thisis an equality in Theorem 4.1.2.1. Coarse geometry.
In the setting of coarse geometry, homeomorphismsare replaced by maps preserving the large-scale geometry. This is a crucialdefinition we will be using in the whole paper.
Definition 2.2.
Let (
X, d ) and (
Y, ∂ ) be metric spaces, and let f and g befunctions X → Y . The map f is called coarse if for all r > s > d ( x, y ) < r implies ∂ ( f ( x ) , f ( y )) < s. We say that f and g are close ifsup x ∂ ( f ( x ) , g ( x )) < ∞ . The map f is called a coarse equivalence if it is coarse and there exists acoarse map h : Y → X so that f ◦ h and h ◦ f are close to Id Y and Id X ,respectively. GELFAND-TYPE DUALITY FOR COARSE METRIC SPACES 7
Notice that a coarse equivalence f : X → Y is automatically cobounded ,i.e., sup y ∈ Y ∂ ( y, f ( X )) < ∞ .Throughout the paper, every time X and Y are metric spaces, we willassume without further notice that d and ∂ are the metrics of X and Y ,respectively.2.2. The canonical maps.
We now present the construction of the canon-ical map associating an automorphism of C ∗ u ( X ) to a bijective coarse equiv-alence of X . We then prove Theorem A. Finally, we associate an automor-phism of C ∗ ( X ) to a coarse equivalence of X . Although such an associationis highly noncanonical we prove that it becomes canonical when Aut(C ∗ ( X ))is replaced by Out(C ∗ ( X )).Let f : X → X be a bijective coarse equivalence. Consider the unitary v f ∈ B ( ℓ ( X )) given by v f δ x = δ f ( x ) for all x ∈ X . Since f and its inverseare coarse, Ad( v f ) is an automorphism of C ∗ u ( X ). If f and g are bijections,then v f ◦ g = v f v g and v f − = v ∗ f . Moreover, v f ∈ C ∗ u ( X ) if and only if f isclose to the identity; this gives an injective homomorphism BijCoa( X ) → Out(C ∗ u ( X )). The proof of Theorem A amounts then to show surjectivityin case property A is assumed. To do so, we recall the work of White andWillett, who proved uniqueness of Cartan masas in uniform Roe algebrasfor property A spaces. We slightly restate their result. Theorem 2.3 (Theorem E of [23]) . Let X be a u.l.f. metric space withproperty A . Let Φ ∈ Aut(C ∗ u ( X )) . Then there is a unitary u ∈ C ∗ u ( X ) suchthat Ad( u ) ◦ Φ( ℓ ∞ ( X )) = ℓ ∞ ( X ) . We restate Theorem A for convenience.
Theorem 2.4.
Let X be a u.l.f. metric space with property A. The canonicalmap BijCoa( X ) → Out(C ∗ u ( X )) is a group isomorphism.Proof. If Φ ∈ Aut(C ∗ u ( X )), then Theorem 2.3 gives a unitary u ∈ C ∗ u ( X )so that Ψ = Ad( u ) ◦ Φ is an automorphism of C ∗ u ( X ) which takes ℓ ∞ ( X )to itself. As every automorphism of C ∗ u ( X ) is implemented by a unitary in B ( ℓ ( X )) (see [21, Lemma 3.1]), let v be this unitary, i.e., Ψ = Ad( v ). As vℓ ∞ ( X ) v ∗ = ℓ ∞ ( X ), there is a bijection f : X → X and a family ( λ x ) x ∈ X inthe unit circle of C so that vδ x = λ x δ f ( x ) for all x ∈ X (see [3, Lemma 8.10]for a proof of that). Hence, Ad( v f ) equals Ψ modulo Inn(C ∗ u ( X )), which inturn, as u ∈ C ∗ u ( X ), equals Φ modulo Inn(C ∗ u ( X )). (cid:3) We now present a map which associates to a coarse equivalence of X anautomorphism of C ∗ ( X ). This construction is well known to specialists but,as we use its specifics in the proof of Theorem B, we give its details in full.Fix metric spaces X and Y , two orthonormal bases of H , ¯ ξ = ( ξ n ) n and¯ ζ = ( ζ n ) n , and let f : X → Y be a coarse equivalence. Let Y = f [ X ]. Foreach y ∈ Y , pick x y ∈ X such that f ( x y ) = y , and let X = { x y | y ∈ Y } . B. M. BRAGA AND A. VIGNATI
Since f is a coarse equivalence, X and Y are cobounded in X and Y ,respectively. By the last statement, we can pick sequences of disjoint finitesets ( X x ) x ∈ X and ( Y y ) y ∈ Y , and r >
0, so that1. X = F x ∈ X X x and Y = F x ∈ Y Y y ,2. x ∈ X x and diam( X x ) ≤ r for all x ∈ X , and3. y ∈ Y y and diam( Y y ) ≤ r for all all y ∈ Y .For each x ∈ X , pick a bijection g x : X x × N → Y f ( x ) × N . Define g = [ x ∈ X g x . So g is a bijection between X × N and Y × N . Let g : X × N → Y and g : X × N → N be the compositions of g with the projections onto the firstand second coordinates, respectively. If x ∈ X , we write g ( x, N ) for the set { g ( x, n ) | n ∈ N } .Define a unitary u = u g : ℓ ( X, H ) → ℓ ( Y, H ) by uδ x ⊗ ξ n = δ g ( x,n ) ⊗ ζ g ( x,n ) for all ( x, n ) ∈ X × N . For a ∈ B ( ℓ ( X, H )), define Ψ : B ( ℓ ( X, H )) →B ( ℓ ( Y, H )) by Ψ( a ) = uau ∗ . Notice that Ψ maps locally compact elements to locally compact elements.We intend to show that the image of C ∗ ( X ) is in C ∗ ( Y ). Claim 2.5. If a has finite propagation so does Ψ( a ). Proof.
Fix r > s ′ such that if d ( z, z ′ ) ≤ r +2 r then ∂ ( f ( z ) , f ( z ′ )) ≤ s ′ whenever z and z ′ are in X . This exists as f is coarse.Suppose now that x and x ′ are elements of X with d ( x, x ′ ) ≤ r . Let y and y ′ be such that y ∈ g ( x, N ) and y ′ ∈ g ( x ′ , N ). Pick z and z ′ such that x ∈ X z and x ′ ∈ X z ′ . Notice that y ∈ Y f ( z ) and y ′ ∈ Y f ( z ′ ) . Since f iscoarse and the diameter of each X z is bounded by r , d ( z, z ′ ) ≤ r + 2 r ,hence ∂ ( f ( z ) , f ( z ′ )) ≤ s ′ . Since the diameter of each Y y is bounded by r ,we have that ∂ ( y, y ′ ) ≤ s ′ + 2 r .Pick now a ∈ BD( X ) with prop( a ) ≤ r , and suppose that y and y ′ areelements on Y with ∂ ( y, y ′ ) > s ′ +2 r . Fix also n, n ′ ∈ N . Let w and w ′ be in X and, let m, m ′ ∈ N be such that g ( w, m ) = ( y, n ) and g ( w ′ , m ′ ) = ( y ′ , n ′ ).Since ∂ ( y, y ′ ) > s ′ + 2 r , then d ( w, w ′ ) > r , hence a ww ′ = 0. In particular0 = h a ( δ w ⊗ ξ n ) , δ w ′ ⊗ ξ n ′ i = h u ∗ Ψ( a ) u ( δ w ⊗ ξ n ) , δ w ′ ⊗ ξ n ′ i = h Ψ( a ) u ( δ w ⊗ ξ n ) , uδ w ′ ⊗ ξ n ′ i = h Ψ( a )( δ y ⊗ ζ m ) , δ y ′ ⊗ ζ m ′ i Since m and m ′ are arbitrary, then Ψ yy ′ = 0. Since y and y ′ were arbitraryelements such that ∂ ( y, y ′ ) > s ′ + 2 r , then prop(Ψ( a )) ≤ s ′ + 2 r . Thisconcludes the proof. (cid:3) GELFAND-TYPE DUALITY FOR COARSE METRIC SPACES 9
Claim 2.5 implies that Ψ(C ∗ ( X )) ⊂ C ∗ ( Y ). By symmetry of the ar-guments, it follows that Ψ − (C ∗ ( Y )) ⊂ C ∗ ( X ). Hence Ψ restricts to anisomorphism between C ∗ ( X ) and C ∗ ( Y ). We setΦ f, ( ξ n ) , ( ζ n ) ,g = Ψ ↾ C ∗ ( X ) . This map is highly noncanonical, as it depends on the choices of ¯ ξ , ¯ ζ , and g .(The latter in turns depends on f , but not canonically.) We want to makethis association canonical.If Φ and Ψ are isomorphisms between C ∗ ( X ) and C ∗ ( Y ), we writeΦ ∼ u, BD Ψ ⇐⇒ ∃ u ∈ BD( Y ) such that Φ = Ad( u ) ◦ Ψ . We show that our association becomes canonical when Aut(C ∗ ( X )) is di-vided by Inn(BD( X )). Proposition 2.6.
Let X and Y be u.l.f. metric spaces, and let f and h be coarse equivalences between X and Y . Suppose that Φ f, ( ξ n ) , ( ζ n ) ,g and Φ h, ( ξ ′ n ) , ( ζ ′ n ) ,g ′ are constructed as above from different parameters. Then Φ f, ( ξ n ) , ( ζ n ) ,g ∼ u, BD Φ h, ( ξ ′ n ) , ( ζ ′ n ) ,g ′ if and only if f is close to h. Proof.
Let u ∈ BD( X ) be the unitary such that for all x ∈ X and n ∈ N , uδ x ⊗ ξ n = δ x ⊗ ξ ′ n , and let w ∈ BD( Y ) be the unitary such that for all y ∈ Y and n ∈ N , w ( δ y ⊗ ζ n ) = δ y ⊗ ζ ′ n . ThenΦ h, ( ξ n ) , ( ζ n ) ,g ′ = Ad( w ∗ ) ◦ Φ h, ( ξ ′ n ) , ( ζ ′ n ) ,g ′ ◦ Ad( u ) . Since w and Φ h, ( ξ ′ n ) , ( ζ ′ n ) ,g ′ ( u ) are in BD( Y ), thenΦ f, ( ξ n ) , ( ζ n ) ,g ∼ u, BD Φ h, ( ξ ′ n ) , ( ζ ′ n ) ,g ′ ⇔ Φ f, ( ξ n ) , ( ζ n ) ,g ∼ u, BD Φ h, ( ξ n ) , ( ζ n ) ,g ′ . Let g ′′ : Y × N → Y × N given by g ′′ = g ◦ g ′− . Since g and g ′ are bijections,so is g ′′ . Define a unitary v ∈ B ( ℓ ( Y, H )) by v ( δ y ⊗ ζ n ) = δ g ′′ ( y,n ) ⊗ ζ g ′′ ( y,n ) . It is routine to check that Φ f, ( ξ n ) , ( ζ n ) ,g = Ad( v ) ◦ Φ h, ( ξ ′ n ) , ( ζ ′ n ) ,g ′ . Hence wejust need to show that v ∈ BD( Y ). This follows from how g and g ′ areconstructed because f and h are close. (cid:3) We have constructed a canonical injective homomorphism Coa( X ) → Out(C ∗ ( X )). As in the proof of Theorem A, our efforts will amount toprove surjectivity.3. Uniform approximability in Roe algebras
This section deals with uniform approximability of maps Φ between C ∗ -subalgebras of B ( ℓ ( X, H )) and B ( ℓ ( Y, H )). Precisely, in this section westudy when maps satisfy coarse-like properties, that is, when morphismsrespect the large-scale geometry of the underlying spaces. This concept wasintroduced for maps between uniform Roe algebras in [3], and formalised in[4]; here we define it in the setting of Roe algebras.
