The coarse Baum-Connes conjecture for certain extensions and relative expanders
aa r X i v : . [ m a t h . K T ] F e b The coarse Baum-Connes conjecture for certain groupextensions and relative expanders ∗Jintao Deng, Qin Wang, and Guoliang Yu
Abstract
Let (1 → N n → G n → Q n → n ∈ N be a sequence of extensions of finitely generated groups withuniformly finite generating subsets. We show that if the sequence ( N n ) n ∈ N with the induced metricfrom the word metrics of ( G n ) n ∈ N has property A, and the sequence ( Q n ) n ∈ N with the quotientmetrics coarsely embeds into Hilbert space, then the coarse Baum-Connes conjecture holds for thesequence ( G n ) n ∈ N , which may not admit a coarse embedding into Hilbert space. It follows that thecoarse Baum-Connes conjecture holds for the relative expanders and group extensions exhibited byG. Arzhantseva and R. Tessera, and special box spaces of free groups discovered by T. Delabie andA. Khukhro, which do not coarsely embed into Hilbert space, yet do not contain a weakly embeddedexpander. This in particular solves an open problem raised by G. Arzhantseva and R. Tessera [3]. Contents Z ⋊ Q F . . . . . . . . . . . . . . . . 62.1.2 Relative expander example 2: a box space of Z ≀ Q F U . . . . . . . . . . . . . . . . . 82.1.3 Relative expander example 3: a box space of Z ≀ Q F . . . . . . . . . . . . . . . . . 92.2 Group extensions à la G. Arzhantseva and R. Tessera . . . . . . . . . . . . . . . . . . . . 102.3 Box spaces of free groups à la T. Delabie and A. Khukhuro . . . . . . . . . . . . . . . . . 12 ∗ The second author is partially supported by NSFC (No. 11771143, 11831006). The third author is partially supportedby NSF (No. 1700021, 2000082), and the Simons Fellows Program. Twisted Roe algebras and twisted localization algebras 24 β and β L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.2 Dirac maps α and α L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.3 The geometric analogue of Bott periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . 48 The coarse Baum-Connes conjecture is a geometric analogue of the Baum-Connes conjecture which hasimportant applications to geometry, topology and analysis. More precisely, it states that a certain assemblymap µ : lim d →∞ K ∗ ( P d ( X )) → K ∗ ( C ∗ ( X )) for a metric space X is an isomorphism, where on the left hand side lim d →∞ K ∗ ( P d ( X )) is the limit of thelocally finite K -homology group of the Rips complexes of X , while on the right hand side K ∗ ( C ∗ ( X )) is the K -theory group of the Roe algebra of X . The left hand side of this conjecture is local andtherefore computable, while the right hand side is global and is the receptacle of higher indices of ellipticoperators. A positive answer to this conjecture would provide a complete solution to the problem ofcomputing K -theoretic indices for elliptic operators on non-compact spaces. In particular, it implies theNovikov conjecture on homotopy invariance of higher signatures for closed manifolds when X is a finitelygenerated group (e.g. the fundamental group of a closed manifold) equipped with a word length metric,and the Gromov’s zero-in-the-spectrum conjecture and the positive scalar curvature conjecture when X is a Riemannian manifold. See [55] for a comprehensive survey for the coarse Baum-Connes conjecture,and [5, 9, 18, 19, 20, 21, 22, 24, 41, 48, 49, 50, 51, 53] for some recent developments.The notion of coarse embedding into Hilbert space was introduced by M. Gromov [25, p. 211] inrelation to the Novikov conjecture (1965). A sequence of metric spaces ( X n , d n ) n ∈ N is said to admit a coarse embedding into Hilbert space , or briefly, CE , if there exist a sequence of maps f n : X n → H from X n to a Hilbert space H , and two proper maps ρ and ρ from [0 , ∞ ) to [0 , ∞ ) such that ρ ( d n ( x, x ′ )) ≤ k f n ( x ) − f n ( x ′ ) k ≤ ρ ( d n ( x, x ′ )) . for all n ∈ N and all x, x ′ ∈ X n . The third author of this paper G. Yu established the coarse Baum-Connesconjecture for all metric spaces with bounded geometry which admit a coarse embedding into a Hilbertspace [54]. In the same paper, G. Yu introduced a geometric version of amenability, called property A ,which implies coarse embeddability into Hilbert space. A sequence of countable discrete metric spaces ( X n ) n ∈ N has Property A if and only if for every
R > and ε > , there exist S > , and a sequence2f functions ξ n : X n → ℓ ( X n ) such that (0) k ξ x k = 1 ; (1) if d ( x, x ′ ) ≤ R , then k ξ x − ξ x ′ k ≤ ε ; (2) Supp( ξ x ) ⊂ Ball X n ( x, S ) , for all n ∈ N and all x, x ′ ∈ X n (see [38]).The main result of this paper is the following theorem. Theorem 1.1.
Let (1 → N n → G n → Q n → n ∈ N be a sequence of extensions of finitely generated groupswith uniformly finite generating subsets. If the sequence ( N n ) n ∈ N with the induced metric from the wordmetrics of ( G n ) n ∈ N has property A, and the sequence ( Q n ) n ∈ N with the quotient metrics coarsely embedsinto Hilbert space, then the coarse Baum-Connes conjecture holds for the sequence ( G n ) n ∈ N . We will say that a sequence of group extensions (1 → N n → G n → Q n → n ∈ N is an "A-by-CE"extension , or has the "A-by-CE" structure , if it satisfies the assumptions in the theorem. Similarly, wemay talk about "CE-by-CE", "CE-by-A", "abelian-by-Haagerup", and so on for obvious meanings. Fora long time, weakly embedded expanders (see Definition 2.1 below) were the only known obstructionfor a metric space with bounded geometry to coarsely embed into Hilbert space [26, 27, 36]. In [3],G. Arzhantseva and R. Tessera introduce the notion of relative expanders (see Definition 2.4 below) toconstruct the first sequence of finite Cayley graphs which does not coarsely embed into any L p -spacefor any ≤ p < ∞ , nor into any uniformly curved Banach space, and yet does not admit any weaklyembedded expander. In [14], T. Delabie and A. Khukhro construct a certain box space of a free groupto answer in the affirmative an open problem asked in [3]: does there exist a sequence of finite graphswith bounded degree and large girth which does not coarsely embed into a Hilbert space and yet doesnot contain a weakly embedded expander? We observe that all these examples by G. Arzhantseva and R.Tessera, and T. Delabie and A. Khukhro have the A-by-CE group extension structure. Hence, we have Corollary 1.2.
The coarse Baum-Connes conjecture holds for all relative expanders exhibited by G.Arzhantseva and R. Tessera, and the special box spaces of free groups discovered by T. Delabie and A.Khukhro, which do not coarsely embed into Hilbert space, yet do not contain a weakly embedded expander.
This in particular solves an open problem raised by G. Arzhantseva and R. Tessera in [3, Section 8Open Problems]. For a single extension of groups, the above theorem may be restated as follows.
Theorem 1.3.
Let → N → G → Q → be a short exact sequence of finitely generated groups. If N has Property A and Q coarsely embeds into Hilbert space, then the coarse Baum-Connes conjecture holdsfor G . In [4], G. Arzhantseva and R. Tessera answer in the negative the following well-known question [12, 28]:Does coarse embeddability into Hilbert spaces is preserved under group extensions of finitely generated groups? Their constructions also provide the first example of finitely generated group which does notcoarsely embed into a Hilbert space and yet does not contain weakly embedded expander, answering inthe affirmative another open problem in [28], see also [3, 38]. The first group Z ≀ G H constructed by G.Arzhatseva and R. Tessera in [4] is an extension of two groups with the Haagerup property. It satisfiesthe strong Baum-Connes conjecture [29] which is strictly stronger than the Baum-Connes conjecturewith coefficients [7], and consequently, the coarse Baum-Connes conjecture. It has been proved in [5,Proposition 2.11] by B. M. Braga, Y. C. Chung and K. Li that the second group Z ≀ G ( H × F n ) constructedin [4] also satisfies the Baum-Connes conjecture with coefficients by applying a permanence result of H.Oyono-Oyono [40], and hence the coarse Baum-Connes conjecture. Since both groups are A-by-CE splitextensions, our result provides an alternative proof to these facts. Notice that the reason why Z ≀ G H and Z ≀ G ( H × F n ) do not coarsely embed into Hilbert space is that both groups contain isometrically3n their Cayley graphs a technical relative expander W n = Z ≀ G n H n , where ( G n ) is a sequence of finitegroups whose Cayley graphs are Ramanujan graphs, and H n is a certain finite-sheeted cover of G n [4,Proposition 2.16]. We observe that this relative expander W n = Z ≀ G n H n also has the A-by-CE structure(see Remark 2.10 below), and hence satisfies the coarse Baum-Connes conjecture as well. So, the coarseBaum-Connes conjecture holds for all currently known examples of spaces or groups which do not coarselyembed into Hilbert space, yet do not contain a weakly embedded expander.When the group G has a classifying space of finite type, Theorem 1.3, together with the decentprinciple, recovers the extension results, in the case where N has property A, on the Novikov conjecturein [15, Theorem 1.1] by J. Deng, and in [17, Theorem 33] (together with [16, Theorem 61]) by H. Emersonand R. Meyer.The basic strategy of the proof of Theorem 1.1 is to apply the fundamental ideas in [54] to the case ofgroup extensions, by using localization algebras, twisted Roe algebras, and a geometric version of infinitedimensional Bott periodicity. We use the coarse embedding of the quotients ( Q n ) n ∈ N into a Hilbertspace to furnish the twisted Roe algebra of ( G n ) n ∈ N and its localization algebra with twisted coefficients,and then apply cutting-and-pasting techniques to decompose these twisted algebras, so as to reduce thecoarse Baum-Connes conjecture for ( G n ) n ∈ N to the coarse Baum-Connes conjecture for ( N n ) n ∈ N . At thispoint, it is natural to expect to complete the proof merely under the assumption that ( N n ) n ∈ N coarselyembeds into a Hilbert space. However, there is a subtle issue about different completions of ideals insidethe maximal twisted Roe algebras, for which we can not settle under this assumption. Fortunately, thistechnical difficulty disappears under the assumption that the sequence ( N n ) n ∈ N has Property A.The paper is organized as follows. In Section 2, we briefly review all the examples of relative expandersand group extensions constructed by G. Arzhantseva and R. Tessera, and a special box space of free groupdiscovered by T. Delabie and A. Khukhro, which do not coarsely embed into Hilbert spaces and yet donot contain a weakly embedded expander. We in particular indicate that all these examples has the "A-by-CE" structure. In Section 3, we recall the concept of the Roe algebras, localization algebras, and thecoarse Baum-Connes conjecture. In Section 4, we introduce uniformly twisted Roe algebras and uniformlytwisted localization algebras for the extension groups with coefficients coming from the coarse embeddingof the quotient groups into Hilbert space, and prove that the twisted coarse Baum-Connes conjecturefor the sequence of extensions. In Section 5, we introduce a version of the geometric analogue of theinfinite-dimensional Bott periodicity of Higson-Kasparov-Trout to complete the proof.As a comparison, we mention that "CE-by-A" implies "CE". This was proved by M. Dadalart andE. Guentner in [12] for a single short exact sequence of groups, and by A. Khukhro [31] for a sequence ofextensions of finite groups with uniformly finite generating subsets. So the following question is natural: Problem.
Does "CE-by-CE" implies the coarse Baum-Connes conjecture?
In this section, we briefly review the recent discoveries of relative expanders and certain groups extensionsdue to G. Arzhantseva and R. Tessera, and of certain box spaces of free groups due to T. Delabie and A.Khukhro, which do not coarsely embed into Hilbert space and yet contain no weakly embedded expanders.4e observe that all these spaces or groups have “A-by-CE" structure as (sequences of) group extensions.For a finite connected graph X with | X | vertices and a subset A ⊂ X , denote by ∂A the set of edgesbetween A and X \ A . The Cheeger constant of X is defined as h ( X ) := min ≤| A |≤| X | / | ∂A || A | . An expander is a sequence ( X n ) n ∈ N of finite connected graphs with uniformly bounded degree, such that | X n | → ∞ , and h ( X n ) ≥ c uniformly over n ∈ N for some constant c > . Definition 2.1 (weakly embedded expander) . Let ( X n ) n ∈ N be an expander and let Y be a discrete metricspace with bounded geometry. A sequence of maps f n : X n → Y is a weak embedding of ( X n ) n ∈ N into Y if there exists D > such that all f n are D -Lipschitz and lim n →∞ sup x ∈ X n f − n ( f n ( x )) | X n | = 0 . Let us recall the concepts of semidirect product and wreath product of groups. Let H be a group, let Q be a group of automorphisms of H , and let K be a group such that there is a surjective homomorphism φ : K ։ Q . The restricted semi-direct product H ⋊ Q K is the product H × K with the multiplication role ( h , k )( h , k ) = (cid:0) h · φ ( k ) h , k k (cid:1) . Note that a semi-direct product is a spit extension.Let H and K be finitely generated groups, and let φ : K ։ Q be a surjective homomorphism from K to a countable discrete group Q . The restricted permutational wreath product of H by K through Q is thesemi-direct product H ≀ Q K := M Q H ⋊ K, where L Q H is the group of finitely supported functions ξ : Q → H with the pointwise multiplication,and K acts on L Q H by permuting the indices by multiplications on the left via φ : K ։ Q .Let S and T be finite generating sets of H and K , respectively. Then (cid:8) ( δ s , K ) : s ∈ S (cid:9) ⊔ (cid:8) ( , t ) : t ∈ T (cid:9) is a finite generating subset of H ≀ Q K , where δ s ( q ) = ( s, if q = 1 Q ;1 H , otherwise , and is the constant function in L Q H such that ( q ) ≡ H for all q ∈ Q . A well known obstruction for a metric space to coarsely embed into a Hilbert space is to admit a weaklyembedded expander [26, 27]. A long standing open problem is: does a weakly embedded expander isthe only obstruction to coarse embeddability into Hilbert space? In a ground breaking article [3], G.Arzhantseva and R. Tessera gave first examples of sequences of finite Cayley graphs of uniformly boundeddegree which do not coarsely embed into a Hilbert space but do not contain any weakly embeddedexpander. 5 heorem 2.2 ([3] Theorem 1 or 7.1) . There exist a finitely generated residually finite group G and abox space ( Y n ) n ∈ N of G which does not coarsely embed into any L p space for p ∈ [1 . ∞ ) , neither into anyuniformly curved Banach space, and yet does not admit any sequence of weakly embedded expanders. The proof of this theorem relies on two major observations. The first one says that there are noexpanders weakly contained in "CE-by-CE" group extensions.
Proposition 2.3 ( Observation 1). (Proposition 2 in [3]) A "CE-by-CE" sequence of group extensionsdoes not contain any weakly embedded expander. More presicely, let (cid:16) −→ N n −→ G n −→ Q n −→ (cid:17) n ∈ N be a sequence of extensions of finitely generated groups with uniformly finite generating subsets, such that(1) for each n ∈ N , the group G n is equipped with the word length induced by a given generating sets S n ;(2) the sequence ( N n ) n ∈ N equipped with the induced metric as subgroups of G n admit a coarse embeddinginto a Hilbert space;(3) the sequence ( Q n ) n ∈ N equipped with the quotient metric from G n admit a coarse embedding into aHilbert space.Then the sequence ( G n ) n ∈ N does not contain any weakly embedded expander. The second observation is a refined strengthening to an early observation of John Roe [43] that relativeproperty (T), as opposite to the Haagerup property, leads to non-embeddability into Hilbert space. Thisis done by introducing a notion of relative expander in terms of Poincaré inequality [3].
Definition 2.4 (Relative expanders [3], 1.4) . Let (cid:0) G n (cid:1) n ∈ N , be a sequence of finite groups with generatingsubsets S n such that sup n | S n | < ∞ , and let Y n ⊆ G n be an unbounded sequence of subsets of G n , i.e.for any R > there exists n such that Y n is not contained entirely in the ball of radius R around theneutral element in G n . Then the sequence of Cayley graphs ( G n , S n ) n ∈ N is said to be a relative expanderwith respect to ( Y n ) if it satisfies the "relative Poincaré inequality": there exists C > such that for every n ∈ N , for every function f : G n → H from G n to a Hilbert space H , and for every y ∈ Y n , we have X g ∈ G n k f ( gy ) − f ( g ) k ≤ C X g ∈ G n ,s ∈ S n k f ( gs ) − f ( g ) k . Proposition 2.5 ( Observation 2). ([3], Proposition 3 and Corollary 1.1) Let G be a finitely generatedresidually finite group, with a finite generating subset S . If G has relative property (T) with respect toan infinite subset Y ⊂ G , then for any filtration ( N n ) of G , with the quotient maps π n : G → G/N n , thesequence of Cayley graphs ( G/N n , π n ( S )) is a relative expander with respect to the unbounded sequence ( π n ( Y )) n . In particular, the box space F n G/N n does not coarsely embed into a Hilbert space. In the paper [3], G. Arzhantseva and R. Tessera provide three examples of relative expanders whichare box spaces of certain carefully defined semi-direct product groups. We observe that all these examplesare "A-by-CE" sequence of extensions of finite groups. Z ⋊ Q F Consider the semi-direct product G := Z ⋊ Q F , Q is the kernel of the surjection SL(2 , Z ) ։ SL(2 , Z ) , which is generated by elements: α = − − ! , β = ! , γ = ! . Take a surjective homomorphism π : F ։ Q mapping the standard generating subset { a, b, c } of F ontosome generating subset { α, β, γ } of Q , so that F acts on Z via the standard action of Q on Z . Thenthe set S := { (1 , , (0 , , a, b, c } is a finite generating subset of G .Define a sequence of semi-direct products (i.e. split extensions) of finite groups as follows: G n := (cid:16) Z n (cid:17) ⋊ Q n (cid:16) F / Γ n − ( F ) (cid:17) where, for all n ≥ ,• Q n < SL(2 , Z n ) is the image of Q <
SL(2 , Z ) under the quotient map π n : SL(2 , Z ) ։ SL(2 , Z n ); Q n is a -group of order n − . SL(2 , Z ) π n / / / / SL(2 , Z n ) F π / / / / q n − (cid:15) (cid:15) (cid:15) (cid:15) Q ?(cid:31) O O π n / / / / Q n ?(cid:31) O O F / Γ n − ( F ) ^ q n − ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ • For any group G and every n ≥ , define inductively the characteristic subgroup Γ n ( G ) of G as: Γ ( G ) = G , and Γ n +1 ( G ) is the subgroup of Γ n ( G ) generated by squares of elements of Γ n ( G ) .A central tool of Arzhantseva-Tessera [3, Lemma 2.3] is the following observation. If G is a finitegroups of order n , then Γ n ( G ) = { e } . In particular, if G admits a finite generating subset U , thenthere is a surjective homomorphism F U / Γ n ( F U ) ։ G preserving the generating subsets U .In the above case of interest, since Q n has order n − and a generating subset of elements, there isa surjective homomorphism ^ q n − : F / Γ n − ( F ) ։ Q n preserving the corresponding generators,so that the semi-direct product of G n is defined. Proposition 2.6 (Section 7 in [3]) . The following statements are true for the sequence of ( G n ) n ∈ N .(1) Let K n be the kernel of the surjective homomorphism p n : G ։ G n . Then ∩ n K n = { e } . Therefore, ⊔ n ( G n , S n ) is a box space of G with respect to the generating subset S and the filtration ( K n ) n , where S n = p n ( S ) .(2) The pair ( G, Z ) has relative property (T), by the proof of M. Burger in [6, Proposition 1]. Itfollows that the sequence of Cayley graphs ( G n , S n ) n is a relative expander with respect to the unboundedsequence of subsets (cid:0) Z n (cid:1) ⊂ G n , n ∈ N . In particular, the sequence ( G n , S n ) n does not coarsely embedinto a Hilbert space.(3) The sequence of subgroups (cid:0) Z n (cid:1) n ∈ N of ( G n ) n ∈ N , equipped with the induced metric from ( G n , S n ) ,are uniformly amenable [3, Lemma 2.5]. Consequently, it satisfies the Haagerup property uniformly, hasProperty A uniformly, and uniformly coarsely embed into a Hilbert space.
4) It follows from an earlier remarkable result of G. Arzhantseva, E. Guentner, and J. Špakula [1]that the box space { F / Γ n − ( F ) } n ∈ N coarsely embed into a Hilbert space. Together with (3), we have thatthe sequence ( G n ) n ∈ N has "A-by-CE" structure. In particular, the sequence ( G n , S n ) n does not contain aweakly embedded expander.(5) The sequence ( G n , S n ) n does not admit a fibred coarse embedding into a Hilbert space [8]. However,the maximal coarse Baum-Connes conjecture holds for the box space F n G n [41, Corollary 4.18]. Note that the maximal coarse Baum-Connes conjecture does not imply the (reduced) coarse Baum-Connes conjecture. Nevertheless, it follows from the main result in this paper, the coarse Baum-Connesconjecture holds for the box space F n G n . This solves an open problem raised by G. Arzhantseva and R.Tessera in [3]. Z ≀ Q F U Consider the generalized wreath product G := A ≀ Q F U := M Q A ⋊ F U , where A = h V i is a finitely generated, residually finite, amenable group, with a finite generating subset V and a filtration { A n } n , and Q := ker (cid:0) SL(3 , Z ) ։ SL(3 , Z ) (cid:1) , equipped with a finite generating subset U .For each v ∈ V and g ∈ Q , let v g : Q → A denote the function such that v g ( g ) = v ∈ A and v g ( h ) = e A ,the identity element of A , if g = h ∈ Q . Then (1) the set S := { v e : v ∈ V } ⊔ U is a finite generatingsubset of G ; (2) the set Y := { v g : v ∈ V, g ∈ Q } is an unbounded subset of L Q A . It follows from aresult by I. Chifan and A. Ioana [10] that the pair ( G, Y ) has relative property (T).Define a sequence of generalized wreath products of finite groups as follows: G n := (cid:0) A/A n (cid:1) ≀ Q n (cid:16) F U / Γ n − ( F U ) (cid:17) := M Q n (cid:0) A/A n (cid:1) ⋊ (cid:16) F U / Γ n − ( F U ) (cid:17) where, for all n ≥ ,• Q n < SL(3 , Z n ) is the image of Q <
SL(3 , Z ) under the quotient map π n : SL(3 , Z ) ։ SL(3 , Z n ); Q n is a -group of order n − . SL(3 , Z ) π n / / / / SL(3 , Z n ) F U π / / / / q n − (cid:15) (cid:15) (cid:15) (cid:15) Q ?(cid:31) O O π n / / / / Q n ?(cid:31) O O F U / Γ n − ( F U ) ^ q n − ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ Q n has order n − and a finite generating subset U , by the central tool of Arzhantseva-Tessera[3, Lemma 2.3] again, there is a surjective homomorphism ^ q n − : F U / Γ n − ( F U ) ։ Q n preservingthe corresponding generating subsets U , so that the generalized wreath product of G n is defined. Proposition 2.7 (Theorem 7.3 in [3]) . The following statements are true for the sequence of ( G n ) n ∈ N .(1) Let K n be the kernel of the surjective homomorphism p n : G ։ G n . Then ∩ n K n = { e } . Therefore, ⊔ n ( G n , S n ) is a box space of G with respect to the generating subset S and the filtration ( K n ) n , where S n = p n ( S ) .(2) The pair ( G, Y ) has relative property (T), by a result of I. Chifan and A. Ioana [10]. It followsthat the sequence of Cayley graphs ( G n , S n ) n is a relative expander with respect to the unbounded sequenceof subsets Y n := p n ( Y ) ⊂ G n , n ∈ N . In particular, the sequence ( G n , S n ) n does not coarsely embed intoa Hilbert space.(3) Since amenability of an amenable group A does not depend on the choice of a proper metric on A , the sequence of finite quotient groups (cid:0) A/A n (cid:1) n ∈ N as subgroups of ( G n ) n ∈ N , equipped with the inducedmetric from ( G n , S n ) , has Yu’s property A by a result of E. Guentner as recorded in the book of J. Roe[43].(4) It follows from an earlier remarkable result of G. Arzhantseva, E. Guentner, and J. Špakula [1] thatthe box space { F U / Γ n − ( F U ) } n ∈ N coarsely embed into a Hilbert space. Together with (3), we have thatthe sequence ( G n ) n ∈ N has "A-by-CE" structure. In particular, the sequence ( G n , S n ) n does not contain aweakly embedded expander.(5) The sequence ( G n , S n ) n does not admit a fibred coarse embedding into a Hilbert space [8]. However,the maximal coarse Baum-Connes conjecture holds for the box space F n G n [41, Corollary 4.18]. Note that the maximal coarse Baum-Connes conjecture does not imply the (reduced) coarse Baum-Connes conjecture. Nevertheless, it follows from the main result in this paper, the coarse Baum-Connesconjecture holds for the box space F n G n . This solves an open problem raised by G. Arzhantseva and R.Tessera in [3]. Z ≀ Q F Consider the generalized wreath product G := A ≀ Q F := M Q A ⋊ F , where A = h V i is a finitely generated, residually finite, amenable group, with a finite generating subset V and a filtration { A n } n , and Q := ker (cid:0) SL(2 , Z ) ։ SL(2 , Z ) (cid:1) , equipped with a generating subset U = { α, β, γ } as in Example No 1.As Q is a subgroup of a group with Haagerup property, by [11, Theorem 1.5], the group G := A ≀ Q F has the Haagerup property.For each v ∈ V and g ∈ Q , let v g : Q → A denote the function such that v g ( g ) = v ∈ A and v g ( h ) = e A ,the identity element of A , if g = h ∈ Q . Then (1) the set S := { v e : v ∈ V } ⊔ U is a finite generatingsubset of G ; (2) the set Y := { v g : v ∈ V, g ∈ Q } is an unbounded subset of L Q A .9efine a sequence of generalized wreath products of finite groups as follows: G n := (cid:0) A/A n (cid:1) ≀ Q n (cid:16) F / Γ n − ( F ) (cid:17) := M Q n (cid:0) A/A n (cid:1) ⋊ (cid:16) F / Γ n − ( F ) (cid:17) . Proposition 2.8 (Section 7.2 in [3]) . The following statements are true for the sequence of ( G n ) n ∈ N .(1) Let K n be the kernel of the surjective homomorphism p n : G ։ G n . Then ∩ n K n = { e } . Therefore, ⊔ n ( G n , S n ) is a box space of G with respect to the generating subset S and the filtration ( K n ) n , where S n = p n ( S ) .(2) Since the sequence ( Q n ) n ∈ N has a uniform spectral gap by a famous result of A. Selberg [33],it follows from the proof of [10, Theorem 3.1] that the sequence ( G n ) n ∈ N has relative property (T) withrespect to Y , in restriction to unitary representations which are direct sums of representations which factorthrough some G n . This restricted version of relative property (T) implies the relative Poincaré inequality.Namely, the sequence of Cayley graphs ( G n , S n ) n is a relative expander with respect to the unboundedsequence of subsets Y n := p n ( Y ) ⊂ G n , n ∈ N . In particular, the sequence ( G n , S n ) n does not coarselyembed into a Hilbert space.(3)The sequence ( G n ) n ∈ N has "A-by-CE" structure. In particular, the sequence ( G n , S n ) n does notcontain a weakly embedded expander.(4) Since A is amenable and Q is a subgroup of a Haagerup group, by [11, Theorem 1.5], the group G = A ≀ Q F has the Haagerup property. It follows that the sequence ( G n , S n ) n admits a fibred coarseembedding into a Hilbert space [8]. Hence, the maximal coarse Baum-Connes conjecture holds for the boxspace F n G n [9]. Note that the maximal coarse Baum-Connes conjecture does not imply the (reduced) coarse Baum-Connes conjecture. Nevertheless, it follows from the main result in this paper, the coarse Baum-Connesconjecture holds for the box space F n G n . This solves an open problem raised by G. Arzhantseva and R.Tessera in [3]. In a very recent work [4], G. Arzhantseva and R. Tessera answer in the negative the following well-knownquestion [12]: Does coarse embeddability into Hilbert space is preserved by group extensions of finitelygenerated groups? Their discoveries also answer in the affirmative another open problem [28]: Does thereexist a finitely generated group which does not coarsely embed into Hilbert space and yet has no weaklyembedded expander? Their examples are in fact "A-by-Haagerup" group extensions.