Definition 3.1.
Let X and Y be metric spaces, A ⊂ B ( ℓ ( X, H )) and
B ⊂ B ( ℓ ( Y, H )) be C ∗ -subalgebras.1. Given ε, r >
0, an element a ∈ A is ε - r -approximable in A if there is c ∈ A with prop( c ) ≤ r so that k a − c k ≤ ε .2. A map Φ : A → B is coarse-like if for all ε, r > s > a ) is ε - s -approximable in B for all contractions in a ∈ A withprop( a ) ≤ r .The following theorem is the starting point for our research on uniformapproximability for Roe algebras. It is a simple consequence of [3, Lemma4.9] (see [5, Proposition 3.3] for a precise proof; cf. [3, Theorem 4.4]). Theorem 3.2.
Let X and Y be u.l.f. metric spaces and let Φ : C ∗ u ( X ) → C ∗ u ( Y ) be a strongly continuous compact preserving linear map. Then Φ iscoarse-like. (cid:3) In layman terms, Theorem 3.2 says that, for uniform Roe algebras, uni-form approximability holds in a very strong sense. This result is false forRoe algebras.
Proposition 3.3.
Given a finite metric space X and a metric space ofinfinite diameter Y , there is a compact preserving strongly continuous em-bedding Φ : C ∗ ( X ) → C ∗ ( Y ) onto a hereditary subalgebra of C ∗ ( Y ) which isnot coarse-like.Proof. For simplicity, we assume X to be a singleton, say X = { x } , and Y to be countable, say Y = N . Fix a metric ∂ on Y of infinite diameter. Let( ξ n ) n be an orthonormal basis for H , and define u : ℓ ( X, H ) → ℓ ( Y, H ) by uδ x ⊗ ξ n = δ n ⊗ ξ , for n ∈ N . The operator u is an isometry, and Ad( u ) is a compact preserving stronglycontinuous embedding of B ( ℓ ( X, H )) into B ( ℓ ( Y, H )). As C ∗ ( X ) = χ { x } ⊗K ( H ), then Ad( u )(C ∗ ( X )) ⊂ K ( ℓ ( Y, H )) ⊂ C ∗ ( Y ). The image of Ad( u )is a hereditary subalgebra of C ∗ ( Y ) since it equals K ( ℓ ( Y )) ⊗ q , where q : H → span { ξ } is the standard projection.We are left to show that Ad( u ) is not coarse-like. Fix n ∈ N , and let m ∈ N be such that ∂ (1 , m ) ≥ n . Let v ∈ B ( ℓ ( X, H )) be the rank 1 partialisometry sending δ x ⊗ ξ to δ x ⊗ ξ m . Then v has propagation 0 but, as h Ad( u ) v ( δ ⊗ ξ ) , δ m ⊗ ξ i = 1 , it follows that Ad( u ) v cannot be 1 / n -approximated. As n is arbitrary, weare done. (cid:3) The map of Proposition 3.3 sends a sequence which is converging in thestrong operator topology to an element outside of C ∗ ( X ) to a sequence whichis converging in the strong operator topology to an element in C ∗ ( Y ). Thisis the only obstacle in generalising Theorem 3.2 to Roe algebras.The following two theorems are our main uniform approximability resultsand most of this section is dedicated to prove them. GELFAND-TYPE DUALITY FOR COARSE METRIC SPACES 11
Theorem 3.4.
Let X and Y be u.l.f. metric spaces. Then every isomor-phism between C ∗ ( X ) and C ∗ ( Y ) is coarse-like. Although Proposition 3.3 shows that Theorem 3.2 cannot be extendedto Roe algebras, the latter can be extended to band-dominated algebras asfollows:
Theorem 3.5.
Let X and Y be u.l.f. metric spaces. Then every stronglycontinuous compact preserving ∗ -homomorphism Φ : BD( X ) → BD( Y ) iscoarse-like. We now proceed to prove Theorems 3.4 and 3.5. First, we show that ε - r -approximability does not depend on the ambient algebra. Proposition 3.6.
Let X be a u.l.f. metric space, r > , ε > , and a ∈ C ∗ ( X ) . The following are equivalent:1. a is ε - r -approximable in BD( X ) ,2. a is ( ε + δ ) - r -approximable in C ∗ ( X ) , for all δ > , and3. a is ( ε + δ ) - r -approximable in BD( X ) , for all δ > .Proof. (1) ⇒ (2) If a is ε - r -approximable in BD( X ), pick b of propagation r with k a − b k ≤ ε . Let p ∈ C ∗ ( X ) be a projection with prop( p ) = 0 andsuch that k pa − a k < δ . This exists by Proposition 2.1. Then pb ∈ C ∗ ( X ),prop( pb ) ≤ r and k a − pb k ≤ k a − pa + pa − pb k < k a − b k + δ/ < ε + δ. (2) ⇒ (3) This is immediate.(3) ⇒ (1) For each n ∈ N , pick b n ∈ BD( X ) with prop( b n ) ≤ r and k a − b n k ≤ ε + 1 /n . Then ( b n ) n is bounded and, by compactness of the unitball of B ( ℓ ( X, H )) with respect to the weak operator topology, by goingto a subsequence if necessary, we can define b = WOT- lim n b n . Clearly,prop( b ) ≤ r , so we only need to notice that k a − b k ≤ ε . Suppose k a − b k > ε .Pick unit vectors ξ and ζ in ℓ ( X, H ), and n ∈ N so that |h ( a − b ) ξ, ζ i| > ε +1 /n . By the definition of b , there is m > n so that |h ( a − b m ) ξ, ζ i| > ε + 1 /n .As m > n , this implies that k a − b m k > ε + 1 /n ; contradiction. (cid:3) The set of ε - r -approximable elements is strongly closed: Proposition 3.7.
Let X be a u.l.f. metric space, r > , ε > , a ∈B ( ℓ ( X, H )) and let ( a n ) n be a sequence in BD( X ) so that a = SOT - lim a n .If each a n is ε - r -approximable in BD( X ) , then a is ε - r -approximable in BD( X ) .Proof. As each a n is ε - r -approximable in BD( X ), pick a sequence ( b n ) n in BD( X ) so that prop( b n ) ≤ r and k a n − b n k ≤ ε for all n ∈ N . As a = SOT- lim n a n , the principle of uniform boundedness implies that ( a n ) n is a bounded sequence, hence so is ( b n ) n . By compactness of the unit ballof B ( ℓ ( X, H )) in the weak operator topology, going to a subsequence ifnecessary, we can assume that b = WOT- lim n b n exists. Clearly, prop( b ) ≤ r and k a − b k ≤ ε , so we are done. (cid:3) Propositions 3.6 and 3.7 together give the following:
Proposition 3.8.
Let X be a u.l.f. metric space, r > , ε > , a ∈ C ∗ ( X ) and let ( a n ) n be a sequence in BD( X ) so that a = SOT - lim a n . If each a n is ε - r -approximable in C ∗ ( X ) , then a is ( ε + δ ) - r -approximable in C ∗ ( X ) forall δ > . (cid:3) Next, we study how strong convergence and ε - r -approximability interact.We prove the Roe algebra and the band-dominated algebra versions of [3,Lemma 4.9]. Let D = { z ∈ C | | z | ≤ } . If ( a n ) n is a sequence of operatorsand ¯ λ ∈ D N is such that SOT- P n λ n a n exists, we write a ¯ λ = SOT- X n λ n a n . When writing a ¯ λ , it is implicit that the limit above exists. Lemma 3.9.
Let X be a u.l.f. metric space and let ( a n ) n be a sequence ofcompact operators in BD( X ) so that a ¯ λ ∈ BD( X ) for all ¯ λ ∈ D N . Then forall ε > there is r > so that a ¯ λ is ε - r -approximable in BD( X ) for all ¯ λ ∈ D N .Proof. Let ( p j ) j be a sequence of finite rank projections on H which stronglyconverges to 1 H . Let 1 X denote the identity on ℓ ( X ) and notice that a = SOT- lim j (1 X ⊗ p j ) a (1 X ⊗ p j ). Given ¯ λ = ( λ n ) n ∈ D N and j ∈ N , let a ¯ λ,j = (1 X ⊗ p j ) a ¯ λ (1 X ⊗ p j ) , and a ¯ λ, ∞ = a ¯ λ , so that a ¯ λ, ∞ = SOT- lim j a ¯ λ,j for all ¯ λ ∈ D N . By Proposition3.8, it is enough to show that( ∗ ) for all ε > r > a ¯ λ,j is ε - r -approximable in BD( X )for all ¯ λ ∈ D N and all j ∈ N .Suppose ( ∗ ) fails for ε >
0. For each finite I ⊂ N , let Z I = n ¯ λ ∈ D N | ∀ j ∈ I, λ j = 0 o and Y I = n ¯ λ ∈ D N | ∀ j I, λ j = 0 o . So Y I is compact in the product space D N . Claim 3.10.
For all r > I ⊂ N , there exist ¯ λ ∈ Z I and j ∈ N so that a ¯ λ,j is not ε/ r -approximable in BD( X ). Proof.
Suppose the claim fails and let r > I ⊂ N witness that. Let N ∞ be the one-point compactification of N . Notice that the map(¯ λ, j ) ∈ Y I × N ∞ a ¯ λ,j ∈ BD( X )is continuous. Indeed, the map is clearly continuous on Y I × N and continuityon Y I × {∞} follows since each a n is compact; therefore, a n = lim j (1 X ⊗ p j ) a n (1 X ⊗ p j ) for all n ∈ N and it follows that a ¯ λ, ∞ = lim j a ¯ λ,j for all¯ λ ∈ Y I . The continuity of this map and the compactness of Y I × N ∞ implythat { a ¯ λ,j | ¯ λ ∈ Y I , j ∈ N ∞ } is a norm compact subset of BD( X ). In GELFAND-TYPE DUALITY FOR COARSE METRIC SPACES 13 particular, the total boundedness of this set gives s > a is ε/ s -approximable in BD( X ) for all a ∈ { a ¯ λ,j | ¯ λ ∈ Y I , j ∈ N } .Let ¯ λ ∈ D N and j ∈ N . Write ¯ λ = ¯ λ + ¯ λ for ¯ λ ∈ Y I and ¯ λ ∈ Z I ,so a ¯ λ,j = a ¯ λ ,j + a ¯ λ ,j . As r and I witness that the claim fails, a ¯ λ ,j is ε/ r -approximable in BD( X ). Moreover, our choice of s implies that a ¯ λ ,j is ε/ s -approximable in BD( X ); hence a ¯ λ,j is ε -max { r, s } -approximable inBD( X ). As ¯ λ ∈ D N and j ∈ N were arbitrary, this contradicts that ( ∗ ) failsfor ε . (cid:3) For r > δ >
0, let U r,δ = n ¯ λ ∈ D N | a ¯ λ,j is δ - r -approximable in BD( X ) for all j ∈ N o . Claim 3.11.