Theorem 2.9 ([4]) . (1) There exists a finitely generated group, e.g. Z ≀ G H , which is a split extensionof an (infinite rank) abelian group by an finitely generated group with the Haagerup property which doesnot coarsely embed into Hilbert space.(2) There exists a finitely generated group, e.g. Z ≀ G ( H × F n ) , which is a split extension of a finitelygenerated group with Property A by a finitely generated group with the Haagerup property that does notcoarsely embed into Hilbert space. In the first case, a certain restricted permutational wreath product Z ≀ G H = M G Z ⋊ H G is a Gromov monster group : a finitely generated group which containsin its Cayley graph an isometrically embedded expander (cf. [39, Theorem 4 or Corollary 3.3]), and H isa Haagerup monster group : a finitely generated group with the Haagerup property which does not haveProperty A (cf. [2, Theorems 1.2 and 5.1] and [39, Theorems 3 and 6.3]), such that there is a surjectivehomomorphism H ։ G which induces the action of H on G by left translations so as to define thewreath product. Recall that L G Z is the abelian group of finitely supported functions ϕ : G → Z withpointwise additions, which is not finitely generated.In the second case, G. Arzhantseva and R. Tessera modify the first example by introducing an ad-ditional action by a finitely generated free group F n . Namely, since G is finitely generated, say by n generators, there is a surjective homomorphism F n ։ G so that one can form the restricted permuta-tional wreath product Z ≀ G (cid:0) H × F n (cid:1) := (cid:0) L G Z (cid:1) ⋊ (cid:16) H × F n (cid:17) = (cid:16)(cid:0) L G Z (cid:1) ⋊ F n (cid:17) ⋊ H = (cid:16)(cid:0) L G Z (cid:1) ⋊ H (cid:17) ⋊ F n , where H acts on G as above by left translations via its surjection onto G , and F n acts on G by righttranslations via its surjection onto G . Since these two actions commute, the group G is an (cid:0) H × F n (cid:1) -setso that the wreath product Z ≀ G (cid:0) H × F n (cid:1) is defined, and it is an "A-by-Haagerup" split extension of thefinitely generated group Z ≀ G F n = (cid:16)(cid:0) M G Z (cid:1) ⋊ F n (cid:17) which has Property A, and the Haagerup monster H .The groups Z ≀ G H and Z ≀ G (cid:0) H × F n (cid:1) do not coarsely embed into Hilbert space because they containan embedded sequence of relative expanders ([4], Theorem 4.1, Corollary 3.3 and 4.2). Both groups donot contain weakly embedded expanders because they are "CE-by-CE" group extensions ([3] Proposition2). The first group Z ≀ G H is an "abelian-by-Haagerup" extension of two groups. It satisfies the StrongBaum-Connes Conjecture [29, Theorem 1.1], which is strictly stronger than the Baum-Connes Conjecturewith coefficients [7, Corollary 3.14]. It is proved in [5, Proposition 2.11] that, by [40, Theorem 7.1], thesecond group Z ≀ G (cid:0) H × F n (cid:1) = (cid:0) Z ≀ G H (cid:1) ⋊ F n = (cid:16)(cid:0) M G Z (cid:1) ⋊ H (cid:17) ⋊ F n also satisfies the Baum-Connes conjecture with coefficients. Consequently, the Coarse Baum-ConnesConjecture holds for both Z ≀ G H and Z ≀ G (cid:0) H × F n (cid:1) . Since both groups are "A-by-CE" extensions, themain result in this paper provides an alternative proof to this fact. Remark . Recall that the groups Z ≀ G H and Z ≀ G (cid:0) H × F n (cid:1) do not coarsely embed into Hilbertspace because they contain an isometrically embedded sequence of relative expanders W n = Z ≀ G n H n ,with generators σ n = { δ Gn } ∪ T n , n ∈ N , with respective to an unbounded sequence of subsets X n = { δ g , g ∈ G n } ⊂ W n , where G n isometrically embeds in G , and H n isometrically embeds in H , see ([4],Theorem 4.1, Corollary 3.3 and 4.2) for more details on these notations. It follows that the sequence ofsplit extensions W n = Z ≀ G n H n has the "A-by-CE" structure. Therefore, by the main theorem of thispaper, the coarse Baum-Connes conjecture holds for the relative expander W n = Z ≀ G n H n inside thegroups Z ≀ G H and Z ≀ G (cid:0) H × F n (cid:1) . 11 / Θ( N ) · · · (cid:15) (cid:15) F / Θ( N ) (cid:15) (cid:15) F / ( N ∩ Θ( N )) · · · o o (cid:15) (cid:15) F / Θ( N ) (cid:15) (cid:15) F / ( N ∩ Θ( N )) o o (cid:15) (cid:15) F / ( N ∩ Θ( N )) · · · (cid:15) (cid:15) o o F / Θ( N ) (cid:15) (cid:15) F / ( N ∩ Θ( N )) o o (cid:15) (cid:15) F / ( N ∩ Θ( N )) (cid:15) (cid:15) o o F / ( N ∩ Θ( N )) · · · (cid:15) (cid:15) o o F /N o o F /N o o F /N o o F /N · · · o o Figure 1:
Delabie-Khukhro’s magic triangle [14]
In [14] T. Delabie and A. Khukhro answer an open question asked by G. Arzhantseva and R. Tesserain [3, Section 8: Open Problems]: Does there exist a sequence of finite graphs with bounded degree andlarge girth that does not coarsely embed into a Hilbert space and yet has no weakly embedded expander(in particular, a box space of the free group F m )? Theorem 2.11 ([14]) . There exists a filtration of the free group F such that the corresponding box spacedoes not coarsely embed into a Hilbert space, but does not admit a weakly embedded expander. The overall structure of the proof is as follows. They first construct a particular sequence of nestedfinite index normal subgroups { N i } of F with trial intersection so that the corresponding box space isan expander, and consider the sequence of q -homology covers, where q is a certain prime number, of thequotients { F /N i } : this gives rise to another sequence of subgroups Θ( N i ) < N i of F such that thecorresponding box space (cid:3) Θ( N n ) F coarsely embeds into a Hilbert space, following a previous work of A.Khukhro [32]. Then they consider the quotients of F by intersections of these two sequences of subgroupsto obtain the following magic triangle, where the arrows represent quotient maps.Finally, they find an ingenuous way to choose a subsequence of quotients n F / (cid:0) N n i ∩ Θ( N k i ) o which lie on some path that moves sufficiently slowly away the horizontal expander sequence in this"triangle" of intersections.They show [14, Corollary 4.4] that there exist increasing sequences k i and n i such that the box space (cid:3) N ni ∩ Θ( N ki ) F contains a generalized expander and therefore does not coarsely embed into Hilbert space, by the charac-terization of R. Tessera [47].On the other hand, T. Delabie and A. Khukhro have essentially proved the following general fact [14,Proposition 2.4]: Let G be a finitely generated residually finite group with two filtrations { N i } and { M i } M i < N i for all i ∈ N . Consider the sequence of group extensions (cid:16) −→ N i /M i −→ G/M i −→ G/N i −→ (cid:17) i ∈ N , where G/M i and G/N i are considered with the metric induced by the restriction of the respective boxspace metrics, and N i /M i is considered with the metric induced by viewing N i /M i as a subspace of G/M i .Then the quotient maps G ։ G/M n ; G ։ G/N n are asymptotically faithful, so that the quotient maps π n : G/M n ։ G/N n is asymptotically faithful as well. It follows that there exists a sequence r n → ∞ such that any two pointsof N n /M n are at distance at least r n from each other. In other words, the asymptotic dimension of thecoarse disjoint union F i N i /M i is , which implies property A. Thus, if { G/N i } i coarsely embeds into aHilbert space, then the sequence { G/M i } i has "A-by-CE" structure, so that it does not contain weakly(and hence coarsely) embedded expanders. Applied to the above magic triangle, by our main Theorem1.1 in this paper, we actually have the following Corollary 2.12.
For all increasing sequences k i and n i of N , the corresponding sequence of extensionsin the Delabie-Khukhro magic triangle: (cid:16) −→ Θ( N k i ) / (cid:0) N n i ∩ Θ( N k i ) (cid:1) −→ F / (cid:0) N n i ∩ Θ( N k i ) (cid:1) −→ F / Θ( N k i ) −→ (cid:17) i ∈ N has "A-by-CE" structure. Therefore, the coarse Baum-Connes conjecture holds for the box space (cid:3) N ni ∩ Θ( N ki ) F := G i ∈ N F / (cid:0) N n i ∩ Θ( N k i ) (cid:1) for all increasing sequences k i and n i of N . In this section, we shall recall the concepts of Roe algebras, localization algebras and the coarse Baum–Connes conjecture.
Recall that a metric space is proper if every closed bounded subset is compact. Let X be a proper metricspace. An ample X -module is a separable Hilbert space H X equipped with a faithful and non-degenerate ∗ -representation of C ( X ) whose range contains no nonzero compact operators, where C ( X ) is the algebraof all complex-valued continuous functions on X vanishing at infinity. Definition 3.1.
Let X be a proper metric space and H X an ample X -module.(1) The support of a bounded linear operator T : H X → H X is defined to be the complement to the setof points ( x, y ) ∈ X × X for which there exist f, g ∈ C ( X ) with f ( x ) = 0 and g ( y ) = 0 such that f T g = 0 . The support of T is denoted by Supp ( T ) .132) For a bounded linear operator T : H X → H X , the propagation of T is defined bypropagation ( T ) := sup { d ( x, y ) : ( x, y ) ∈ Supp ( T ) } . An operator T is said to have finite propagation if propagation ( T ) < ∞ .(3) An operator T is locally finite if the operators f T and T f are compact operators on H X for all f ∈ C ( X ) .In the case when X is a countable metric space, we can choose a specific ample X -module as follows.Let H be any separable infinite-dimensional Hilbert space. Consider the Hilbert space ℓ ( X ) ⊗ H . Foreach f ∈ C ( X ) , we define a bounded linear map on ℓ ( X ) ⊗ H by linearly extending the map f · ( ξ ⊗ v ) = ( f ξ ) ⊗ v where ξ ∈ ℓ ( X ) and v ∈ H .Since ℓ ( X ) ⊗ H = L x ∈ X C · δ x ⊗ H , we can express each bounded operator T ∈ B ( ℓ ( X ) ⊗ H ) asan X -by- X matrix, T = ( T x,y ) x,y where T x,y is a bounded linear operator on H . We have thatpropagation ( T ) = sup { d ( x, y ) : T x,y = 0 } . If T is locally compact, then T x,y is a compact operator on H for all x, y ∈ X .Now we are ready to recall the definition of Roe algebras due to John Roe [44]. Definition 3.2.
Let X be a proper metric space and H X an ample X -module.(1) The algebraic Roe algebra, denoted by C [ X ] , is the ∗ -algebra of all locally compact, finite propagationoperators on H X .(2) The Roe algebra, denoted by C ∗ ( X ) , is defined to be the completion of C [ X ] under the operator normon H X .Let us now recall some basic properties of Roe algebras. Let X and Y be proper metric spaces. A map f : X → Y is said to be a coarse embedding if there exist two non-decreasing functions ρ , ρ : [0 , ∞ ) → [0 , ∞ ) such that(1) lim t →∞ ρ i ( t ) = ∞ for i = 1 , ;(2) ρ ( d ( x, y )) ≤ d ( f ( x ) , f ( y )) ≤ ρ ( d ( x, y )) , for all x, y ∈ X .A metric space X is said to be coarsely equivalent to another metric space Y , if there exist a coarseembedding f : X → Y such that Y is equal to the C -neighborhood of the image f ( X ) for some C > .Let us recall the Construction 4.2.5 in [52] about the homomorphism between Roe algebras inducedby a coarse embedding between metric spaces. Let f : X → Y be a coarse embedding. Let { U i } i ∈ I be aBorel cover of Y with the following properties: 141) { U i } i ∈ I is mutually disjoint;(2) each U i has non-empty interior;(3) for any compact K ⊆ Y , the set { i ∈ I | U i ∩ K = ∅} is finite;(4) the diameter of U i is uniformly bounded over i ∈ I .Note that the ∗ -representation of C ( Y ) on H Y extends to a ∗ -representation of the algebra of all boundedBorel functions on Y . Since the range of the representation C ( Y ) contains no non-zero compact operators, χ U i H Y is an infinite dimensional subspace of H Y for each i , where χ U i is the projection of the characteristicfunction on U i . Hence, for each i ∈ I , there always exists an isometry V i : χ f − ( U i ) H X → χ U i H Y . Consequently, we have an isometry V = ⊕ i ∈ I V i : H X = ⊕ i χ f − ( U i ) H X → H Y = ⊕ i χ U i H Y . For any operator T ∈ C ∗ alg ( X ) with finite propagation, Condition (3) above implies that the operator V T V ∗ has finite propagation, and it follows form Condition (4) above that the operator V T V ∗ is locallycompact. Therefore, the map Ad V : C [ X ] → C [ Y ] defined by T V T V ∗ is well-defined.It is obvious that when f : X → X is the identity map, the isometry V can be chosen as a unitary.Thus we have that the definition of C [ X ] is independent of the choice of H X up to ∗ -isomorphisms.Accordingly, we obtain a homomorphismAd V ∗ : K ∗ ( C ∗ ( X )) → K ∗ ( C ∗ ( Y )) on K -theory induced by Ad V , where V : H X → H Y is induced by the coarse embedding f : X → Y . Themap induced by Ad V on K -theory does not depend on the choice of V . Moreover, it is obvious that if f is a coarse equivalence, the isometry V can be chosen as a unitary. Remark . Let Z and Z be countable metric spaces and f : Z → Z an injective coarse map. Thenwe have an explicit construction of the homomorphism between the algebraic Roe algebras: C [ Z ] → C [ Z ] by ( Ad V ( T )) y,y ′ = ( T x,x ′ if y = f ( x ) , y ′ = f ( y )0 otherwise . , let us briefly recall the definition of analytic K -homology groups. More details can also be found in[51].Let X be a proper metric space. The K -homology groups K ( X ) and K ( X ) are groups generated bycertain cycles modulo a certain homotopy equivalence relation.151) A cycle for K ( X ) is a pair ( H X , F ) where H X is an X -module and F is a bounded linear operatoron H X such that f ( F ∗ F − , f ( F F ∗ − and f F − F f are compact operators for all f ∈ C ( X ) ;(2) A cycle for K ( X ) is a pair ( H X , F ) where H X is an X -module and F is a self-adjoint operator on H X such that f ( F − and f F − F f are compact operators for all f ∈ C ( X ) .In the above description of of cycles, the X -module can be chosen to be ample.Next, we shall recall the assembly map µ : K i ( X ) → K i ( C ∗ ( X )) . An open cover { U j } j ∈ J of X is locally finite if every point x ∈ X is contained in only finitely manyelements of the cover { U j } j ∈ J . Let { U j } j ∈ J be a locally finite and uniformly bounded open cover of X .Let { φ j } j ∈ J be a partition of unity subordinate to the open cover { U j } j ∈ J . Let ( H X , F ) be a cycle for K ( X ) such that H X is an ample X -module. Define F ′ = X j φ j F φ j , where the infinite sum converges in strong operator topology. Note that ( H X , F ′ ) is equivalent to ( H X , F ) via ( H X , (1 − t ) F + tF ′ ) , where t ∈ [0 , . Note that F ′ has finite propagation, so F ′ is a multiplier of C ∗ ( X ) . It is obvious that F ′ − ∈ C ∗ ( X ) . Thus, F ′ is invertible modulo C ∗ ( X ) . Hence F ′ gives rise toan element, denoted by ∂ [ F ′ ] in K ( C ∗ ( X )) , where ∂ : K ( M ( C ∗ ( X )) /C ∗ ( X )) → K ( C ∗ ( X )) is the boundary map on K -theory, and M ( C ∗ ( X )) is the multiplier algebra of C ∗ ( X ) . We define µ ([ H X , F ]) = ∂ [ F ′ ] . Similarly, we can define the index map from K ( X ) to K ( C ∗ ( X )) .Recall that a discrete metric space is said to have bounded geometry if for any r > there exists N > such that any ball of radius r in X contains at most N elements. Now we are ready to introducethe coarse Baum–Connes conjecture for a metric space with bounded geometry. Definition 3.4.
Let Γ be a metric space with bounded geometry. For each d ≥ , the Rips complex P d (Γ) is defined to be the simplical complex where the set of vertices is Γ , and a finite subset { γ , γ , · · · , γ n } ⊂ Γ spans a simplex if and only if d ( γ i , γ j ) ≤ d for all ≤ i, j ≤ n. The Rips complex P d (Γ) is equipped with the spherical metric that is the maximal metric whoserestriction to each simplex { P ni =0 t i γ i : t i ≥ , P i t i = 1 } is the metric obtained by identifying the simplexwith S n + via the map: n X i =0 t i γ i → t pP ni =0 t i , · · · , t n pP ni =0 t i ! , where S n + = (cid:8) ( x , · · · , x n ) ∈ R n +1 : x i ≥ , P ni =0 x i = 1 (cid:9) endowed with the standard Riemannian metric.The distance between a pair of points in different connected components is defined to be infinity.For each s ≥ r , note that P r (Γ) ⊆ P s (Γ) . Then the canonical inclusion map P r (Γ) i sr ֒ −→ P s (Γ) inducesa ∗ -homomorphism from C ∗ ( P r (Γ)) to C ∗ ( P s (Γ)) . Furthermore, for s ≥ r ≥ d , the inclusion map i sd andthe composition i sr ◦ i rd induce the same ∗ -homomorphism from C ∗ ( P d (Γ)) to C ∗ ( P s (Γ)) .16 he coarse Baum–Connes conjecture. For any discrete metric space Γ with bounded geometry, thecoarse assembly map µ : lim d →∞ K ∗ ( P d (Γ)) → lim d →∞ K ∗ ( C ∗ ( P d (Γ))) ∼ = K ∗ ( C ∗ (Γ)) is an isomorphism.Now, we shall recall the definition of localization algebras [52] and the relation between its K -theoryand the K -homology groups. Definition 3.5.
Let X be a proper metric space.(1) The algebraic localization algebra, denoted by C L [ X ] , is defined to be the ∗ -algebra of all uniformlybounded and uniformly norm-continuous functions g : [0 , ∞ ) → C [ X ] such that propagation ( g ( t )) → as t → ∞ .(2) The localization algebra C ∗ L ( X ) is the norm closure of C L [ X ] under the norm k f k = sup t ∈ [0 , ∞ ) k f ( t ) k . Naturally, the evaluation-at-zero map e : C ∗ L ( X ) → C ∗ ( X ) defined by e ( g ) = g (0) for g ∈ C ∗ L ( X ) is a ∗ -homomorphism.Next, we recall some basic properties of localization algebras. A Borel map f : X → Y is said to beLipschitz, if there exists a positive constant C such that d ( f ( x ) , f ( y )) ≤ Cd ( x, y ) for all x, y ∈ X . Followingthe construction in [52], a Lipschitz map f : X → Y induces a ∗ -homomorphism Ad ( V f ) : C ∗ L ( X ) → C ∗ L ( Y ) as follows.Let { ǫ n } n ∈ N be a sequence of positive numbers with lim n →∞ ǫ n = 0 . For each k , there exists an isometry V k : H X → H Y such that Supp ( V k ) ⊂ { ( y, x ) ∈ Y × X : d ( y, f ( x )) ≤ ǫ k } .Define a family of isometries ( V f ( t )) t ∈ [0 , ∞ ) : H X ⊕ H X → H Y ⊕ H Y by V f ( t ) = R ( t − k + 1)( V k ⊕ V k +1 ) R ∗ ( t − k + 1) , for t ∈ [ k − , k ) , where R ( t ) = cos ( πt/ sin ( πt/ − sin ( πt/ cos ( πt/ ! . Then V f ( t ) induces a homomorphism on unitizationAd ( V f ( t )) : C ∗ L ( X ) → C ∗ L ( Y ) ⊗ M ( C ) by Ad ( V f ( t ))( u ( t ) + cI ) = V f ( t )( u ( t ) ⊕ V ∗ f ( t ) + cI for all u ( t ) ∈ C ∗ L ( X ) and c ∈ C . 17 efinition 3.6. Let X and Y be two proper metric spaces and f, g two Lipschitz maps from X to Y . Acontinuous homotopy F ( t, x ) : [0 , × X → Y between f and g is said to be strongly Lipschitz if(1) F ( t, x ) is coarse map from X to Y for each t;(2) there exists a positive constant C such that d ( F ( t, x ) , F ( t, y )) ≤ Cd ( x, y ) for all x, y ∈ X and t ∈ [0 , ;(3) for any ǫ > , there exists δ > such that d ( F ( t , x ) , F ( t , x )) < ǫ for all x ∈ X and | t − t | < δ ;(4) F (0 , x ) = f ( x ) and F (1 , x ) = g ( x ) for all x ∈ X . Definition 3.7.
A metric space X is said to be strongly Lipschitz homotopy equivalent to Y , if thereexist two Lipschitz maps f : X → Y and g : Y → X such that f ◦ g and g ◦ f are strongly Lipschitzhomotopy equivalent to the identity maps id X and id Y , respectively.The K -theory of localization algebras is invariant under strongly Lipschitz homotopy equivalence (see[52]). The following Mayer-Vietoris sequence was introduced by the third author, and more details canbe found in [52]. Proposition 3.8.
Let X be a simplical complex endowed with the spherical metric, and X , X ⊂ X its simplical subcomplexes endowed with the subspace metric. Then we have the following six-term exactsequence: K ( C ∗ L ( X ∩ X )) / / K ( C ∗ L ( X )) ⊕ K ( C ∗ L ( X )) / / K ( C ∗ L ( X ∪ X )) (cid:15) (cid:15) K ( C ∗ L ( X ∪ X )) O O K ( C ∗ L ( X )) ⊕ K ( C ∗ L ( X )) o o K ( C ∗ L ( X ∩ X )) o o We then recall a local index map from the K -homology group K ∗ ( X ) to the K -theory group K ∗ ( C ∗ L ( X )) .For every positive integer n , let { U n,i } i be a locally finite open cover for X with diameter ( U n,i ) ≤ n for all i . Let { φ n,i } i be the partition of unity subordinate to the open cover { U n,i } i . For any [ H X , F ] ∈ K ( X ) ,we define a family of operators ( F ( t )) t ∈ [0 , ∞ ) by F ( t ) = X i (cid:16) ( n − t ) φ n,i F φ n,i + ( t − n + 1) φ n +1 ,i F φ n +1 ,i ) (cid:17) , for all t ∈ [ n − , n ] , where the infinite sum converges under the strong operator topology. Sincediameter ( U n,i ) → as n → ∞ , we have that propagation ( F ( t )) → as t → . Moreover, we havethat the path ( F ( t )) t ∈ [0 , ∞ ) is a multiplier of C ∗ L ( X ) and it is a unitary modulo C ∗ L ( X ) . Hence we candefine a local index map ind L : K ( X ) → K ( C ∗ L ( X )) by ind L ([ H X , F ]) = ∂ ([ F ( t )]) , where ∂ : K ( M ( C ∗ L,max ( X )) /C ∗ L ( X )) → K ( C ∗ L,max ( X )) is the boundary map in K -theory, and M ( C ∗ L,max ( X )) is the multiplier algebra of C ∗ L,max ( X ) . Similarly we can define the local index map ind L : K ( X ) → K ( C ∗ L,max ( X )) . The following result establishes the relation between the K -homology groups and the K -theory of local-ization algebras. 18 heorem 3.9. [52] For any finite-dimensional simplical complex X endowed with the spherical metric,the local index map ind L : K ∗ ( X ) → K ∗ ( C ∗ L ( X )) is an isomorphism. In general, Roe and Qiao [42] prove the local index map is isomorphism for any proper metric space.If Γ is a discrete metric space with bounded geometry, we have the following commutative diagram: lim d →∞ K ∗ ( C ∗ L ( P d (Γ))) e ∗ (cid:15) (cid:15) lim d →∞ K ∗ ( P d (Γ)) ind L ∼ = ❧❧❧❧❧❧❧❧❧❧❧❧❧ µ / / lim d →∞ K ∗ ( C ∗ ( P d (Γ))) . Therefore, the coarse Baum–Connes conjecture is a consequence of the result that the map e ∗ : lim d →∞ K ∗ ( C ∗ L ( P d (Γ))) → lim d →∞ K ∗ ( C ∗ ( P d (Γ))) induced by evaluation-at-zero map on K -theory is an isomorphism. In this paper, we denote I to be either the singleton { } or N . Let { ( X m , d m ) } m ∈ I be a sequence of metricspaces with uniform bounded geometry in the sense that for each R > , the number of the elements inthe set B X m ( x, R ) = { y ∈ X m : d ( x, y ) ≤ R } is at most M R for some M R > which does not depend on m . In the case when the sequence ( X m ) m is a sequence of finitely generated groups ( G m ) m , ( G m ) m hasuniform bounded geometry if the numbers of generating subsets for all groups are uniformly bounded.For each d > , and each m ∈ I , we choose a countable dense subset X md ⊂ P d ( X m ) such that X md ⊂ X md ′ for each m if d < d ′ . Let K be the algebra of compact operators on the infinite-dimensionalseparable Hilbert space H . Definition 3.10.
For each d > , the algebraic Roe algebra C u [( P d ( X m )) m ∈ I ] is the collection of tuples T = (cid:0) T ( m ) (cid:1) m ∈ I satisfying(1) each T ( m ) is a bounded function from X md × X md to K ;(2) there exists r > such that for each m , T ( m ) x,y = 0 for all x, y ∈ X md with d ( x, y ) ≥ r ;(3) there exists L > such that for each m and x ∈ X md , ♯ { y ∈ X md : T ( m ) x,y = 0 } ≤ L, and ♯ { y ∈ X md : T ( m ) y,x = 0 } ≤ L ; (4) for each m and each bounded subset B ⊂ P d ( G m ) , the set n ( x, y ) ∈ ( B × B ) ∩ ( X md × X md ) : T ( m ) x,y = 0 o is finite. 19e can then view C u [( P d ( X m )) m ∈ I ] as a ∗ -subalgebra of Q m C ∗ u (( P d ( X m )) m ∈ I ) . Let E = M ℓ ( X md ) ⊗ H. The algebraic Roe algebra C u [( P d ( X m )) m ∈ I ] admits a ∗ -representation on E by the restriction of the ∗ -representation of Q m C ∗ ( P d ( X m )) .The Roe algebra for the sequence ( P d ( G m )) m , denoted by C ∗ u (( P d ( X m )) m ∈ I ) , is the completion of C u ( P d ( X m )) m ∈ I ] under the operator norm on E .The following definition was essentially defined by the third author in [52]. Definition 3.11.