The set U r,δ is closed for all r > δ > Proof.
Suppose the claim fails for r > δ >
0. Then there is ¯ λ ∈ U ∁ r,δ ∩ U r,δ . As ¯ λ U r,δ , there is j ∈ N so that a ¯ λ,j is not δ - r -approximablein BD( X ). Proposition 3.8 implies that there is a finite F ⊂ X so that χ F a ¯ λ,j χ F is not δ - r -approximable in BD( X ).Fix γ >
0. By the definition of a ¯ θ,j , χ F a ¯ θ,j χ F = ( χ F ⊗ p j ) a ¯ θ ( χ F ⊗ p j ); since χ F ⊗ p j is compact, then there exists a finite I ⊂ N so that k χ F a ¯ θ,j χ F k < γ for all ¯ θ ∈ Z I . Indeed, this follows since χ F a ¯ θ,j χ F = SOT- X n θ n ( χ F ⊗ p j ) a n ( χ F ⊗ p j )for all ¯ θ ∈ D N . Let ¯ λ ∈ Y I and ¯ λ ∈ Z I be so that ¯ λ = ¯ λ + ¯ λ . As¯ λ ∈ U r,δ , there exists ¯ θ ∈ Y I and ¯ θ ∈ Z I so that ¯ θ = ¯ θ + ¯ θ ∈ U r,δ and k a ¯ λ ,j − a ¯ θ ,j k ≤ γ . As ¯ θ ∈ U r,δ , a ¯ θ,j is δ - r -approximable in BD( X ).In particular, as χ F ⊗ p m is a projection with propagation 0, χ F a ¯ θ,j χ F is δ - r -approximable in BD( X ). Therefore, since a ¯ λ,j = a ¯ θ,j + a ¯ λ ,j − a ¯ θ ,j + a ¯ λ ,j − a ¯ θ ,j , then χ F a ¯ λ,j χ F is ( δ + 3 γ )- r -approximable in BD( X ). As γ was arbitrary,Proposition 3.6 implies that χ F a ¯ λ,j χ F is δ - r -approximable in BD( X ); con-tradiction. (cid:3) Fix δ = ε/ Claim 3.12.
For all r >
0, the set U r,δ has empty interior. Proof.
Fix r > λ ∈ U r,δ . Fix a finite I ⊂ N and write ¯ λ = ¯ λ + ¯ λ ,for ¯ λ ∈ Y I and ¯ λ ∈ Z I . Pick s > r so that a ¯ λ ,j is δ - s -approximable inBD( X ). By Claim 3.10, there is ¯ θ ∈ Z I and j ∈ N so that a ¯ θ ,j is not2 δ - s -approximable in BD( X ). Hence, letting ¯ θ = ¯ λ + ¯ θ , we have that a ¯ θ,j is not δ - s -approximable in BD( X ). As s > r , a ¯ θ,j is not δ - r -approximable inBD( X ), so ¯ θ U r,δ . Since I was an arbitrary finite subset of N , this showsthat ¯ λ is not an interior point of U r,δ . (cid:3) By Claim 3.11 and 3.12, U r,δ is nowhere dense for all r >
0. However D N = [ n ∈ N U n,δ . Indeed, if ¯ λ ∈ D N , then there is n ∈ N so that a ¯ λ, ∞ is δ - n -approximable inBD( X ). In particular, a ¯ λ,j is δ - n -approximable in BD( X ) for all j ∈ N , so¯ λ ∈ U n,δ . Since D N is a Baire space, we have a contradiction. (cid:3) Remark . The Baire categorical nature of the proof of Lemma 3.9 impliesthat its statement holds outside the scope of metrizable coarse spaces (forbrevity, we refer the reader to [3] for definitions). Indeed, if ( X, E ) is acoarse space which is small (see [3, Definition 4.2]), Lemma 3.9 still holds.In particular, Theorems 3.4 and 3.5 also hold for small coarse spaces.The following is the Roe algebra version of Lemma 3.9. Lemma 3.14.
Let X be a u.l.f. metric space and let ( a n ) n be a sequenceof compact operators in C ∗ ( X ) so that a ¯ λ ∈ C ∗ ( X ) for all ¯ λ ∈ D N . Thenfor all ε > there is r > so that a ¯ λ is ε - r -approximable in C ∗ ( X ) for all ¯ λ ∈ D N .Proof. By Lemma 3.9, we know that for all ε > r such that a ¯ λ is ε/ r -approximable in BD( X ) for all ¯ λ ∈ D N . Given ¯ λ ∈ D N , Proposition 3.6implies that a ¯ λ is ε - r -approximable in C ∗ ( X ). Since ε and ¯ λ are arbitrary,we are done. (cid:3) Lemma 3.15.
Let X and Y be u.l.f. metric spaces and let Φ : C ∗ ( X ) → C ∗ ( Y ) be a strongly continuous compact preserving linear map. If Φ ↾ χ F C ∗ ( X ) χ F is coarse-like for all finite F ⊂ X , then Φ is coarse-like.Proof. Suppose that Φ ↾ χ E C ∗ ( X ) χ E is coarse-like for all finite E ⊆ X butΦ is not coarse-like. Let ε > r > s > a s ∈ C ∗ ( X ) of propagation at most r such that Φ( a s ) is not ε - s -approximable. Claim 3.16.
For all cofinite F ⊂ X and all s > a ∈ χ F C ∗ ( X ) χ F with finite support and propagation at most r so that Φ( a )is not ε/ s -approximable. Proof.
Fix F and s . Let E = n x ∈ X | d ( x, X \ F ) ≤ r o . As X \ F is finite and X is u.l.f., E is finite, and therefore Φ ↾ χ E C ∗ ( X ) χ E is coarse-like. Pick s ′ > s so that Φ( χ E aχ E ) is ε/ s ′ -approximable for allcontractions a ∈ C ∗ ( X ).By the definition of E , we have that { ( x, y ) ∈ X | d ( x, y ) ≤ r } ⊂ ( E × E ) ∪ ( F × F ) . GELFAND-TYPE DUALITY FOR COARSE METRIC SPACES 15
Hence, if a ∈ C ∗ ( X ) has propagation at most r then there is b ∈ χ E C ∗ ( X ) χ E such that a = b + χ F aχ F . If a is a contraction, then k b k ≤ b ) ≤ r .Let b ∈ χ E C ∗ ( X ) χ E be as above for a = a s ′ . By our choice of s ′ , Φ( b ) is ε/ s ′ -approximable. As s ′ > s , if Φ( χ F aχ F ) is ε/ s -approximable, then Φ( a )is ε - s ′ -approximable. This contradicts our choice of a s , so Φ( χ F a s χ F ) is not ε/ s -approximable. By Proposition 3.8, we can obtain a finite G ⊂ X \ F so that Φ( χ G a s χ G ) is not ε/ s -approximable. This finishes that claim. (cid:3) By the previous claim, there exists a disjoint sequence of finite subsets( E n ) n of X and a sequence ( a n ) n in C ∗ ( X ) so that a n ∈ B ( ℓ ( E n , H )),prop( a n ) ≤ r and Φ( a n ) is not ε/ n -approximable for all n ∈ N . Sincethe E n ’s are disjoint, for all ¯ λ ∈ D N , a ¯ λ is a well defined element of C ∗ ( X ).By Lemma 3.14, there is s > a n ) is ε/ s -approximable for all n ∈ N , a contradiction. (cid:3) If we substitute instances of C ∗ ( X ) and of C ∗ ( Y ) with BD( X ) and BD( Y )in the proof of Lemma 3.15, we obtain the following. Lemma 3.17.
Let X and Y be u.l.f. metric spaces and let Φ : BD( X ) → BD( Y ) be a strongly continuous compact preserving linear map. If Φ ↾ χ F BD( X ) χ F is coarse-like for all finite F ⊂ X , then Φ is coarse-like. (cid:3) We are ready to prove our uniform approximability results.
Proof of Theorem 3.5.
Fix u.l.f metric spaces X and Y . Recall that we needto show that any strongly continuous compact preserving ∗ -homomorphismΦ : BD( X ) → BD( Y ) is coarse-like. By Lemma 3.17, it is enough to showthat Φ ↾ χ F BD( X ) χ F is coarse-like for all such Φ and all finite F ⊆ X .As χ { x } BD( X ) χ { y } ∼ = B ( H ) for all x and y in X , it is enough to show thatany strongly continuous compact preserving ∗ -homomorphism Φ : B ( H ) → BD( Y ) is coarse-like. Fix such Φ.Let ( ξ n ) n be an orthonormal basis for H . If F ⊆ N , let p F be the projec-tion onto span { ξ i | i ∈ F } . We write p n for p { ,...,n } . By Proposition 3.7, it isenough to show that for all ε > s > n ∈ N and allcontractions a ∈ B ( H ), Φ( p n ap n ) is ε - s -approximable in BD( Y ). Supposethat this fails for ε ∈ (0 , p n B ( H ) p n ,we then have that( ∗ ) For all finite E ⊂ N and all s > a ∈ B ( H ) withfinite support so that p E ap E = 0 and Φ( a ) is not ε/ s -approximablein BD( Y ). Claim 3.18.
For all cofinite F ⊂ N and all s > a ∈ B ( p F H ) with finite support so that Φ( a ) is not ε/ s -approximable inBD( Y ). Proof.
Suppose the claim fails for a cofinite F ⊂ N and s >
0. Let A = N \ F . By ( ∗ ) there are an increasing sequence of finite subsets ( A n ) n of N and a sequence of contractions ( a n ) n in B ( H ) so that supp( a n ) ⊂ A n \ A n −
16 B. M. BRAGA AND A. VIGNATI and Φ( a n ) is not ε/ n -approximable in BD( Y ) for all n ∈ N . Since theclaim fails, going to a subsequence and redefining ( a n ) n , we can assume thatsupp( a n ) ⊂ ( A n \ A n − ) × A for all n ∈ N and that Φ( a n ) is not ε/ n -approximable in BD( Y ) for all n ∈ N (otherwise, we could assume thatsupp( a n ) ⊂ A × ( A n \ A n − ) and the proof would proceed similarly). Let F n = A n \ A n − .Let ( E n ) n be a disjoint sequence of finite subsets of N so that | E n | = | A | for all n ∈ N . For each n ∈ N , let b n ∈ B ( p A H, p E n H ) be a unitary. Foreach n ∈ N , let k ( n ) ∈ N be so that Φ( b n ) can be ε/ k ( n )-approximablein BD( Y ) and let m ( n ) ∈ N be so that Φ( a m ( n ) ) is not ε/ k ( n ) + n )-approximable in BD( Y ). Without loss of generality ( m ( n )) n is strictly in-creasing.Notice that supp( b n a m ( n ) ) ⊂ F m ( n ) × E n for all n ∈ N . As both ( E n ) n and ( F m ( n ) ) n are disjoint sequences, we have thatSOT- X n ∈ N λ n b n a m ( n ) ∈ B ( H )for all λ n ∈ D N . As Φ is strongly continuous and compact preserving, eachΦ( b n a m ( n ) ) is compact andSOT- X n ∈ N λ n Φ( b n a m ( n ) ) ∈ BD( Y )for all ( λ n ) n ∈ D N . By Lemma 3.9, there exists s ′ > b n a m ( n ) )is ε/ s ′ -approximable in BD( Y ) for all n ∈ N . Fix a sequence ( c n ) n inBD( Y ) so that prop( c n ) ≤ s ′ and k Φ( b n a m ( n ) ) − c n k ≤ ε/
16 for all n ∈ N .As b n is unitary, each Φ( b − n ) is ε/ k ( n )-approximable in BD( Y ). Fix( d n ) n in BD( Y ) so that prop( d n ) ≤ k ( n ) and k Φ( b − n ) − d n k ≤ ε/
17 for all n ∈ N . Then, for all n ∈ N , k Φ( a m ( n ) ) − d n c n k == (cid:13)(cid:13) Φ( b − n )Φ( b n a m ( n ) ) − d n c n (cid:13)(cid:13) ≤k Φ( b − n )Φ( b n a m ( n ) ) − Φ( b n ) − c n k + k Φ( b n ) − c n − d n c n k≤k Φ( b n a m ( n ) ) − c n k + k Φ( b n ) − − d n k · k c n k≤ ε
16 + ε (cid:16) ε (cid:17) ≤ ε . As prop( d n c n ) ≤ k ( n ) + s ′ , then Φ( a m ( n ) ) is ε/ k ( n ) + s ′ )-approximable inBD( Y ) for all n ∈ N . For n > s ′ , this gives a contradiction. (cid:3) By the previous claim we can pick mutually disjoint finite sets E n ⊆ N and a sequence of contractions ( a n ) n so that a n ∈ B ( P E N H ) and Φ( a n ) isnot ε/ n -approximable for all n ∈ N . Since the E n ’s are mutually disjoint, a ¯ λ ∈ B ( H ) for all ¯ λ ∈ D N . By Lemma 3.9, there exists s > a n )is ε/ s -approximable in BD( Y ) for all n ∈ N , a contradiction. (cid:3) GELFAND-TYPE DUALITY FOR COARSE METRIC SPACES 17
Proof of Theorem 3.4.
Fix u.l.f metric spaces X and Y , and an isomor-phism Φ : C ∗ ( X ) → C ∗ ( Y ). By Lemma 3.15, it is enough to show thatΦ ↾ χ F C ∗ ( X ) χ F is coarse-like for all finite F ⊆ X . Therefore, finiteness of F implies that we only need to show that Φ ↾ χ { x } C ∗ ( X ) χ { y } is coarse-likefor all x and y in X . To simplify the notation, assume x = y .We prove the following stronger statement:( ∗ ) For every ε > F ⊆ Y such that k χ F Φ( a ) χ F − Φ( a ) k < ε for all positive contractions a ∈ χ { x } C ∗ ( X ) χ { x } .Notice that, since χ F Φ( a ) χ F has propagation at most diam( F ), for all a ∈ C ∗ ( X ), ( ∗ ) implies the desired result. We proceed by contradiction, soassume the statement in ( ∗ ) fails for ε > ξ n ) n be an orthonormal base for H , and let p n be the projection ontospan { ξ i | i ≤ n } . For each n ∈ N , let q n = χ { x } ⊗ p n . Given a finite I ⊂ N ,write q I = q max I − q min I . Claim 3.19.
For every finite F ⊆ Y and n ∈ N there is a positive contrac-tion a ∈ χ { x } C ∗ ( X ) χ { x } with aq n = q n a = 0 and (cid:13)(cid:13) χ Y \ F Φ( a ) χ Y \ F (cid:13)(cid:13) > ε . Proof.
Fix a finite F ⊂ Y and n ∈ N . Since q n C ∗ ( X ) q n is finite dimensionaland each element of q n C ∗ ( X ) q n has finite rank, there is a finite G ⊆ Y suchthat whenever a ∈ q n C ∗ ( X ) q n is a contraction and G ′ ⊃ G , then k Φ( a ) − χ G ′ Φ( a ) k < ε . Fix G ′ = G ∪ F . By our choice of ε , there is a positive contraction b ∈ χ { x } C ∗ ( X ) χ { x } such that k χ G ′ Φ( b ) χ G ′ − Φ( b ) k > ε . In particular, thetriangular inequality implies that (cid:13)(cid:13) χ Y \ G ′ Φ( b ) (cid:13)(cid:13) = k χ G ′ Φ( b ) − Φ( b ) k > ε. Let a = ( χ { x } − q n ) b ( χ { x } − q n ). So a is a positive contraction. Assume fora contradiction that (cid:13)(cid:13) χ Y \ F Φ( a ) χ Y \ F (cid:13)(cid:13) ≤ ε . Then (cid:13)(cid:13) χ Y \ G ′ Φ( a ) χ Y \ G ′ (cid:13)(cid:13) ≤ ε .Since b = a + q n b q n + q n b ( χ { x } − q n ) + ( χ { x } − q n ) b q n , we have that (cid:13)(cid:13) χ Y \ G ′ Φ( b ) χ Y \ G ′ (cid:13)(cid:13) ≤ ε , so (cid:13)(cid:13) χ Y \ G ′ Φ( b ) (cid:13)(cid:13) ≤ ε . This is acontradiction. (cid:3) Notice that a = SOT- lim m ( q m − q n ) a ( q m − q n ), for all a ∈ χ { x } C ∗ ( X ) χ { x } with q n a = aq n = 0. Therefore, the previous claim can be used to producea sequence ( a n ) n of contractions in χ { x } C ∗ ( X ) χ { x } , a sequence of naturalnumbers ( k ( n )) n , and sequences ( F n ) n and ( I n ) n of finite subsets of Y and N , respectively, so that ( F n ) n is a disjoint sequence, max I n < max I n +1 forall n ∈ N , and (cid:13)(cid:13) ( χ F n ⊗ p k ( n ) )Φ( q I n a n a ∗ n q I n )( χ F n ⊗ p k ( n ) ) (cid:13)(cid:13) > ε / for all n ∈ N . Hence, the C ∗ -equality gives that (cid:13)(cid:13) χ F n ⊗ p k ( n ) Φ( q I n a n ) (cid:13)(cid:13) > ε/ n ∈ N .As both ( χ F n ⊗ p k ( n ) ) n and (Φ( q I n a n )) n are sequences of compact operatorsconverging to zero in the strong operator topology, passing to a subsequenceif necessary, we can assume that (cid:13)(cid:13) χ F n ⊗ p k ( n ) Φ( q I m a m ) (cid:13)(cid:13) ≤ − n − ε for all n = m in N . As ( F n ) n is a disjoint sequence, b = SOT- P n χ F n ⊗ p k ( n ) exists and it clearly belongs to C ∗ ( Y ). Let c = Φ − ( b ); in particular c ∈ C ∗ ( X ). Then k cq I m k ≥ k cq I m a m k = k b Φ( q I m a m ) k≥ (cid:13)(cid:13) χ F n ⊗ p k ( n ) Φ( q I m a m ) (cid:13)(cid:13) − X n = m (cid:13)(cid:13) χ F n ⊗ p k ( n ) Φ( q I m a m ) (cid:13)(cid:13) ≥ ε/ m ∈ N . As ( I n ) n are disjoint, this contradicts the fact that c ∈ C ∗ ( X ),i.e., that c is locally compact. (cid:3) Remark . We used positivity and the fact that the q n ’s are projection inthe above proof. We would not need it, by playing with functional analysis.So if we ever want, we could follow the strategy highlighted above to showthat if Φ : C ∗ ( X ) → C ∗ ( Y ) is a strongly continuous linear compact preserv-ing map with the property that for every sequence of operators ( a n ) wehave that if Φ( a n ) converges strongly to b in C ∗ ( Y ) then there is c ∈ C ∗ ( X )such that a n converges strongly to c . This shows that having a stronglycontinuous sequence which is converging outside C ∗ ( X ) and which is sent toa strongly converging sequence converging inside C ∗ ( Y ) is indeed the onlyobstruction to generalising Theorem 3.2 to the Roe algebra setting (see, e.g.,the discussion after Proposition 3.3).We finish the section introducing a weaker version of coarse-likeness forwhich Theorem 3.2 has an equivalent for Roe algebras. Definition 3.21.
Let X and Y be metric spaces. A map Φ : C ∗ ( X ) → C ∗ ( Y ) is asymptotically coarse-like if for all ε > r > s > X ′ ⊂ X so that Φ( a ) is ε - s -approximable in C ∗ ( Y ) for allcontractions in a ∈ C ∗ ( X ′ ) with prop( a ) ≤ r . Theorem 3.22.
Let X and Y be u.l.f. metric spaces. Every strongly contin-uous compact preserving linear map Φ : C ∗ ( X ) → C ∗ ( Y ) is asymptoticallycoarse-like.Proof. Suppose this fails. So there is ε > r > n ∈ N and all cofinite X ′ ⊂ X , there is a contraction a ∈ C ∗ ( X ′ ) with prop( a ) ≤ r so that Φ( a ) is not ε - n -approximable. Then, by Proposition 3.8, we can GELFAND-TYPE DUALITY FOR COARSE METRIC SPACES 19 pick a disjoint sequence ( X n ) n of finite subsets of X and a sequence ( a n ) n of contractions so that a n ∈ C ∗ ( X n ), prop( a n ) ≤ r , and Φ( a n ) is not ε - n -approximable for all n ∈ N .Since each X n is finite and Φ preserves the compacts, each Φ( a n ) is com-pact. Hence, as ( X n ) n is a disjoint sequence and as each a n has propagationat most r , strong continuity of Φ gives that SOT- P n λ n Φ( a n ) ∈ C ∗ u ( Y ) forall ( λ n ) n ∈ D N . Therefore, Lemma 3.14 implies that there is s > a n ) is δ - s -approximable; contradiction. (cid:3) Remark . Although we will prove Theorems B and 2.3 below usingTheorem 3.4, we point out that both those results could be obtained (in avery similar way) using Theorem 3.22 above instead of Theorem 3.4.4.
The multiplier algebra of C ∗ ( X )In this short section, we characterize BD( X ) as the multiplier algebra ofC ∗ ( X ). As a consequence, this shows that Inn(BD( X )) is a normal subgroupof Aut(C ∗ ( X )). Since all automorphisms of C ∗ ( X ) are strongly continuous,being induced by a unitary in B ( ℓ ( X, H )) (see e.g., [21, Lemma 3.1]), wehave that all automorphisms of C ∗ ( X ) extend to automorphisms of BD( X ).An operator algebraist used to work with multipliers would not be surprisedby this result, and may even find it obvious. However, we do not know of aproof that does not use uniform approximability in some way. Theorem 4.1.
Let X be a u.l.f. metric space. Then BD( X ) = M (C ∗ ( X )) .Proof. We use the characterisation of the multiplier algebra given in [1,II.7.3.5]. As C ∗ ( X ) is already represented faithfully on ℓ ( X, H ), the multi-plier algebra of C ∗ ( X ) coincides with its idealizer, that is, M (C ∗ ( X )) = n b ∈ B ( ℓ ( X, H )) | ∀ a ∈ C ∗ ( X ) , ba, ab ∈ C ∗ ( X ) o . As C ∗ ( X ) is an ideal in BD( X ), we clearly have that BD( X ) ⊆ M (C ∗ ( X )). Claim 4.2.