The algebraic localization algebra, C u,L [( P d ( X m )) m ∈ I ] , is the ∗ -algebra of all uniformlybounded and uniformly continuous functions f : [0 , ∞ ) → C u [( P d ( X m )) m ∈ I ] such that f ( t ) is of the form f ( t ) = ( f m ( t )) m ∈ I for all t ∈ [0 , ∞ ) , where the path of the tuples ( f m ( t )) m ∈ I satisfy the conditions in Definition 3.10 with uniform constants and there is a bounded function r :[0 , ∞ ) → [0 , ∞ ) with lim t →∞ r ( t ) = 0 , such that ( f ( m ) ( t )) x,y = 0 whenever d ( x, y ) > r ( t ) for all m ∈ I , x, y ∈ X md and t ∈ [0 , ∞ ) .The uniform localization algebra C ∗ u,L (( P d ( X m )) m ∈ I ) is defined to be the completion of C u,L [( P d ( X m )) m ∈ I ] under the norm k f k = k f ( t ) k C ∗ (( P d ( X m )) m ∈ I ) for all f ∈ C u,L ( P d ( X m )) m ∈ I ] .Naturally, we have a ∗ -homomorphism e : C ∗ u,L (( P d ( X m )) m ∈ I ) → C ∗ u (( P d ( X m )) m ∈ I ) defined by the evaluation-at-zero map. Moreover, this homomorphism induces a map e ∗ : lim d →∞ K ∗ ( C ∗ u,L (( P d ( X m )) m ∈ I )) → lim d →∞ K ∗ ( C ∗ u (( P d ( X m )) m ∈ I )) on the K -theory level. The coarse Baum–Connes conjecture for a sequence of metric spaces.
For any sequence ofdiscrete metric spaces ( X m , d m ) m with uniform bounded geometry, the map e ∗ : lim d →∞ K ∗ ( C ∗ u,L (( P d ( X m )) m ∈ I )) → lim d →∞ K ∗ ( C ∗ u (( P d ( X m )) m ∈ I )) induced by the evaluation-at-zero map on K -theory is an isomorphism.Let ( X m , d m ) m ∈ N be a sequence of metric spaces with uniform bounded geometry. The sequence ofmetric spaces ( X m , d m ) m is said to be coarsely embeddable into Hilbert space if for each m ∈ I there existsa map f m from X m to a Hilbert space H m , and two non-decreasing functions ρ − , ρ + : [0 , ∞ ) → [0 , ∞ ) such that 20 lim t →∞ ρ ± ( t ) = ∞ • ρ − ( d ( x, y )) ≤ k f m ( x ) − f m ( y ) k H m ≤ ρ + ( d ( x, y )) for all x, y ∈ X m .Note that a sequence ( X m ) m ∈ I is coarsely embeddable into Hilbert space if and only if the metricspace of separated disjoint union of the sequence ( X m ) m is coarsely embeddable into Hilbert space. Bythe third author’s result in [54], the coarse Baum–Connes conjecture for the sequence ( X m ) m ∈ I is truewhen the sequence ( X m ) m can be coarsely embedded into Hilbert space.The concept of Property A was introduced by the third author in [54]. As a generalization of amenabil-ity in the context of metric spaces, Property A has many equivalent characterizations, see [38]. In thispaper, we will consider the concept of Property A for a sequence of metric spaces. A sequence of metricspace { ( X m , d m ) m ∈ I } is said to have Property A, if for each ǫ > and each R > , there exists a sequenceof functions { k m : X m × X m → R } and a constant S > , such that(1) for each m , the function k m is of positive definite type, i.e., P Ni k m ( x i , x j ) t i t j ≥ for each finitesubset { x i } Ni =1 ⊂ X m and { t i } Ni =1 ⊂ R ;(2) for each m ∈ I , | − k m ( x, y ) | ≤ ǫ for all x, y ∈ X m with d ( x, y ) < R ;(3) for each m ∈ I , k m ( x, y ) = 0 for all x, y ∈ X m with d ( x, y ) ≥ S .In the rest of this paper, we shall prove the following result. Theorem 1.3 and 1.1 follows from thefollowing result. Theorem 3.12.
Let (1 → N m → G m → Q m → m ∈ I be a sequence of exact sequences of groups suchthat the sequences ( N m ) m , ( G m ) m and ( Q m ) have uniform bounded geometry. If ( N m ) m has Property Aand ( Q m ) m is coarsely embeddable into Hilbert space, then the coarse Baum–Connes conjecture holds forthe sequence ( G m ) m ∈ I , that is, the map e ∗ : lim d →∞ K ∗ ( C ∗ u,L (( P d ( G m )) m ∈ I )) → lim d →∞ K ∗ ( C ∗ u (( P d ( G m )) m ∈ I )) induced by the evaluation-at-zero map on K -theory is an isomorphism. When I = { } , Theorem 1.3 follows from the Theorem 3.12.Let us consider I = N and (1 → N m → G m → Q m → m ∈ I be a sequence of extensions of finitegroups. A coarse disjoint union ( X, d ) is the disjoint union F m G m as a set endowed with a metric d satisfying the following conditions:• for each m , d ( x, y ) = d m ( x, y ) for all x, y ∈ G m ;• dist ( G m , G m ′ ) → ∞ , as m + m ′ → ∞ for all m = m ′ .The separated disjoint union is the metric space ( Y, d ′ ) be the disjoint union equipped with the metric d ′ satisfying the following conditions:• for each m , d ′ ( x, y ) = d m ( x, y ) for all x, y ∈ G m ;• dist ( G m , G m ′ ) = ∞ for all m = m ′ . 21ecall that the index set I is neither the singleton { } or the natural numbers N . Note that when I = { } , then the coarse Baum–Connes conjecture for the sequence ( X m ) m ∈ I is the coarse Baum–Connesconjecture for a metric space X with bounded geometry, while the coarse Baum–Connes conjecture forthe sequence ( X m ) m ∈ I is the coarse Baum–Connes conjecture for the separated disjoint union of X m ’swhen I = N . Theorem 3.13.
Let ( G m ) m ∈ I be a sequence of finite groups endowed with word length metrics such thatthe metric spaces X and Y defined above have bounded geometry. If the coarse Baum–Connes conjectureholds for the separated disjoint union ( Y, d ′ ) , then the coarse Baum–Connes conjecture holds for the coarsedisjoint union ( X, d ) .Proof. We have the following commutative diagram: (cid:15) (cid:15) (cid:15) (cid:15) L m K ∗ ( C ∗ L ( P d ( G m ))) (cid:15) (cid:15) / / L m K ∗ ( C ∗ u (( P d ( G m )) m ∈ I )) (cid:15) (cid:15) K ∗ ( C ∗ L ( P d ( Y ))) (cid:15) (cid:15) / / K ∗ ( C ∗ ( P d ( Y ))) (cid:15) (cid:15) K ∗ ( C ∗ L ( P d ( Y ))) L m K ∗ ( C ∗ L ( P d ( G m ))) / / (cid:15) (cid:15) K ∗ ( C ∗ ( P d ( Y ))) L m K ∗ ( C ∗ L ( P d ( G m ))) (cid:15) (cid:15) where the horizontal maps are induced by evaluation-at-zero maps and the vertical maps are induced byinclusions. Note that the diagram is compatible with increasing of the Rips complex scale, so we have thecommutative diagram: (cid:15) (cid:15) (cid:15) (cid:15) lim d →∞ M m K ∗ ( C ∗ L ( P d ( G m ))) (cid:15) (cid:15) / / lim d →∞ M m K ∗ ( C ∗ ( P d ( G m ))) (cid:15) (cid:15) lim d →∞ K ∗ ( C ∗ L ( P d ( Y ))) (cid:15) (cid:15) / / lim d →∞ K ∗ ( C ∗ ( P d ( Y ))) (cid:15) (cid:15) lim d →∞ K ∗ ( C ∗ L ( P d ( Y ))) L m K ∗ ( C ∗ L ( P d ( G m ))) / / (cid:15) (cid:15) lim d →∞ K ∗ ( C ∗ ( P d ( Y ))) L m K ∗ ( C ∗ L ( P d ( G m ))) (cid:15) (cid:15) G m is finite, so the Rips complex P d ( G m ) is strongly Lipschitz homotopy equivalent toa point when d is larger than the diameter of G m . As a result, the map e ∗ : lim d →∞ M m K ∗ ( C ∗ L ( P d ( G m ))) → lim d →∞ M m K ∗ ( C ∗ ( P d ( G m ))) induced by the evaluation-at-zero map is an isomorphism. In addition, we assume that the middle hori-zontal map in the above diagram is an isomorphism. As a consequence of diagram chasing, we have thatthe map e ∗ : lim d →∞ K ∗ ( C ∗ L ( P d ( Y ))) L m K ∗ ( C ∗ L ( P d ( G m ))) → lim d →∞ K ∗ ( C ∗ ( P d ( Y ))) L m K ∗ ( C ∗ ( P d ( G m ))) is an isomorphism.Let H d be an ample ( ⊔ m P d ( G m )) -module. We have a commutative diagram: (cid:15) (cid:15) (cid:15) (cid:15) K ∗ ( C ∗ L (∆ d )) L m>N d K ∗ ( C ∗ L ( P d ( G m ))) (cid:15) (cid:15) / / K ∗ ( K ( H d )) (cid:15) (cid:15) K ∗ ( C ∗ L ( P d ( X ))) (cid:15) (cid:15) / / K ∗ ( C ∗ ( P d ( X ))) (cid:15) (cid:15) K ∗ ( C ∗ L ( P d ( X ))) K ∗ ( C ∗ L (∆ d )) L m>N d K ∗ ( C ∗ L ( P d ( G m ))) / / (cid:15) (cid:15) K ∗ ( C ∗ ( P d ( X ))) K ∗ ( K ( H d )) (cid:15) (cid:15) The diagram is also compatible with increasing of the Rips complex scale d , so we may take the limitas d goes to infinity. Note that ∆ d is strongly homotopy to a point, we have that the restriction of thetop horizontal map e ∗ : K ∗ ( C ∗ L (∆ d )) → K ∗ ( K ( H d )) is an isomorphism. As d tends to infinity, we have that the top horizontal map is an isomorphism.For each d > , we have that C ∗ ( P d ( X )) = K ( H d ) + C ∗ ( P d ( Y )) and C ∗ ( P d ( Y )) ∩ K ( H d ) = M m C ∗ ( P d ( G m )) . Therefore, we have C ∗ ( P d ( X )) K ( H d ) ∼ = C ∗ ( P r ( Y )) L m C ∗ ( P d ( G m )) In addition, lim d →∞ K ∗ ( C ∗ L ( P d ( Y ))) L m K ∗ ( C ∗ L ( P d ( G m ))) ∼ = lim d →∞ K ∗ ( C ∗ ( P d ( Y ))) L m K ∗ ( C ∗ ( P d ( G m )))
23s a result, the bottom horizontal map is an isomorphism when taking the limit on d as d tends toinfinity. By Five Lemma, the middle horizontal map is an isomorphism as d tends to infinity, i.e., themap e ∗ : lim d →∞ K ∗ ( C ∗ L ( P d ( X ))) → lim d →∞ K ∗ ( C ∗ ( P d ( X ))) induced by the evaluation-at-zero map is an isomorphism.For a sequence of finite groups ( N m ) m ∈ N , we have that the coarse disjoint union has Property A if andonly if the sequence ( N m ) m ∈ N has Property A.In the constructions of relative expanders by Arzhantseva–Tessera (in [3]) and Delabie–Khukhro(in[14]), the normal subgroups are a sequence of quotient groups of an abelian group. As a result of Propo-sition 11.39 in [43], the coarse disjoint union F N m has property A, which implies that the sequenceof finite metric spaces ( N m ) m ∈ N has Property A. In addition, the coarse disjoint union of the quotientgroups ( Q m ) m ∈ N in the relative expanders constructed by Arzhantseva–Tessera and Delabie–Khukhro areall coarsely embeddable into Hilbert space which implies the sequence of the quotient groups ( Q m ) m ∈ N are uniformly coarsely embeddable into Hilbert space. Therefore, Theorem 3.12 holds. By Theorem 3.13,we have that the coarse Baum–Connes conjecture holds for the coarse disjoint union F G m , that is, themap e ∗ : lim d →∞ K ∗ ( C ∗ L ( P d ( ⊔ m G m ))) → lim d →∞ K ∗ ( C ∗ ( P d ( ⊔ m G m ))) ia an isomorphism. In this section, we will define twisted Roe algebras and twisted localization algebras for a sequence ofmetric spaces. The constructions of these algebras are similar to those defined by the third author in [54]and by the second and third author with X. Chen in [9].