Let b ∈ M (C ∗ ( X )), ε >
0, and let F ⊆ X be finite. Then thereis a finite G ⊂ X such that k χ G bχ F − bχ F k < ε and k χ F bχ G − χ F b k < ε .In particular, bχ F and χ F b belong to BD( X ). Proof.
Suppose not. Then, without loss of generality, we assume that thereis a sequence ( G n ) n of disjoint finite subsets X so that k χ G n bχ F k > ε/ n ∈ N . For each n ∈ N , fix a unit vector ξ n such that k χ G n bχ F ξ n k > ε/ p n be a finite rank projection in B ( H ) so that k ( χ G n ⊗ p n ) bχ F ξ n k >ε/
2. Then SOT- P n ( χ G n ⊗ p n ) ∈ C ∗ ( X ), so, as bχ F ∈ M (C ∗ ( X )), thenSOT- P n ( χ G n ⊗ p n ) bχ F ∈ C ∗ ( X ). Fix k ∈ N such that SOT- P n ( χ G n ⊗ p n ) bχ F can be ε/ k -approximated in C ∗ ( X ) and m large enough so that d ( G m , F ) > k . Since χ G m ⊗ p m and χ F have propagation 0, ( χ G m ⊗ p m ) bχ F can be ε/ k -approximated in C ∗ ( X ). Let c ∈ C ∗ ( X ) be an element withpropagation at most k so that k c − ( χ G m ⊗ p m ) bχ F k < ε/
2. Since c haspropagation at most k and d ( G m , F ) > k , then ( χ G m ⊗ p m ) cχ F = 0. Hence k ( χ G m ⊗ p m ) bχ F k < ε/
2, a contradiction. (cid:3)
In order to get a contradiction, suppose b ∈ M (C ∗ ( X )) is such that thereis ε > b cannot be ε - n -approximated for every n ∈ N . Wewill construct two sequences ( F n ) n and ( p n ) n of finite subsets of X andfinite-rank projections in B ( H ), respectively, with the following properties: • the F n are disjoint, • for all n , (cid:13)(cid:13)(cid:13) ( χ S m = n F m ⊗ H ) b ( χ F n ⊗ p n ) (cid:13)(cid:13)(cid:13) < − n , • each ( χ F n ⊗ p n ) b ( χ F n ⊗ p n ) cannot be ε/ n -approximated.We do this by induction. Let ( q n ) n be a sequence of finite-rank projec-tions in B ( H ) which is converging strongly to 1 H . Since b cannot be ε -0-approximated and b = SOT- lim F ⊆ X finite n ∈ N ( χ F ⊗ q n ) b ( χ F ⊗ q n ) , Proposition 3.7 gives a finite F ⊆ X and n ∈ N such that ( χ F ⊗ q n ) b ( χ F ⊗ q n )cannot be ε -0-approximated. Let p = q n .We now make the inductive step: suppose that p , . . . , p n and F , . . . , F n have been defined. Let b n = bχ S m ≤ n F n . Using the previous claim, pick afinite G ⊆ X such that k χ G b n − b n k < − n − . By Claim 4.2, χ X \ G bχ G + χ G bχ X \ G + χ G bχ G ∈ BD( X ), so there is n ′ > n such that χ X \ G bχ G + χ G bχ X \ G + χ G bχ G can be ε/ n ′ -approximated. As b = χ X \ G bχ X \ G + χ X \ G bχ G + χ G bχ X \ G + χ G bχ G cannot be ε - n ′ -approximated, χ X \ G bχ X \ G cannot be ε/ n ′ -approximated.As χ X \ G = SOT- lim F ⊆ X \ G finite i ∈ N χ F ⊗ q i , Proposition 3.7 gives a finite F n +1 and i ∈ N such that ( χ F n +1 ⊗ q i ) b ( χ F n +1 ⊗ q i ) cannot be ε/ n ′ -approximated. Setting p n +1 = q i concludes the con-struction.Let now c = SOT- P n ( χ F n ⊗ p n ) and notice that cbc = SOT- X n ( χ F n ⊗ p n ) b ( χ F n ⊗ p n ) + d where d = SOT- X n ∈ N (cid:16) ( χ F n ⊗ p n ) b X m = n χ F m ⊗ p m (cid:17) . By our choice of ( F n ) n and ( p n ) n , we have that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:16) X ⊗ H − X m ≤ n χ F m ⊗ p m (cid:17) d (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ X m>n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( χ F m ⊗ p m ) b X n ′ = m χ F n ′ ⊗ p n ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ X m>n − m ≤ − n +1 GELFAND-TYPE DUALITY FOR COARSE METRIC SPACES 21 for all n ∈ N . Hence d is compact, so d ∈ C ∗ ( X ). Let b ′ = SOT- X n ( χ F n ⊗ p n ) b ( χ F n ⊗ p n ) . As b ∈ M (C ∗ ( X )), it follows that cbc ∈ C ∗ ( X ). Hence, as d ∈ C ∗ ( X ),we have that b ′ ∈ C ∗ ( X ). Pick n such that b ′ can be ε/ n -approximated.Since χ F n ⊗ p n has propagation 0, ( χ F n ⊗ p n ) b ′ ( χ F n ⊗ p n ) can be ε/ n -approximated. This is a contradiction since ( χ F n ⊗ p n ) b ′ ( χ F n ⊗ p n ) = ( χ F n ⊗ p n ) b ( χ F n ⊗ p n ). (cid:3) The following is a simple consequence of Theorem 4.1.
Corollary 4.3.
Let X be a u.l.f. metric space. Any Φ ∈ Aut(C ∗ ( X )) ex-tends to an automorphism of BD( X ) . Moreover, Inn(BD( X )) is a normalsubgroup of Aut(C ∗ ( X )) . (cid:3) Remark . If one is only interested in Corollary 4.3, Theorem 4.1 is notnecessary. In fact, Theorem 3.4 gives us an easy proof of Corollary 4.3. Weoutline it here as an example of the power of Theorem 3.4.Fix Φ ∈ Aut(C ∗ ( X )) and let u ∈ B ( ℓ ( X, H )) be a unitary so thatΦ = Ad( u ) (e.g., [21, Lemma 3.1]). Fix r > ε > a ∈ BD( X )be a contraction with prop( a ) ≤ r . Then, a can be easily written as a = SOT- lim n a n where ( a n ) n is a sequence of contractions in C ∗ ( X ) withprop( a n ) ≤ r for all n ∈ N . By Theorem 3.4, there is s = s ( r, ε ) > b n ) n in C ∗ ( X ) so that prop( b n ) ≤ s and k Φ( a n ) − b n k ≤ ε forall n ∈ N . Using weak operator compactness and going to a subsequence,we can assume that b = WOT- lim n b n exists. Clearly, prop( b ) ≤ r and k Ad( u )( a ) − b k ≤ ε . As ε was arbitrary, this shows that Ad( u )( a ) ∈ BD( X ).We leave the remaining details to the reader.5. Proof of the main result
We use the uniform approximability results of § Technical lemmas.
We prove several technical lemmas in this sub-section. Their proofs are inspired by techniques in [23, Section 6].
Definition 5.1.
A u.l.f. metric space X has the operator norm localisationproperty ( ONL ) if for all s > ρ ∈ (0 ,
1) there is r > a ∈ B ( ℓ ( X, H )) has propagation at most s then there exists a unit vector ξ ∈ ℓ ( X, H ) with diam(supp( ξ )) ≤ r so that k aξ k ≥ ρ k a k . By [19, Theorem 4.1], a u.l.f. metric space has property A if and only ifit has ONL. The following assumption will be recurrent: Recall, supp( ξ ) = { x ∈ X | ξ ( x ) = 0 } for a given ξ : X → H . Assumption 5.2.
Let X and Y be u.l.f. metric spaces with ONL, H be aseparable infinite-dimensional Hilbert spaces and let Φ : C ∗ ( X ) → C ∗ ( Y ) bean isomorphism. For δ >
0, and projections p ∈ C ∗ ( X ) and q ∈ C ∗ ( Y ) ofrank 1, we let Y p,δ = { y ∈ Y | k Φ( p ) χ { y } k ≥ δ } and X q,δ = { x ∈ X | k Φ − ( q ) χ { x } k ≥ δ } . We point out that isomorphisms between Roe algebras are automaticallystrongly continuous and rank preserving (see [21, Lemma 3.1]). This will beused with no further mention in the proofs of the lemmas below.
Lemma 5.3.
In the setting of Assumption 5.2, for all r > and ε > there is t > so that for all projections p ∈ C ∗ u ( X ) and q ∈ C ∗ ( Y ) withpropagation at most r there is E ⊂ X with diam( E ) ≤ t so that k Φ( p ) qχ E k ≥ (1 − ε ) k Φ( p ) q k − ε. Proof.
Fix ε >
0. Theorem 3.4 gives s > p ) is ε/ s -approximablefor all projections p ∈ C ∗ ( X ) with prop( p ) ≤ r . Fix projections p ∈ C ∗ ( X )and q ∈ C ∗ ( Y ) with propagation at most r . So there is b ∈ C ∗ ( Y ) withprop( b ) ≤ s + r so that k Φ( p ) q − b k ≤ ε/
2. As Y has ONL, there exists t > ε and r ) and a unit vector ξ ∈ ℓ ( Y, H ) so thatsupp( ξ ) ≤ t and k bξ k ≥ (1 − ε ) k b k . Let E = supp( ξ ). Hence, k Φ( p ) qχ E k ≥ k bχ E k − k Φ( p ) qχ E − bχ E k≥ (1 − ε ) k b k − ε ≥ (1 − ε ) k Φ( p ) q k − (1 − ε ) k Φ( p ) q − b k − ε ≥ (1 − ε ) k Φ( p ) q k − ε, and we are done. (cid:3) Given n ∈ N , r ≥
0, we writeProj n,r ( X ) = n p ∈ Proj(C ∗ ( X )) | rank( p ) ≤ n and prop( p ) ≤ r o . We define Proj n,r ( Y ) analogously. Lemma 5.4.
In the setting of Assumption 5.2, for all r > and δ > , wehave that sup p ∈ Proj ,r ( X ) diam( Y p,δ ) < ∞ . Proof.
By Lemma 5.3, there is t > p ∈ Proj ,r ( X ) there is E ⊂ Y with diam( E ) ≤ t so that k Φ( p ) χ E k > − δ . Fix p ∈ Proj ,r ( X )and let E be as above. As Φ is rank preserving, Φ( p ) has rank 1. Pick aunit vector ξ ∈ ℓ ( Y, H ) in the range of Φ( p ). Then k Φ( p ) χ F k = k χ F Φ( p ) k = k χ F ξ k GELFAND-TYPE DUALITY FOR COARSE METRIC SPACES 23 for all F ⊂ Y . In particular, k χ E ξ k > − δ . If y E , then k Φ( p ) χ { y } k = k χ { y } ξ k ≤ k ξ k − k χ Y \{ y } k ≤ − k χ E ξ k < δ . So, y Y p,δ . This shows that Y p,δ ⊂ E and we must have diam( Y p,δ ) ≤ t . (cid:3) Lemma 5.5.
In the setting of Assumption 5.2, for all r > and ε > thereis δ > so that inf p ∈ Proj ,r ( X ) k Φ( p ) χ Y p,δ k ≥ − ε. Proof.
Fix r > ε ∈ (0 , t > p ∈ Proj ,r ( X ) there is E ⊂ Y with diam( E ) ≤ t such that k Φ( p ) χ E k > − ε . As Y is u.l.f., N = sup y ∈ Y | B t ( y ) | is finite. Let δ ∈ (0 , p ( ε − ε ) /N )and fix p ∈ Proj ,r ( X ). Let ξ ∈ ℓ ( Y, H ) be a unit vector in the range of therank 1 projection Φ( p ). Then, picking E as above, we have that k Φ( p ) χ Y p,δ k = k χ Y p,δ ξ k ≥ k χ E ∩ Y p,δ ξ k ≥ k χ E ξ k − k χ E \ Y p,δ ξ k = k Φ( p ) χ E k − k Φ( p ) χ E \ Y p,δ k ≥ − ε − δ N ≥ (1 − ε ) , and we are done. (cid:3) Before stating the next lemma, we need to introduce some technical no-tation. Given positive reals t , r and k , we denote by D t,r,k ( X ) the set of allfamilies ( p i ) i ∈ N of orthogonal projections in C ∗ ( X ) satisfying:1. each p i has rank at most 1,2. each p i has propagation at most r , and3. |{ i ∈ N | p i χ E = 0 }| ≤ k for any E ⊂ X with diam( E ) ≤ t .If ( p i ) i ∈ N ∈ D t,r,k ( X ), then SOT- P i ∈ N p i ∈ C ∗ ( X ). Let ¯ p = SOT- P i ∈ N p i .By abuse of notation, we identify ¯ p with ( p i ) i ∈ N and write ¯ p ∈ D t,r,k ( X ). Wedefine D t,r,k ( Y ) analogously, and write ¯ q ∈ D t,r,k ( Y ) if ¯ q = SOT- P i ∈ N q i where ( q i ) i ∈ N is in D t,r,k ( Y ). Remark . A word on the prototypical elements of D t,r,k ( X ) is useful:If ( P x ) x ∈ X is a family of projections on H and p x = χ { x } ⊗ P x , then( p x ) x ∈ X belongs to D t,r,k ( X ) for any t and r ≥
0, where k = sup x ∈ X | B t ( x ) | .More generally, let ( X n ) n be a sequence of disjoint subsets of X with r =sup n diam( X n ) < ∞ , ℓ ∈ N , and for each n ∈ N let ( p n,i ) ℓi =1 be a familyof orthogonal projections in B ( ℓ ( X n , H )) of rank at most 1. Then, for any t >
0, there is k > p = (( p n,i ) ℓi =1 ) n ∈ N ∈ D t,r,k ( X ). Indeed, firstnotice that the propagation of each p n,i is at most r . Moreover, since X isu.l.f., there is k ∈ N so that any E ⊂ X with diam( E ) ≤ t intersects atmost k -many X n ’s. Therefore, p n,i χ E = 0 for at most k ℓ -many ( n, i )’s. So ¯ p ∈ D t,r,k ( X ) for k = k ℓ . Notice that k depends only on ℓ and t (i.e., itdepends on neither ¯ p not r ).Let t, r, k and δ be positive reals, ¯ p ∈ D t,r,k ( X ) and ¯ q = ( q i ) i ∈ N ∈D t,r,k ( Y ), we write Y ¯ p,δ = [ i ∈ N Y p i ,δ and X ¯ q,δ = [ i ∈ N X q i ,δ . Lemma 5.7.
In the setting of Assumption 5.2, for all r > and ε > ,there is t > so that for all k ∈ N , there is δ > so that sup ¯ p ∈D t,r,k ( X ) k Φ(¯ p ) χ Y \ Y ¯ p,δ k ≤ ε. Proof.
Let θ = ε/ (2+ ε ). By Lemma 5.3 applied to Φ − , r and θ , there exists t > p ∈ C ∗ ( X ) and q ∈ C ∗ ( Y ) with propagationat most r there is E ⊂ X with diam( E ) ≤ t so that k Φ( pχ E ) q k = k Φ − ( q ) pχ E k ≥ (1 − θ ) k Φ − ( q ) p k − θ = (1 − θ ) k Φ( p ) q k − θ. Fix k ∈ N . By Lemma 5.5, pick δ > k Φ( p ) χ Y p,δ k ≥ − ( θ/k ) forall p ∈ Proj ,r ( X ).Fix ¯ p ∈ D t,r,k ( X ). For each i ∈ N with p i = 0, p i has rank 1; hence sohas Φ( p i ). For each i ∈ N , pick a unit vector ξ i ∈ ℓ ( Y, H ) in the range ofΦ( p i ). Then k Φ( p i )(1 − χ Y pi,δ ) k = k (1 − χ Y pi,δ ) ξ i k = k ξ i k − k χ Y pi,δ ξ i k = 1 − k Φ( p i ) χ Y pi,δ k ≤ ( θ/k ) . In particular, for all i ∈ N k Φ( p i )(1 − χ Y ¯ p,δ ) k ≤ k Φ( p i )(1 − χ Y pi,δ ) k ≤ θ/k. Let C = Y \ Y ¯ p,δ and Q be a finite rank projection on H . So q = χ C ⊗ Q is a projection in C ∗ ( Y ) and prop( q ) = 0. As prop(¯ p ) ≤ r , our choice of t gives E ⊂ Y with diam( E ) ≤ t so that k Φ(¯ p ) q k ≤ k Φ(¯ pχ E ) q k + θ − θ . Therefore, as ¯ p ∈ D t,r,k ( X ), we must have k Φ(¯ p ) q k ≤ k · sup i ∈ N k Φ( p i ) q k + θ − θ ≤ k · sup i ∈ N k Φ( p i )(1 − χ Y ¯ p,δ ) k + θ − θ ≤ θ − θ As Q was an arbitrary finite rank projection on H , this shows that k Φ(¯ p )(1 − χ Y ¯ p,δ ) k ≤ θ − θ ≤ ε, GELFAND-TYPE DUALITY FOR COARSE METRIC SPACES 25 and we are done. (cid:3)
We need one more lemma before presenting the proof of Theorem C.
Lemma 5.8.
In the setting of Assumption 5.2, for every positive reals δ and t there exists s > so that for all E ⊂ X with diam( E ) ≤ t and allrank 1 projections p and q in B ( ℓ ( E, H )) we have that ∂ ( Y p,δ , Y q,δ ) ≤ s .Proof. Suppose the lemma fails for δ and t . Without loss of generality,assume δ ∈ (0 , E n ) n of X ,and sequences of rank 1 projections ( p n ) n and ( q n ) n so that1. diam( E n ) ≤ t for all n ∈ N ,2. p n , q n ∈ B ( ℓ ( E n , H )) for all n ∈ N , and3. ∂ ( Y p n ,δ , Y q n ,δ ) > n for all n ∈ N .For each n ∈ N , let a n ∈ B ( ℓ ( E n , H )) be a partial isometry so that a n a ∗ n = p n and a ∗ n a n = q n .Let γ > δ = γ (2 − γ ). As Φ is coarse-like (Theorem 3.4), thereis s > a ) is γ/ s -approximable for all contractions a ∈ C ∗ ( X )with prop( a ) ≤ t . Notice that all operators in each B ( ℓ ( E n , H )) musthave propagation at most t . Hence, for each n ∈ N pick b n ∈ C ∗ ( Y ) sothat prop( b n ) ≤ s and k Φ( a n ) − b n k ≤ γ/
4. Since Y has ONL, there are s ′ > A n ) n of Y so that diam( A n ) ≤ s ′ and k b n χ A n k ≥ − γ/ n ∈ N . As prop( b n χ A n ) ≤ s for all n ∈ N , wecan use that Y has ONL once again in order to obtain a sequence of subsets( B n ) n of Y so that prop( B n ) ≤ s ′ and k χ B n b n χ A n k ≥ − γ/ n ∈ N .Hence, for all n ∈ N , k χ B n Φ( a n ) χ A n k ≥ k χ B n b n χ A n k − k Φ( a n ) − b n k > − γ. (1)Hence, as Φ( a n ) is γ/ s -approximable for all n ∈ N , this implies that d ( A n , B n ) ≤ s for all n ∈ N . Therefore,diam( A n ∪ B n ) ≤ s + 2 s ′ for all n ∈ N . As each a n is a partial isometry, it follows that k Φ( q n ) χ A n k ≥ k Φ( a n q n ) χ A n k = k Φ( a n a ∗ n a n ) χ A n k = k Φ( a n ) χ A n k > − γ for all n ∈ N . Similarly, we have that k χ B n Φ( p n ) k ≥ − γ for all n ∈ N .Let ( ξ n ) n and ( ζ n ) n be sequences of unit vectors in ℓ ( Y, H ) so that, forall n ∈ N , ξ n and ζ n belong to the range of Φ( p n ) and Φ( q n ), respectively.Given n ∈ N and y A n , we have that k Φ( q n ) χ { y } k = k χ { y } ζ n k ≤ k ζ n k −k χ A n ζ n k ≤ −k Φ( q n ) χ A n k < γ (2 − γ ) . As δ = γ (2 − γ ), this implies that y Y q n ,δ . As y was an arbitrary elementin X \ A n , this shows that Y q n ,δ ⊂ A n . Analogous arguments applied to( ξ n ) n and ( B n ) n give us that Y p n ,δ ⊂ B n for all n ∈ N . Therefore, we musthave that ∂ ( Y p n ,δ , Y q n ,δ ) ≤ s + 2 s ′ for all n ∈ N . This contradicts our choice of ( p n ) n and ( q n ) n . (cid:3) Cartan masas.
We now prove Theorem C; even better, we prove astronger version of it in Theorem 5.10. Recall, given an orthonormal basis¯ ξ = ( ξ n ) n of H , we denote by ℓ ∞ ( X, ¯ ξ ) the masa of C ∗ ( X ) consisting of alloperators a ∈ B ( ℓ ( X, H )) such that for all x ∈ X there is ( λ n ) n ∈ c with a ( δ x ⊗ ξ n ) = λ n δ x ⊗ ξ n for all n ∈ N . This algebra is a masa in the algebra of those operators in B ( ℓ ( X, H )) suchthat each entry is locally compact, and therefore it is such in C ∗ ( X ). Definition 5.9.
Given metric spaces X and Y , a map Φ : C ∗ ( X ) → C ∗ ( Y )is strongly coarse-like if for all r > s > a )) ≤ s for all a ∈ C ∗ ( X ) with prop( a ) ≤ r . Theorem 5.10.
Let X and Y be u.l.f. metric spaces and assume that Y hasproperty A . Let Φ : C ∗ ( X ) → C ∗ ( Y ) be an isomorphism. Let ¯ ξ = ( ξ n ) and ¯ ζ = ( ζ n ) be orthonormal bases of H . Then there exists a unitary v ∈ BD( Y ) such that Ad( v ) ◦ Φ : C ∗ ( X ) → C ∗ ( Y ) is strongly coarse-like and Ad( v ) ◦ Φ( ℓ ∞ ( X, ¯ ξ )) = ℓ ∞ ( Y, ¯ ζ ) . Proof.