In this subsection, we shall recall a C ∗ -algebra associated with an infinite-dimensional Euclidean spaceintroduced by Higson, Kasparov and Trout [30].Let H be a countably infinite-dimensional Euclidean space. Denote by V a , V b the finite-dimensionalaffine subspaces of H . Let V a be the finite dimensional linear subspace of H consisting of differencesof elements in V a . Let Cliff ( V a ) be the complexified Clifford algebra on V a and C ( V a ) be the graded C ∗ -algebra of continuous functions from V a to Cliff ( V a ) vanishing at infinity. Let S be the C ∗ -algebraof all continuous functions on R vanishing at infinity. Then S is graded according to the odd and evenfunctions. Define the graded tensor product A ( V a ) = S b ⊗C ( V a ) . If V a ⊆ V b , we have a decomposition V b = V ba + V a , where V ba is the orthogonal complement of V a in V b .For each v b ∈ V b , we have a unique decomposition v b = v ba + v a , where v ba ∈ V ba and v a ∈ V a . Everyfunction h on V a can be extended to a function ˜ h on V b by the formula ˜ h ( v ba + v a ) = h ( v a ) .24 efinition 4.1. (1) If V a ⊆ V b , we define C ba to be the Clifford algebra-valued function V b → Cliff ( V b ) , v b v ba ∈ V ba ⊂ Cliff ( V b ) . Let X be the function multiplication by x on R , considered as a degreeone and unbounded multiplier of S . Define a homomorphism β ba : A ( V a ) → A ( V b ) by β ba ( g b ⊗ h ) = g ( X b ⊗ b ⊗ C ba )(1 b ⊗ ˜ h ) for all g ∈ S and h ∈ A ( V a ) , and g ( X b ⊗ b ⊗ C ba ) is the functional calculus of g on the unbounded,essentially self-adjoint operator X b ⊗ b ⊗ C ba .(2) We define a C ∗ -algebra A ( H ) by: A ( H ) = lim −→ A ( V a ) , where the direct limit is taken over the directed set of all finite-dimensional affine subspaces V a ⊂ H . Remark . If V a ⊂ V b ⊂ V c , then we have β cb ◦ β ba = β ca . Therefore, the above homomorphisms give adirected system ( A ( V a )) V a as V a ranges over finite dimensional affine subspaces of V .The set R + × H is equipped with a topology as follows. Let { ( t i , v i ) } be a net in R + × H , it convergesto a point ( t, v ) ∈ R + × H if(1) t i + k v i k → t + k v k , as i → ∞ ; (2) h v i , u i → h v, u i for any u ∈ H , as i → ∞ . It is obvious that R + × H is a locally compact Hausdorff space. Note that for each v ∈ H and each r > , B ( v, r ) = { ( t, w ) ∈ R + × H : t + k v − w k < r } is an open subset of R + × H . For finitedimensional subspaces V a ⊆ V b ⊂ H , since β ba takes C ( R + × V a ) into C ( R + × V b ) , then the C ∗ -algebra lim −→ C ( R + × V a ) is ∗ -isomorphic to C ( R + × H ) . Definition 4.3.
The support of an element a ∈ A ( H ) is the complement to the set of all points ( t, v ) ∈ R + × H such that there exists g ∈ C ( R + × H ) with g ( t, v ) = 0 and g · a = 0 .Let G be a finitely generated group with a finite symmetric generating subset S ⊂ G . Recall that aset is symmetric in the sense that s − ∈ S for all s ∈ S . We then can define the word length | g | S of g ∈ G as follows: | g | S = min { n | g = g · · · g n , where g , · · · , g n ∈ S } . Definition 4.4.
Let G be a finitely generated discrete group with a finite symmetric generating subset S . The word length metric d associated with S , is defined by d ( g, h ) = | gh − | S for all g, h ∈ G .Note that a finitely generated group G equipped with any word length metric is a metric space withbounded geometry. In addition, the metric spaces of G associated with different word length metrics arecoarsely equivalent. In general, any countable discrete group admits a proper metric and different propermetrics on such a group are coarsely equivalent.Let → N → G → Q → be a short exact sequence of finitely generated discrete groups. Let π : G → Q be the quotient map, and S ⊂ G a finite symmetric generating subset of G . It is easy to verify25hat the image π ( S ) is a generating subset of the quotient group Q . As a result, we obtain a right invariantword length metric on Q . In addition, the normal subgroup N admits a metric which is a restriction ofthe metric on G .Let f : Q → H be a coarse embedding, and G = ⊔ g ∈ Λ gN a coset decomposition of G where Λ ⊂ G is aset of representatives of cosets in Q . For each d > , we shall extend the map f to a map from the Ripscomplex P d ( Q ) to the Hilbert space H as follows. For any point z = P g ∈ Λ c g gN ∈ P d ( Q ) , define f ( z ) = X g ∈ Λ c g f ( gN ) . For each z = P g ∈ Λ c g gN ∈ P d ( Q ) and each n ∈ N , define a subspace W n ( z ) of H as W n ( g ) = span { f ( g ′ N ) : g ′ N ∈ Q, d Q ( g ′ N, gN ) ≤ n } for all g such that c g > . The bounded geometry property of Q implies that W n ( g ) is finite dimensional.The quotient map π : G → Q induces a map π : P d ( G ) → P d ( Q ) by π ( P ki c i g i ) = P ki c i π ( g i ) for all P ki c i g i ∈ P d ( G ) . For any element x ∈ P d ( G ) , for any n ∈ N , we define W n ( x ) = W n ( π ( x )) to be thefinite-dimensional Euclidean space of Hilbert space H . Remark . (1) Since f : Q → H is a coarse embedding and G has bounded geometry, then for any n ∈ N , there exists R n,d > such that dim ( W n ( x )) ≤ R n,d for all x ∈ P d ( G ) .(2) For each r > , there exists N > such that W n ( x ) ⊂ W n +1 ( y ) for all n ≥ N and x, y ∈ P d ( G ) satisfying d ( x, y ) ≤ r. In this subsection, we shall define the twisted Roe algebras and twisted localization algebras for thesequence ( G m ) m using the coarse embeddability of the sequence ( Q m ) m .Let (1 → N m → G m → Q m → m ∈ I be a sequence of extensions of discrete groups. For each m ∈ I ,we have word length metrics on N m , G m and Q m . Here, we do not assume the groups N m , G m , and Q m are finite. Assume that the sequences of metric spaces ( N m ) m ∈ I , ( G m ) m ∈ I and ( Q m ) m ∈ I have uniformbounded geometry.Recall that a sequence of maps ( f m : Q m → H m ) m ∈ I is a coarse embedding for the sequence ( Q m ) m if there exist two non-decreasing maps, ρ i : [0 , ∞ ) → [0 , ∞ ) , satisfying• lim t →∞ ρ i ( t ) = ∞ as t → ∞ ;• for each m ∈ I , ρ ( d ( x, y )) ≤ k f m ( x ) − f m ( y ) k ≤ ρ ( d ( x, y )) for all x, y ∈ Q m .For each m ∈ I and d > , let f m : P d ( Q m ) → H m be the linear extension of f m : Q m → H m , and π : P − d ( G m ) → P d ( Q m ) the linear extension of the quotient map π : G m → Q m . For each m ∈ I and x ∈ P d ( G m ) , let W n ( x ) is the linear subspace of H m defined by W n ( x ) = span (cid:8) f m ( π ( y )) : d ( π ( x ) , π ( y )) ≤ n (cid:9) . d > and each m ∈ I , we firstly defined a spherical metric on each P d ( G m ) , then choose acountable dense subset X md of P d ( G m ) such that X md ⊆ X md if d ≤ d for each m ∈ I . Definition 4.6.
For each d > , the algebraic twisted Roe algebra C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) is theset of all tuples T = (cid:0) T ( m ) (cid:1) m ∈ I satisfying(1) for each m , T ( m ) is a bounded function from X md × X md to A ( H m ) b ⊗ K where K is the algebra of allcompact operators on an infinite-dimensional separable Hilbert space;(2) there exists an integer N such that T ( m ) ( x, y ) ∈ ( β N ( x ))( A ( W N ( x )) ˆ ⊗ K ) ⊆ A ( H m ) ˆ ⊗ K for each m and x, y ∈ X md , where β N ( x ) : A ( W N ( x )) → A ( H m ) is the ∗ -homomorphism associatedwith the inclusion of W N ( x ) into H m ;(3) there exists L > such that, for each m and each y ∈ X md , ♯ { x : T ( m ) ( x, y ) = 0 } ≤ L, ♯ { x : T ( m ) ( y, x ) = 0 } ≤ L ; (4) for each m and each bounded subset B ⊂ P d ( G m ) , the set ♯ { ( x, y ) ∈ ( B × B ) ∩ ( X md × X md ) : T ( m ) ( x, y ) = 0 } is finite;(5) there exists r > such that T ( m ) ( x, y ) = 0 for all m ∈ I and x, y ∈ X md with d ( x, y ) > r (The leastsuch r is the propagation of the tuple T = ( T ( m ) ) m .);(6) there exists r > such that Supp ( T ( m ) ( x, y )) ⊆ B ( f m ( π ( x )) , r ) for all x, y ∈ X d where B ( f m ( π ( x )) , r ) = (cid:8) ( s, h ) ∈ R + × H : s + k h − f m ( π ( x )) k < r (cid:9) ; (7) there exists c > such that if T ( m ) ( x, y ) = β N ( x )( S ( m ) ( x, y )) for some S ( m ) ( x, y ) ∈ A ( W N ( x )) ˆ ⊗ K ,then sup m k D Y ( S ( m ) ( x, y )) k ≤ c for each m ∈ I , x, y ∈ X md and all Y = ( s, h ) ∈ R + × W N ( x ) satisfying k Y k = p s + k h k ≤ , where D Y ( S ( m ) ( x, y )) is the derivative of the function S ( m ) ( x, y ) : R + × W N ( x ) → Cliff ( W N ( x )) ˆ ⊗ K , in thedirection of Y . Definition 4.7.
For each tuple T = (cid:0) T ( m ) (cid:1) m ∈ I , the propagation of T is defined to bepropagation ( T ) = sup m propagation ( T ( m ) ) = sup m sup x,y ∈ X md { d ( x, y ) : T ( m ) ( x, y ) = 0 } . We define the multiplication and adjoint operations on C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) by: ( T S ) ( m ) ( x, y ) = X z ∈ X d T ( m ) ( x, z ) S ( m ) ( z, y ) , and ( T ∗ ) ( m ) ( x, y ) = (cid:16) T ( m ) ( y, x ) (cid:17) ∗ , T = (cid:0) T ( m ) (cid:1) m ∈ I = ( T (1) , T (2) , · · · ) and S = (cid:0) S ( m ) (cid:1) m ∈ I = ( S (1) , S (2) , · · · ) in C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) .It is obvious that C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) is a ∗ -algebra.For each m ∈ I , define E m = X x ∈ X md a x [ x ] : X x a ∗ x a x converges in strong operator topology, where a x ∈ A ( H m ) b ⊗ K . The space E m can be equipped with a Hilbert A ( H m ) ⊗ K -module by h X x ∈ X md a x [ x ] , X x ∈ X md b x [ x ] i = X x ∈ X md a ∗ x b x , for all P x ∈ X md a x [ x ] , P x ∈ X md b x [ x ] ∈ E m . Let B ( E m ) be the algebra of all adjointable A ( H m ) ⊗ K -linearhomomorphisms. We then define a ∗ -representation φ m : C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) → B ( E m ) by T · ( X x ∈ X md a x [ x ]) = X x ∈ X md X y ∈ X md T ( m ) x,y a y [ x ] , for all T = (cid:0) T ( m ) (cid:1) m ∈ I ∈ C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) and P x ∈ X md a x [ x ] ∈ E m . Note that this is a ∗ -representation. Taking direct sum of these ∗ -representations, we obtain a faithful ∗ -representation forthe ∗ -algebra C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) by M φ m : C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) → B ( M m E m ) . The uniform twisted Roe algebra C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) is defined to be the completion of C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) under the operator norm in B ( L m E m ) .Let C ∗ u,L,alg (( P d ( G m ) , A ( H m )) m ∈ I ) be the ∗ -algebra of all uniformly bounded and uniformly norm-continuous functions g : R + → C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) such that:(1) there exists N such that ( g ( t )) ( m ) ( x, y ) ∈ ( β N ( x ))( A ( W N ( x )) ˆ ⊗ K ) ⊆ A ( H m ) ˆ ⊗ K for all t ∈ R + , m ∈ I and x, y ∈ X md ;(2) there exists a bounded function r ( t ) : R + → R + such that lim t →∞ r ( t ) = 0 and if d ( x, y ) > r ( t ) , then ( g ( t )) ( m ) ( x, y ) = 0; (3) there exists R > such that Supp (( g ( t )) ( m ) )( x, y )) ⊆ B ( f m ( π ( x )) , R ) for all t ∈ R + , m ∈ I and x, y ∈ X md ;(4) there exists c > such that k D Y ( h ( m ) ( t )( x, y )) k ≤ c for all t ∈ R + , m ∈ I , x, y ∈ X md and Y ∈ R × W N ( x ) satisfying k Y k ≤ , where h m ( t )( x, y ) ∈ A ( W N ( x )) satisfying ( β N ( x ))( h ( m ) ( t )( x, y )) =( g ( t )) ( m ) ( x, y ) . 28 efinition 4.8. The twisted localization algebra C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) is defined to be the normcompletion of C ∗ u,L,alg (( P d ( G m ) , A ( H m )) m ∈ I ) with respect to the norm k g k = sup t ∈ R + k g ( t ) k . There is a natural evaluation-at-zero map e : C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) → C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) by e (( g (1) ( t ) , g (2) ( t ) , · · · , )) = ( g (1) (0) , g (2) (0) , · · · , ) for all ( g (1) ( t ) , g (2) ( t ) , · · · , ) ∈ C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) . The map e is a ∗ -homomorphism, then itinduces a homomorphism e ∗ : lim d →∞ K ∗ ( C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I )) → lim d →∞ K ∗ ( C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I )) on K -theory. In this this section, we will prove the twisted coarse Baum–Connes conjecture.
Theorem 4.9.
Let (1 →→ N m → G m → G m /N m → m ∈ I be a sequence of extensions of discrete groups.Assume that the sequences of metric spaces ( N m ) m ∈ I , ( G m ) m ∈ I and ( Q m ) m ∈ I have uniformly boundedgeometry. If the sequence of metric spaces ( N m ) m ∈ I has Property A and the sequence ( Q m ) m ∈ I is coarselyembeddable into Hilbert space. Then the homomorphism e ∗ : lim d →∞ K ∗ ( C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I )) → lim d →∞ K ∗ ( C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I )) on K -theory is an isomorphism, where C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) and C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) arethe C ∗ -algebras defined in Section 4.2. To prove this result, we need to discuss ideals of the twisted algebras associated with open subsets of R + × H m . Definition 4.10.