By [21, Theorem 4.1], X and Y are coarsely equivalent. Hence X has property A and, as property A and ONL are equivalent for u.l.f. metricspaces [19, Theorem 4.1], we conclude that X , Y and Φ satisfy Assumption5.2.For n ∈ N , let P n and Q n be the projections onto span { ξ n } and span { ζ n } ,respectively. For x ∈ X , y ∈ Y and n ∈ N , let p x,n = χ { x } ⊗ P n and q y,n = χ { y } ⊗ Q n . By [21, Lemma 4.6] (or [2, Lemma 3.1]), there is δ > X q y, ,δ and Y p x, ,δ are nonempty for all x ∈ X and all y ∈ Y ; such δ isfixed for the remainder of the proof. Hence, we can pick a map f : X → Y so that f ( x ) ∈ Y p x, ,δ for all x ∈ X . By the proof of [21, Theorem 4.1], f isa coarse equivalence.Let X ⊂ X , Y ⊂ Y , r >
0, ( X x ) x ∈ X and ( Y y ) y ∈ Y be given as in § f , i.e.,1. f : X → Y is a bijection,2. X = F x ∈ X X x and Y = F x ∈ Y Y y ,3. x ∈ X x and diam( X x ) ≤ r for all x ∈ X , and4. y ∈ Y y and diam( Y y ) ≤ r for all all y ∈ Y .Let g : X × N → Y × N and u = u g : ℓ ( X, H ) → ℓ ( Y, H ) be obtained asin § x ∈ X , g restricts to a bijection X x × N → Y f ( x ) × N and uδ x ⊗ ξ n = δ g ( x,n ) ⊗ ζ g ( x,n ) for all ( x, n ) ∈ X × N . Therefore, the discussion in § u ) : C ∗ ( X ) → C ∗ ( Y ) is a strongly coarse-like isomorphism.By [21, Lemma 3.1], we can pick a unitary w : ℓ ( X, H ) → ℓ ( Y, H ) sothat Φ = Ad( w ). Let v = uw ∗ , so Ad( v ) ◦ Φ = Ψ. Therefore, as noticed
GELFAND-TYPE DUALITY FOR COARSE METRIC SPACES 27 above, Ad( v ) ◦ Φ : C ∗ ( X ) → C ∗ ( Y ) is strongly coarse-like. Moreover, it isclear from the definition of u thatAd( v ) ◦ Φ (cid:16) ℓ ∞ (cid:0) X, c ( ¯ ξ ) (cid:1)(cid:17) ⊂ ℓ ∞ (cid:0) Y, c (¯ ζ ) (cid:1) . Therefore, in order to conclude the proof, we only need to notice that v ∈ BD( Y ). For that, we now use the technical results proven in § Claim 5.11.
For all δ ′ ∈ (0 , δ ], there is r > y ∈ Y and allrank 1 projections q ∈ B ( ℓ ( { y } , H )) we have Y Ψ − ( q ) ,δ ′ ⊂ B r ( y ). Proof.
Fix δ ′ ∈ (0 , δ ]. Let s > δ ′ and r . Lemma 5.4 gives k > Y p,δ ′ ) ≤ k for all p ∈ Proj ,r ( X ).Fix y ∈ Y and a rank 1 projection q ∈ B ( ℓ ( { y } , H )). Let y ′ ∈ Y and x ′ ∈ X be so that y ∈ Y y ′ and y ′ = f ( x ′ ). Since g : X × N → Y × N restricts to a bijection X x ′ × N → Y y ′ × N , the definition of Ψ clearlyimplies that Ψ − ( q ) ∈ B ( ℓ ( X x ′ , H )). Hence, as p x ′ , ∈ B ( ℓ ( X x ′ , H )),diam( X x ′ ) ≤ r , our choice of s implies that ∂ ( Y Ψ − ( q ) ,δ ′ , Y p x ′ , ,δ ′ ) ≤ s . Bythe defining property of f : X → Y , we have that y ′ = f ( x ′ ) ∈ Y p x ′ , ,δ . As δ ′ ≤ δ , y ′ ∈ Y p x ′ , ,δ ′ . Therefore, as diam( Y y ′ ) ≤ r we have ∂ ( y, y ′ ) ≤ r ,and our choices of ℓ and k imply that Y Ψ − ( q ) ,δ ′ ⊂ B r + s +2 k ( y )The claim then follows by letting r = r + s + 2 k . (cid:3) We now show that v = uw ∗ belongs to BD( Y ). By [22, Theorem 3.3], as Y has property A, it is enough to show that v is quasi-local . Fix ε >
0. Let t > ε , r and Φ. Then, for all k ∈ N there is δ ′ ∈ (0 , δ ] so that k Φ(¯ p ) χ Y \ Y ¯ p,δ ′ k ≤ ε (2)for all ¯ p = ( p i ) i ∈ N ∈ D t,r ,k ( X ).Before finishing the proof, we introduce some notation: given C ⊂ Y and a sequence of projections ( Q y ) y ∈ C on H of rank at most 1, we write q y = χ { y } ⊗ Q y for each y ∈ C . Let X C = n x ∈ X | ∃ x ′ ∈ X with x ∈ X x ′ and C ∩ Y f ( x ′ ) = ∅ o . By the definition of Ψ, it follows that Ψ − ( q y ) ∈ B ( ℓ ( X x ′ , H )) for all y ′ ∈ Y with y ∈ Y y ′ and x ′ = f − ( y ′ ). Remark 5.6 then implies that there is k > − (¯ q ) = (Ψ − ( q y )) y ∈ C ∈ D t,r ,k ( X C ). Moreover, as noticedin Remark 5.6, k depends only on t and sup y ∈ Y | Y y | (i.e., it depends onneither C nor ( Q y ) y ∈ C ). Fix such k . An operator a ∈ B ( ℓ ( Y, H )) is quasi-local if for all ε > r > ∂ ( A, B ) > r implies k χ A aχ B k < ε for all A, B ⊂ Y . Notice that q y and Q y (for y ∈ C ) are distinct from q y,n and Q n (for n ∈ N ). Webelieve that, since the indices are different, this abuse of notation will cause no confusion. By the defining property of t , pick δ ′ ∈ (0 , δ ] so that (2) holds for k .Claim 5.11 gives r > Y Ψ − ( q ) ,δ ′ ⊂ B r ( y ) for all y ∈ Y and allrank 1 projections q ∈ B ( ℓ ( { y } , H )). Therefore, if C ⊂ Y and ( Q y ) y ∈ C is asequence of projections on H of rank at most 1, it follows that Y Ψ − (¯ q ) ,δ = [ y ∈ C Y Ψ − ( q y ) ,δ ⊂ B r ( C ) . Fix C ⊂ Y , ( Q y ) y ∈ C and ( q y ) y ∈ C as above. Then the previous inequalitiesgive that k Φ(Ψ − (¯ q )) χ Y \ B r ( C ) k ≤ ε. As Ψ − = Ad( u ∗ ) and Φ = Ad( w ), this implies that k ¯ qvχ Y \ B r ( C ) k = k ¯ quw ∗ χ Y \ B r ( C ) k = k wu ∗ ¯ quw ∗ χ Y \ B r ( C ) k ≤ ε. The arbitrariness of ¯ q = ( q y ) y ∈ C (i.e., the arbitrariness of C ⊂ Y and( Q y ) y ∈ C ) gives that k χ C vχ Y \ B r ( C ) k ≤ ε for all C ⊂ Y .Let A, B ⊂ Y be so that d ( A, B ) > r . Then k χ A vχ B k ≤ k χ A ∩ Y ′ vχ Y \ B r ( A ) k ≤ ε As ε was arbitrary, this shows that v is quasi-local, so we are done. (cid:3) Remark . Although we chose to work with fixed bases ¯ ξ and ¯ ζ insideeach H -coordinate of ℓ ( X, H ) in Theorem C (Theorem 5.10), we chose somerely for simplicity. The same proof holds if for each x ∈ X we choosebasis ¯ ξ x = ( ξ xn ) n and ¯ ζ x = ( ζ xn ) n of H . Precisely, given those choices,let Q x ∈ X c ( ¯ ξ x ) be the ℓ ∞ -sum of ( c ( ¯ ξ x )) x ∈ X , where c ( ¯ ξ x ) consists of all a ∈ B ( H ) so that there is ( λ n ) n ∈ c for which aξ xn = λ n ξ xn for all n ∈ N ; Q x ∈ X c (¯ ζ x ) is defined analogously. Then, if Φ ∈ Aut(C ∗ ( X )), there existsa unitary v ∈ BD( X ) so thatAd( v ) ◦ Φ (cid:16) Y x ∈ X c (cid:0) ¯ ξ x (cid:1)(cid:17) = Y x ∈ X c (cid:0) ¯ ζ x (cid:1) . We are ready to prove Theorem B, which we restate for convenience.
Theorem 5.13.
Let ( X, d ) be a u.l.f. metric space with property A. Thecanonical homomorphism Coa( X ) → Out(C ∗ ( X )) described in § Let T : Coa( X ) → Out(C ∗ ( X )) be the injective homomorphism con-structed in § ∈ Aut(C ∗ ( X )). Let v ∈ BD( X ) be given by The-orem C for Φ. Moreover, let f : X → X , u : ℓ ( X, H ) → ℓ ( X, H ) and w : ℓ ( X, H ) → ℓ ( X, H ) be as in the proof of Theorem C for X = Y and¯ ξ = ¯ ζ . Hence, Φ = Ad( w ), v = uw ∗ . Notice that T ( f ) = [Ad( u )] when thelatter is computed in Out(C ∗ ( X )). GELFAND-TYPE DUALITY FOR COARSE METRIC SPACES 29
We are left to show that T ( f ) = [Φ], that is, that Ad( u ) ◦ Φ − ∈ Inn(BD( X )). But this follows sinceAd( u ) ◦ Φ − = Ad( u ) ◦ Ad( w ∗ ) = Ad( v )and v ∈ BD( X ). (cid:3) Applications
In this section, we use Theorem A and Theorem B in order to compute— or at least better understand — Out(C ∗ u ( X )) and Out(C ∗ ( X )) for somespecific spaces X . In § § § { n | n ∈ N } , N n , and Z n , while in § § B (1 , n ) and the lamplighter group F ≀ Z for a finite group F . For brevity, we skip some definitions in these subsections and refer thereader to an appropriate source.6.1. Outer automorphisms of the (uniform) Roe algebra of { n | n ∈ N } . Denote the group of permutations on N by S ∞ . Let ∼ be theequivalence relation on S ∞ given by π ∼ π ′ if there is n ∈ N so that π ( n ) = π ′ ( n ) for all n ≥ n . Clearly, N = { π ∈ S ∞ | π ∼ Id N } is a normalsubgroup of S ∞ , so S ∞ / ∼ = S ∞ /N is a group. Corollary 6.1.
Let X = { n | n ∈ N } . Then BijCoa( X ) is isomorphic to S ∞ / ∼ . In particular, Out(C ∗ u ( X )) is isomorphic to S ∞ / ∼ .Proof. First, notice that X has property A. For this, notice that C ∗ u ( X ) isgenerated by ℓ ∞ ( X ) and K ( ℓ ( X )). Moreover the uniform Roe corona of X ,the algebra C ∗ u ( X ) / K ( ℓ ( X )) is isomorphic to ℓ ∞ /c , and it is therefore nu-clear. Since K ( ℓ ( X )) is nuclear, so is C ∗ u ( X ), and therefore X has property A .A map f : X → X is a bijective coarse equivalence if and only if f is abijection. So the group of bijective coarse equivalences of X is isomorphic to S ∞ . Moreover, two maps f, g : X → X are close if and only if they eventuallycoincide, i.e., there is n ∈ N so that f ( n ) = g ( n ) for all n > n . Theresult now follows. (cid:3) Denote the group of cofinite partial bijections on N by S ∗∞ , i.e., S ∗∞ = n ( π, A, B ) ∈ N N × P ( N ) × P ( N ) : | A ∁ | , | B ∁ | < ∞ and π ↾ A : A → B is a bijection o . By a slight abuse of notation, we denote by ∼ the equivalence relation on S ∗∞ given by ( π, A, B ) ∼ ( π ′ , A ′ , B ′ ) if there is n ∈ N so that π ( n ) = π ′ ( n )for all n ≥ n . Corollary 6.2.
Let X = { n | n ∈ N } . Then Coa( X ) / ∼ is isomorphic to S ∗∞ / ∼ . In particular, Out(C ∗ ( X )) is isomorphic to S ∗∞ / ∼ . Proof.
Clearly, a map f : X → X is a coarse equivalence if and only if f there are cofinite A, B ⊂ N so that f ↾ A : A → B is a bijection. Moreover,maps f, g : X → X are close if and only if they eventually coincide. Theresult follows. (cid:3) Outer automorphisms of the uniform Roe algebras of N and Z .Corollary 6.3. The group
BijCoa( N ) is trivial and BijCoa( Z ) is isomorphicto Z . In particular, Out(C ∗ u ( N )) is trivial and Out(C ∗ u ( Z )) is isomorphicto Z .Proof. Fix a bijective coarse equivalence f : N → N . We are going to provethat f is close to the identity Id N . Suppose this is not the case. Then thereis a sequence ( x n ) n in N so that | f ( x n ) − x n | > n for all n ∈ N . Withoutloss of generality, we can assume that ( x n ) n is strictly increasing. Moreover,replacing f by f − if necessary, we can also assume that f ( x n ) + n < x n for all n ∈ N . For each n ∈ N , let z n = max { z ∈ N | f − ( z ) ≤ x n } . As f is a bijection and f ( x n ) + n < x n , it follows that z n − f ( x n ) > n for all n ∈ N . As f − is expanding, it follows that lim n ( f − ( z n + 1) − x n ) = ∞ .So, as f − ( z n ) ≤ x n , we have that lim n ( f − ( z n + 1) − f − ( z n )) = ∞ . Thiscontradicts coarseness of f − . This shows that BijCoa( N ) is the trivialgroup.Now fix a bijective coarse equivalence f : Z → Z . So either lim n →∞ f ( n ) = ∞ or lim n →∞ f ( n ) = −∞ . Assume that lim n →∞ f ( n ) = ∞ and let x =min( f ( N )). Claim 6.4.
There are bijective coarse equivalences h : N → N and h : Z \ N → Z \ N so that f is close to the bijection h ∪ h : Z → Z . Proof.
Notice that f is close to g = f − x and that g ( N ) is a cofinite subsetof N , say n = | N \ g ( N ) | . Pick bijections i : {− n , . . . , − } → N \ g ( N )and j : g − ( N \ g ( N )) → g ( {− n , . . . , − } ) , and notice that g is close to h ( x ) = g ( x ) , x ∈ N ,i ( x ) , x ∈ {− n , . . . , − } ,j ( x ) , x ∈ g − ( N \ g ( N )) , For each x ∈ Z , let h ( x ) = h ( x − n ) and let h = ↾ N and h = h ↾ Z \ N .Since h is close to h , the result follows. (cid:3) Let h and h be given by the claim above. As BijCoa( N ) is the trivialgroup, it follows that h is close to Id N and h is close to Id Z \ N . So f isclose to Id Z .If lim n →∞ f ( n ) = −∞ , then proceeding analogously as above, we obtainbijective coarse equivalences h : N → Z \ N and h : Z × N → N so that GELFAND-TYPE DUALITY FOR COARSE METRIC SPACES 31 h is close to the map x ∈ N → − x − ∈ Z \ N ,2. h is close to the map x ∈ Z \ N → − x − ∈ N , and3. f is close to h ∪ h .As h ∪ h is close to − Id Z , so is f .We have then shown that BijCoa( Z ) is isomorphic to {− Id Z , Id Z } . Thiscompletes the proof.The last statement follows from the above and Theorem A. (cid:3) Outer automorphisms of the Roe algebra of Z n . Recall, givenmetric spaces (
X, d ) and (
Y, ∂ ), a map f : X → Y is a coarse Lipschitzequivalence if it is cobounded and there is L > L − d ( x, y ) − L ≤ ∂ ( f ( x ) , f ( y )) ≤ Ld ( x, y ) + L for all x, y ∈ X . DefineCoaLip( X ) = n f : X → X | f is a coarse Lipschitz equivalence o / ∼ , where ∼ is the closeness relation on functions X → X . Clearly, CoaLip( X )is a group under composition, i.e., [ f ] ◦ [ g ] = [ f ◦ g ].Given n ∈ N , the inclusion Z n ֒ → R n is a coarse equivalence (coarseLipschitz equivalence even). Therefore Coa( Z n ) ∼ = Coa( R n ). Moreover,notice that a map R → R is coarse if and only if it is coarse Lipschitz [15,Theorem 1.4.13]. Therefore, we have thatCoa( Z n ) ∼ = CoaLip( R n ) . As a consequence of that, results in the literature give us the next corollariesof Theorem B. We denote by PL δ ( Z ) the group of piecewise linear homeo-morphisms f : R → R so that {| f ′ ( x ) | | x ∈ R } ⊂ [ M − , M ] for some M > δ ( Z ) / ∼ . Corollary 6.5.
The group
Out(C ∗ ( Z )) has trivial center. Moreover, Out(C ∗ ( Z )) is isomorphic to PL δ ( Z ) / ∼ .Proof. This follows immediately from Theorem B, the discussion precedingthe corollary, [8, Theorem 1.1], and [20, Theorem 1.2]. (cid:3)
Corollary 6.6.
The group
Out(C ∗ ( Z )) contains isomorphic copies of thefollowing groups:1. PL κ ( R ) , the group of piecewise linear homeomorphisms f : R → R sothat { x ∈ R | f ( x ) = x } is compact,2. Thompson’s group F , and3. the free group of rank the continuum.Proof. This follows immediately from Theorem B, the discussion above and[20, Theorem 1.3]. (cid:3) Coarse Lipschitz equivalences are also referred to as quasi-isometries in the literature. We refer the reader to [7, Section 1] for the definition of Thompson’s group F . Given a metric space (
X, d ), a map f : X → X is a bi-Lipschitz equiva-lence if there is L > L − d ( x, y ) ≤ d ( f ( x ) , f ( y )) ≤ Ld ( x, y )for all x, y ∈ X . We letBiLip( X ) = n f : X → X | f is a bi-Lipschitz equivalence o . So BiLip( X ) is a group under composition (notice that we do not mod outthe bi-Lipschitz equivalences by closeness). Corollary 6.7.
Given n ∈ N , the group Out(C ∗ ( Z n )) contains isomorphiccopies of the following groups:1. BiLip( S n − ) , where S = { z ∈ C | | z | = 1 } , and2. BiLip( D n , S n − ) .Proof. This follows immediately from Theorem B, the discussion above and[14, Theorem 1.1]. (cid:3)
Solvable Baumslag-Solitar groups.
Given n ∈ N , Q n denotes the n -adic rationals and B (1 , n ) denotes the solvable Baumslag-Solitar group ,i.e., the group generated by elements a and b subject to the relation aba − = b n . We endow B (1 , n ) with the metric given by its Cayley graph structure(see [15, Definition 1.2.7] for definitions). Corollary 6.8.
Given n ∈ N , the group Out(C ∗ ( B (1 , n ))) is isomorphic to BiLip( R ) × BiLip( Q n ) .Proof. Since B (1 , n ) is solvable, it is amenable. Therefore, as B (1 , n ) isfinitely generated, it must also have property A [15, Theorem 4.14]. ByTheorem B, we only need to compute Coa( B (1 , n )). As B (1 , n ) is a finitelygenerated group, we have that Coa( B (1 , n )) = CoaLip( B (1 , n )) [15, Corol-lary 1.4.15]. The result then follows since it is known that CoaLip( B (1 , n ))is isomorphic to BiLip( R ) × BiLip( Q n ) (see [11, Theorem 8.1]). (cid:3) The lamplighter group F ≀ Z . Given a group F , we denote the wreathproduct of F and Z by F ≀ Z (we refer the reader to [15, Definition 2.6.2] fora precise definition). This group is commonly called the lamplighter group F ≀ Z , and we endow F ≀ Z with the metric given by its Cayley graph structure[15, Definition 1.2.7].Consider the semidirect product (BiLip( Q n ) × BiLip( Q n )) ⋊ Z given bythe action of Z on BiLip( Q n ) × BiLip( Q n ) of switching factors. Corollary 6.9.
Let F be a group with | F | = n . Then the group Out(C ∗ ( F ≀ Z )) is isomorphic to (cid:16) BiLip( Q n ) × BiLip( Q n ) (cid:17) ⋊ Z Recall that if N and H are groups and α : H y N is an action, then N ⋊ H denotesthe semidirect product , i.e., the set N × H endowed with the product ( n, h ) · ( n ′ , h ′ ) =( nα ( h ) n ′ , hh ′ ). GELFAND-TYPE DUALITY FOR COARSE METRIC SPACES 33
Proof.
Since F and Z are finitely generated, so is F ≀ Z [15, Chapter 2,Exercise 2.5]. The lamplighter group F ≀ Z is amenable (see [24, Corollary2.5]), and since it is finitely generated, it has property A [15, Theorem 4.14].Therefore, we only need to compute Coa( F ≀ Z ) (Theorem B). As F ≀ Z isfinitely generated, Coa( F ≀ Z ) = CoaLip( F ≀ Z ), by [15, Theorem 1.4.13].Moreover, F ≀ Z is quasi-isometric to the Diestel-Leader graph D ( n, n ) (werefer to [9, Section 1] for both the definition of DL( n, n ) and this fact). So itis enough to compute CoaLip(DL( n, n )). The result then follows since it isknown that CoaLip(DL( n, n )) is isomorphic to (BiLip( Q n ) × BiLip( Q n )) ⋊Z [9, Theorem 2.1] (see the discussion at the end of [9, Section 2]). (cid:3) Acknowledgements.
The current paper started from a question asked by RalfMeyer to the authors about whether Theorem B was true for the metricspace Z n . The authors would like to thank Ralf Meyer for the interestingquestion and for several comments on a previous version of this paper. AV ispartially supported by the ANR Project AGRUME (ANR-17-CE40-0026). References [1] B. Blackadar.
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University of Virginia, 141 Cabell Drive, Kerchof Hall, P.O.Box 400137, Charlottesville, USA
E-mail address : [email protected] URL : https://sites.google.com/site/demendoncabraga (A. Vignati) Institut de Math´ematiques de Jussieu - Paris Rive Gauche (IMJ-PRG), Universit´e Paris Diderot, Bˆatiment Sophie Germain, 8 Place Aur´elieNemours, 75013 Paris, France
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