Let O = ( O m ) m ∈ I be a sequence of subsets such that each O m is an open subset of R + × H m .(1) Denoted by C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O the C ∗ -subalgebra of C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) consistingof all tuples T = (cid:0) T ( m ) (cid:1) m ∈ I such thatSupp ( T ( m ) ( x, y )) ⊂ O m for all m ∈ I and x, y ∈ X md .(2) Denoted by C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) O the C ∗ -subalgebra of C ∗ L (( P d ( G m ) , A ( H m )) m ∈ I ) consist-ing of all tuples g = (cid:0) g ( m ) ( t ) (cid:1) m ∈ I such thatSupp ( g ( m ) ( t )( x, y )) ⊂ O m for all t ∈ [0 , ∞ ) , m ∈ I and x, y ∈ X md . 29or any two sequence of open sets O = ( O m ) m and O ′ = ( O ′ m ) m , we say O ⊂ O ′ , if O m ⊂ O ′ m for all m , and denote O ∪ O ′ = ( O m ∪ O ′ m ) m and O ∩ O ′ = ( O m ∩ O ′ m ) m ∈ I . Note that the C ∗ -subalgebras C ∗ (( P d ( G m ) , A ( H m )) m ∈ I ) O and C ∗ L (( P d ( G m ) , A ( H m )) m ∈ I ) O are closed two-sided ideals of C ∗ (( P d ( G m ) , A ( H m )) m ∈ I ) O ′ and C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) O ′ , respectively.Given a sequence of open sets O = ( O m ) m , we have a homomorphism e ∗ : K ∗ ( C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) O ) → K ∗ (( C ∗ u ( P d ( G m ) , A ( H m )) m ∈ I ) O ) induced by the evaluation-at-zero map on K -theory. Lemma 4.11.
Let O (1) = ( O (1) m ) and O (2) = ( O (2) m ) be two sequences of sets such that each O ( i ) m is anopen subset of R + × H m for i = 1 , and for all m ∈ I . Then we have(1) C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O (1) ∪ O (2) = C ∗ (( P d ( G m ) , A ( H m )) m ∈ I ) O (1) + C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O (2) , (2) C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) O (1) ∪ O (2) = C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) O (1) + C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) O (2) , (3) C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O (1) ∩ O (2) = C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O (1) ∩ C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O (2) , (4) C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) O (1) ∩ O (2) = C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) O (1) ∩ C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) O (2) . Proof.
We shall prove the first equality and others can be dealt with similarly. To prove the first equality,it suffices to show that C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) O (1) ∪ O (2) = C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) O (1) + C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) O (2) , Given T ∈ C ∗ alg (( P d ( G m ) , A ( H m )) m ∈ I ) O (1) ∪ O (2) , without loss of generality, assume thatSupp ( T ( m ) ) ⊂ ( Z md × Z md ) × ( C m ∪ C m ) ⊂ ( Z md × Z md ) × ( O (1) m ∪ O (2) m ) , where C m ⊂ O (1) m and C m ⊂ O (2) m are closed subsets. Taking smooth functions φ ∈ C ( O (1) m ) and φ ∈ C ( O (2) m ) such that φ m + φ m = 1 on C m ∪ C m . Since the space R + × H m is locally compact, theexistence of the functions φ m and φ m is guaranteed for each m .Define ( T ( m )1 )( x, y ) = φ m T ( m ) ( x, y )( T ( m )2 )( x, y ) = φ m T ( m ) ( x, y ) for x, y ∈ X md . Then we have T = (cid:16) T ( m )2 (cid:17) m ∈ I ∈ C ∗ alg (( P d ( G m ) , A ( H m )) m ∈ I ) O (1) , and T = (cid:16) T ( m )2 (cid:17) m ∈ I ∈ C ∗ alg (( P d ( G m ) , A ( H m )) m ∈ I ) O (2) . In addition, we have T = T + T . 30 roposition 4.12. Let r > . If O = ( O m ) m ∈ I is a sequence of sets such that each O m is the union ofa family of open subsets { O m,j } j ∈ J of R + × H m satisfying:(1) for each j ∈ J , O m,j ⊆ B ( f m ( π ( x mj )) , r ) for some x mj ∈ G m , where B ( f m ( π ( x mj )) , r ) = (cid:8) ( t, h ) ∈ R + × H m : t + k h − f m ( π ( x mj )) k < r (cid:9) , and f m is the coarse embedding f m : Q m → H m ;(2) for each m , O m,j ∩ O m,j ′ = ∅ if j = j ′ .then the map e ∗ : lim d →∞ K ∗ ( C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) O ) → lim d →∞ K ∗ ( C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O ) induced by the evaluation-at-zero map on K -theory is an isomorphism. In order to prove this result, we need to analyze the algebraic structure of the twisted localizationalgebras and twisted Roe algebras.Let T = (cid:0) T ( m ) (cid:1) m ∈ I ∈ C ∗ (( P d ( G m ) , A ( H m )) m ∈ I ) O . Since each O m is the disjoint union of a family ofopen subsets { O m,j } j ∈ J , then for each m the operator T ( m ) can be expressed as T ( m ) = (cid:16) T ( m ) j (cid:17) m,j , where T ( m ) j is a X md × X md -matrix with Supp ( T ( m ) j ) ⊂ X md × X md × O m,j . Furthermore, by Condition (6)in Definition 4.6, we know that each T mj has support contained in P d ( B G m ( x mj N m , R )) × P d ( B G m ( x mj N m , R )) × O m,j for some R > which is independent on m , j .For a fixed R > , the collection of metric spaces (cid:0) B G m ( x mj N m , R ) (cid:1) m,j satisfies the following properties:(1) the sequence (cid:0) B G m ( x mj N m , R ) (cid:1) m,j has uniform bounded geometry;(2) the sequence (cid:0) B G m ( x mj N m , R ) (cid:1) m,j has Property A.Let N m,j = N m for each m and each j . The sequence of metric spaces (cid:0) B G m ( x mj N m , R ) (cid:1) m,j isuniformly coarsely equivalent to the metric spaces ( N m,j ) m,j in the sense that there exists a constant R > such that N m,j is an R -net of B G m ( x mj N m , R ) for each m and each j .Let A ( H m ) O m,j be the C ∗ -subalgebra of A ( H m ) with support contained in O m,j for each m and each j .For the sequence of metric spaces (cid:0) B G m ( x mj N m , R ) (cid:1) m,j and C ∗ -algebras (cid:0) A ( H m ) O m,j (cid:1) m,j , we can definea ∗ -subalgebra of Roe algebra C ∗ (( P d ( G m ) , A ( H m )) m ∈ I ) O .For brevity, set up Z md,j,R = P d ( B G m ( x mj N m , R )) . Given R > , for each d > , let X md,j,R = X md ∩ P d ( B G m ( x mj N m , R )) be the countable dense subset of P d ( B G m ( x mj N m , R )) , where X md ⊂ P d ( G m ) is thecountable sense subset in Definition 4.6. 31 efinition 4.13. Let C ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) be the set of tuples T = (cid:16) T ( m ) j (cid:17) m,j where each T ( m ) j is a bounded function on X md,j,R × X md,j,R such that(1) there exists an integer N such that T ( m ) j ( x, y ) ∈ ( β N ( x ))( A ( W N ( x )) b ⊗ K ) ⊆ A ( H m ) b ⊗ K for all x, y ∈ X md,j,R , where β N ( x ) : A ( W N ( x )) → A ( H m ) is the ∗ -homomorphism associated to theinclusion of W N ( x ) into H m , and K is the algebra of compact operators on an infinite-dimensionalseparable Hilbert space;(2) there exists L > such that, for each m and each y ∈ X md , ♯ { x : T ( m ) j ( x, y ) = 0 } ≤ L, ♯ { x : T ( m ) j ( x, y ) = 0 } ≤ L ; (3) for each m, j and each bounded subset B ⊂ Z md,j,R , the set ♯ { ( x, y ) ∈ ( B × B ) ∩ ( X md,j,R × X md,j,R ) : T ( m ) j ( x, y ) = 0 } is finite;(4) there exists r > , T ( m ) j ( x, y ) = 0 for all m, j and x, y ∈ X md,j,R with d ( x, y ) > r ;(5) there exists r > such that Supp ( T ( m ) j ( x, y )) ⊆ B ( f m ( π ( x mj )) , r ) for all x, y ∈ X md,j,R where B ( f m ( π ( x )) , r ) = (cid:8) ( s, h ) ∈ R + × H : s + k h − f ( π ( x )) k < r (cid:9) ; (6) there exists c > such that if β N ( x )( S ( m ) j ( x, y )) = T ( m ) j ( x, y ) for some S ( m ) j ( x, y ) ∈ A ( W N ( x )) , then D Y ( S ( m ) j ( x, y )) ∈ A ( W N ( x )) ˆ ⊗ K exists and sup m k D Y ( S ( m ) j ( x, y )) k ≤ c for all x, y ∈ X md,j,R and Y = ( s, h ) ∈ R + × W N ( x ) satisfying k Y k = p s + k h k ≤ .The algebra C ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) admits a ∗ -algebra structure defined by coordinate-wise multiplication and adjoint.Let E mj = ℓ ( X md,j,R ) ⊗ K ⊗ A ( H m ) O m,j . X md,j,R ⊂ X md , O m,j ∩ O m,j ′ = ∅ for all j = j ′ , and A ( H m ) O m,j ⊂ A ( H m ) is an ideal, we have that E mj is sub-module of E m . Moreover, we have that thedirect sum L j E mj is a sub-module of E m . Naturally, the ∗ -algebra C ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) admits by faithful ∗ -representation by matrix operation on the Hilbert module L m,j E mj .Let C ∗ u (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) be the completion of C ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) underthe norm on the Hilbert module L m,j E mj . Note that C ∗ u (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) is a C ∗ -subalgebraof C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O .Similarly, we have the localization algebras C ∗ u,L (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) and it is a C ∗ -subalgebraof C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) O . 32y the Condition (6) in Definition 4.6, we have that C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O = lim R →∞ C ∗ u (( P d ( B G m ( x j N m , R )) , A ( H m ) O m ,j ) m ∈ I,j ∈ J ) and C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) O = lim R →∞ C ∗ u,L (( P d ( B G m ( x j N m , R )) , A ( H m ) O m ,j ) m ∈ I,j ∈ J ) . For each
R > , we have the map e ∗ : lim d →∞ K ∗ ( C ∗ u,L (( Z md,j,R , A ( H m ) O m ,j ) m ∈ I,j ∈ J )) → lim d →∞ K ∗ ( C ∗ u (( Z md,j,R , A ( H m ) O m ,j ) m ∈ I,j ∈ J )) induced by the evaluation-at-zero map on K -theory, where Z md,j,R = P d ( B G m ( x mj N m , R )) is the Ripscomplex of B G m ( x mj N m , R ) for each m, j .Now, let us recall the result of the third author about the coarse Baum–Connes conjecture in [54]. Forevery discrete metric space Z with bounded geometry, the third author showed that if Z admits a coarseembedding into a Hilbert space, then the coarse Baum–Connes conjecture holds for Z , i.e, the map e ∗ : lim d C ∗ L ( P d ( Z )) → lim d C ∗ ( P d ( Z ))) induced by evaluation-at-zero map on K -theory is an isomorphism.In this paper, we need the following generalized version of the third author’s result in [54]. Theorem 4.14 (Yu, [54]) . Let
R > be any positive number. Then the map e ∗ : lim d →∞ K ∗ ( C ∗ u,L (( Z md,j,R , A ( H m ) O m ,j ) m ∈ I,j ∈ J )) → lim d →∞ K ∗ ( C ∗ u (( Z md,j,R , A ( H m ) O m ,j ) m ∈ I,j ∈ J )) induced by the evaluation-at-zero map on K -theory is an isomorphism. Although the algebras in this theorem are different from the ones in [54], it can be proved by the sameideas. Using Dirac-dual-Dirac method reduces the above version of coarse Baum–Connes conjecture toa twisted version of coarse Baum–Connes conjecture which can be proved using the same cutting-and-pasting techniques in [54].
Proof of Proposition 4.12.
For each
R > , we have the commutative diagram: K ∗ ( C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) O ) e ∗ / / ∼ = (cid:15) (cid:15) K ∗ ( C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O ) ∼ = (cid:15) (cid:15) lim R →∞ K ∗ ( C ∗ u,L (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) e ∗ / / lim R →∞ K ∗ ( C ∗ u ( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J )) . By Theorem 4.14, the bottom horizontal map is an isomorphism, as a result, the map e ∗ : lim d →∞ K ∗ ( C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) O ) → lim d →∞ K ∗ ( C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O ) is an isomorphism. 33 roof of Theorem 4.9: For any r > and for each m , let O m,r = ∪ x ∈ G m /N m B ( f m ( x ) , r ) ⊂ R + × H m ,where f m : Q m → H m is the coarse embedding and B ( f m ( x ) , r ) = { ( t, h ) ∈ R + × H m : t + k h − f ( x ) k < r } . Let O r = ( O m,r ) m ∈ I be the sequence of open subsets. By definition, we have C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) = lim r →∞ C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O r , and C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) = lim r →∞ C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) O r . So it suffices to show that, for any r > , the map e ∗ : lim d →∞ C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) O r = lim d →∞ C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O r induced by the evaluation-at-zero map on K -theory is an isomorphism. Since the coarse disjoint union ⊔ m Q m has bounded geometry property and f m : Q m → H m are coarse embedding uniformly, then thereexist finitely many, say k r , independent on m , mutually disjoint subsets { Λ m,i } k r i =1 of Q m such that G m /N m = k r ⊔ i =1 Λ m,i , where for each i and each m , d ( f m ( g ) , f m ( g ′ ))) > r for different elements g and g ′ in Λ m,i .Set O m,r,i = F g ∈ Λ m,i B ( f m ( g ) , r ) , then we have a sequence of open sets O r,i = ( O m,r,i ) m for each r and each i .By Proposition 4.12, we have e ∗ : lim d →∞ K ∗ ( C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) O r,i ) → lim d →∞ K ∗ ( C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O r,i ) is an isomorphism.By the Mayer–Vietoris sequence and Five Lemma, we have that the map e ∗ : lim d →∞ C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) O r = lim d →∞ C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O r induced by the evaluation-at-zero map on K -theory is an isomorphism.Passing to infinity, we have that the map e ∗ : lim d →∞ K ∗ ( C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I )) → lim d →∞ K ∗ ( C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I )) induced by the evaluation-at-zero map on K -theory is an isomorphism. In this subsection, we shall show that the K -theory of the maximal uniform twisted Roe algebra isisomorphic to the K -theory of uniform twisted Roe algebra when the sequence of metric spaces ( N m ) m ∈ I has Property A and the sequence of metric spaces ( Q m ) m ∈ I is uniformly coarsely embeddable into Hilbertspaces. This is important in the definition of the Dirac map in the last section.34 emma 4.15. For each T ∈ C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) , there exists a non-negative number N T suchthat k φ ( a ) k ≤ N T · sup m,x,y k T mx,y k A ( H m ) b ⊗ K for any ∗ -representation φ : C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) → B ( H φ ) . Let φ : C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) → B ( H φ ) be any faithful ∗ -representation. By Lemma 4.15, wecan define the maximal norm on the ∗ -algebra C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) by k T k max = sup (cid:8) k φ ( T ) k| φ : C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) → B ( H φ ) is a ∗ -representation (cid:9) . The maximal twisted Roe algebra C ∗ u,max (( P d ( G m ) , A ( H m )) m ∈ I ) is the norm completion of C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) under the maximal norm.By the universal property of the maximal norm, the identity map on C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) continuously extends to the canonical quotient λ : C ∗ u,max (( P d ( G m ) , A ( H m )) m ∈ I ) → C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) . In the rest of this subsection, we shall follow the arguments in Section 4.3 to analyze the ideal structureof the algebras C ∗ u,max (( P d ( G m ) , A ( H m )) m ∈ I ) and C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) . Using a similar cutting-and pasting machinery, we will show that this canonical quotient map is an isomorphism when the sequenceof metric spaces ( N m ) m has Property A and the sequence ( Q m ) m is uniformly coarsely embeddable intoHilbert space.Let r > . If O = ( O m ) m ∈ I is a sequence of sets such that each O m is a union of a family of opensubsets { O m,j } j ∈ J of R + × H m satisfying:(1) for each j ∈ J , O m,j ⊆ B ( f m ( π ( x mj )) , r ) for some x mj ∈ G m , where B ( f m ( π ( x mj )) , r ) = (cid:8) ( t, h ) ∈ R + × H m : t + k h − f m ( π ( x mj )) k < r (cid:9) , and f m is the coarse embedding f m : G/N → H m ;(2) for each m , O m,j ∩ O m,j ′ = ∅ if j = j ′ .Recall in Definition 4.6 of Section 4.2, for the sequence of open sets O = ( O m ) m ∈ I , we define a ∗ -algebra C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) O . Then C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) O is a ∗ -subalgebra of the maximalRoe algebra C ∗ u,max (( P d ( G m ) , A ( H m )) m ∈ I ) , and the completion of C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) O underthe norm in maximal twisted Roe algebra is denoted by C ∗ u,φ (( P d ( G m ) , A ( H m )) m ∈ I ) O . We can also view C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) O as a ∗ -subalgebra of C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) and let C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O be the completion of C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) O . λ restrict to a ∗ -homomorphism λ : C ∗ u,φ (( P d ( G m ) , A ( H m )) m ∈ I ) O → C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O Recall that Z md,j,R = P d ( B G m ( x j N m , R )) for each m , each j and each R . For each d > , let X md,j,R = X md ∩ P d ( B G m ( N m , R )) be the countable dense subset of P d ( B G m ( N m , R )) , where X md ⊂ P d ( G m ) be the countable sense subsetin Definition 4.6.Let C ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) be the ∗ -algebra defined in Definition 4.13. We have that C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) O = lim R →∞ C ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) . The inductive limit can be viewed as the union of the nested ∗ -algebras. For each R > , the ∗ -algebra C ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) is a subalgebra of C ∗ φ (( P d ( G m ) , A ( H m )) m ∈ I ) O , and the completionof C ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) is denoted by C ∗ u,φ (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) . Note that C ∗ u,φ (( P d ( G m ) , A ( H m )) m ∈ I ) O = lim R →∞ C ∗ u,φ (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) . When viewing C ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) as a ∗ -subalgebra of C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O ,we get a completion of C ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) , denoted by C ∗ u (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) .We have that C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O = lim R →∞ C ∗ u (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) . To prove that the restriction of quotient map λ : C ∗ u,φ (( P d ( G m ) , A ( H m )) m ∈ I ) O → C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O is an isomerism, it suffices to show that the restriction of the quotient map λ : C ∗ u,φ (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) → C ∗ u (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) is an isomorphism.Let us now recall the concept of Property A. A metric space with this property was introduced by thethird author in [54] as an example of metric spaces which are coarsely embeddable into Hilbert space. Definition 4.16 ([38]) . A metric space X with bounded geometry is said to have Property A if for any R > , ǫ > , there exists a map ξ : X → ℓ ( X ) and S > such that• for all x, y ∈ X , ≥ ξ x ( y ) ≤ ;• for all x ∈ X , k ξ x k = 1 ; 36 for all x ∈ X , Supp ( ξ x ) ⊂ B X ( x, S ) ;• if d ( x, y ) ≤ R , then | − h ξ x , ξ y i| < ǫ .Then we can define a kernel k : X × X → R by k ( x, y ) = h ξ x , ξ y i for all x, y ∈ X . The kernel k is support on a stripe with length S and it satisfies that | − k ( x, y ) | < ǫ for all x, y ∈ X with d ( x, y ) ≤ R . For each d , we can extend the kernel to Rips complex k : P d ( X ) × P d ( X ) → [0 , by k ( X a i x i , X b i x i ) = X a i b i k ( x i , x i ) for all P a i x i , P b i x i ∈ P d ( X ) . So we do not distinguish the domain of the kernel in the Definition ofProperty A.Assume that the sequence of metric space ( N m ) m ∈ I has Property A, since for each fixed R , the sequenceof metric spaces (cid:16) Z md,j,R = B G m ( x mj N m , R ) (cid:17) m,j is uniformly coarse equivalent to ( N m ) m ∈ I , we have that (cid:16) Z md,j,R = B G m ( x mj N m , R ) (cid:17) m,j also has Property A.Then for each m , each ǫ > and R > we can find a kernel k mj : X md,j,R × X md,j,R → [0 , and S > such that k mj is supported on a stripe of length S uniformly.Now we can define a map (cid:16) M k mj (cid:17) m,j : C ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m,j ) → C ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m,j ) by (cid:16) M k mj (cid:17) m,j : (cid:16) T ( m ) j (cid:17) m,j (cid:16) M k mj · T ( m ) j (cid:17) m,j for all (cid:0) T ( m ) j (cid:1) m,j ∈ C ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m,j ) , where (cid:16) M k mj · T ( m ) j (cid:17) ( x, y ) = k mj ( x, y ) T ( m ) ( x, y ) for all m , j and all x, y ∈ X md,j,R .We then have the following general result which can be proved by the same arguments as Corollary2.3 in [46]. Lemma 4.17.
Let ( k mj ) m,j be the sequence of kernels defined from Property A of the sequence of metricspaces (cid:16) X md,j,R (cid:17) m,j . Then the map (cid:16) M k mj (cid:17) m,j : C ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m,j ) → C ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m,j ) extends continuously to a contractive completely positive map on any C ∗ -algebraic completion of C ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m,j ) . C ∗ u,φ (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) λ (cid:15) (cid:15) (cid:16) M kmj (cid:17) φm ∈ I,j ∈ J / / C ∗ u,φ (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) λ (cid:15) (cid:15) (cid:15) (cid:15) C ∗ u (( Z md,j,R , A ( H m ) O m,j ) m,j ) (cid:16) M kmj (cid:17) m,j / / C ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) where (cid:16) M k mj (cid:17) φm,j and (cid:16) M k mj (cid:17) m,j are the contractive completely positive maps given by Lemma 4.17.For each s > , let C s, ∗ alg (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) be the subspace of all tuples T = (cid:16) T ( m ) j (cid:17) m,j with propagation of T ( m ) j at most s for all m and all j . Then C s, ∗ u,alg (( Z md,j,R , A ( H m ) O m ∈ I,j ∈ J ) m ∈ I,j ∈ J ) is closed in C ∗ φ (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) under the norm topology. Consider a sequence of tuples n T n = (cid:16) T ( m ) j,n (cid:17)o ∞ n =1 in C s, ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) which converges in norm to an operator T = (cid:16) T ( m ) j (cid:17) m,j ∈ C ∗ u,φ (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) .Let λ : C ∗ u,φ (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) → C ∗ u (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) be the restriction of the canonical quotient map from the maximal twisted uniform Roe algebra to thetwisted uniform Roe algebra. It follows that λ ( T n ) → λ ( T ) as n → ∞ . Note that λ ( T n ) = T n since λ : C ∗ u,φ (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) → C ∗ u (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) is the identity map restricted to C ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) . By Lemma 4.15, we have thateach entry T ( m ) j,n ( x, y ) of T ( m ) j,n converges to the entry λ ( T ( m ) j )( x, y ) uniformly in the norm topology of A ( H m ) ⊗ K . As a result of Lemma 4.15, T ( m ) j ( x, y ) = 0 for all x, y ∈ X md,j,R with d ( x, y ) ≥ s . In addition,we have that (cid:16) M k mj (cid:17) m,j (cid:16) C s, ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) (cid:17) ⊂ C s, ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) . Proposition 4.18.
Let
R > . Assume that the sequence of metric spaces (cid:8) B G m ( x mj N m , R ) (cid:9) m,j hasProperty A. Then the canonical quotient map λ : C ∗ u,φ (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) → C ∗ u (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) is an isomorphism.Proof. Using property A, we obtain a kernels ( k mj,n ) m,j associated to R = n and ǫ = 1 /n for each n ∈ N in Definition 4.16. By Lemma 4.15, we have that for each S ∈ C ∗ u,φ (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) , (cid:16) M k mj,n (cid:17) m,j ( S ) converges in the C ∗ u,φ (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) norm to S as n → ∞ .In addition, for each T ∈ C ∗ u,φ (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) , the image (cid:16) M k mj,n (cid:17) m,j ( T ) is an elementin C ∗ u,alg (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) .It suffices to show that λ is injective. Let T ∈ C ∗ u,φ (( Z md,j,R , A ( H m ) O m,j ) m ∈ I,j ∈ J ) in the kernel of λ .Then we have that T = lim n (cid:16) M k mj,n (cid:17) m,j ( T ) = λ (cid:18)(cid:16) M k mj,n (cid:17) m,j ( T ) (cid:19) = lim n (cid:16) M k mj,n (cid:17) m,j ( λ ( T )) = 0 . Therefore, λ is injective. 38 heorem 4.19. Let (1 → N m → G m → Q m → m ∈ I be a sequence of extensions of discrete groups.Assume that the sequence of metric spaces ( N m ) m , ( G m ) m and ( Q m ) m have uniform bounded geometry.If the sequence of metric spaces ( N m ) m has Property A and the sequence ( Q m ) m is coarsely embeddableinto Hilbert space, then the canonical quotient map λ : C ∗ u,max (( P d ( G m ) , A ( H m )) m ∈ I ) → C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) is an isomorphism.Proof. By the definition the maximal twisted Roe algebra and twisted Roe algebra, and the proof ofProposition 4.12, we have that C ∗ u,max (( P d ( G m ) , A ( H m )) m ∈ I ) = lim r C ∗ u,φ (( P d ( G m ) , A ( H m )) m ∈ I ) O r and C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) = lim r C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O r , where O r = ( O m,r ) m is the sequence of open sets as in Theorem 4.9 and C ∗ u,φ (( P d ( G m ) , A ( H m )) m ∈ I ) O r is the completion of C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) O r under the norm obtained by the inclusion C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) O r ֒ → C ∗ u,max (( P d ( G m ) , A ( H m )) m ∈ I ) O r . Moreover, for each r > , as in the proof of Theorem 4.9, there are finitely many sequences of opensets O r,i = ( O m,r,,i,j ) m,j , for ≤ i ≤ k r for some k r ∈ N , where each O m,r,i,j is an open subsets of R + × H m such that• O m,r,i,j ⊂ B ( f m ( π ( x mj ))) for some x mj ∈ G m ;• O m,r,i,j ∩ O m,r,i,j ′ = ∅ for all j = j ′ .For each r and each ≤ i ≤ k r , the completion of the ∗ -algebra C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) O r,i isdenoted by C ∗ u,φ (( P d ( G m ) , A ( H m )) m ∈ I ) O r,i under the ∗ -representation induced by the inclusion C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) O r,i → C ∗ u,max (( P d ( G m ) , A ( H m )) m ∈ I ) O r,i . Note that C ∗ u,φ (( P d ( G m ) , A ( H m )) m ∈ I ) O r,i = lim R →∞ C ∗ u,φ (( Z md,j,R , A ( H m ) O m,r,i,j ) m ∈ I,j ∈ J ) , and C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O r,i = lim R →∞ C ∗ u (( Z md,j,R , A ( H m ) O m,r,i,j ) m ∈ I,j ∈ J ) . By Proposition 4.18, we have that the restriction of the canonical quotient map λ : C ∗ u,φ (( Z md,j,R , A ( H m ) O m,r,i,j ) m ∈ I,j ∈ J ) → C ∗ u (( Z md,j,R , A ( H m ) O m,r,i,j ) m ∈ I,j ∈ J )
39s an isomorphism. It follows that the map λ : C ∗ u,φ (( P d ( G m ) , A ( H m )) m ∈ I ) O r.i → C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O r.i is an isomorphism for each r and each i . By the Mayer–Vietoris sequence and Five Lemma, we have thatthe restriction of the canonical map λ : C ∗ u,φ (( P d ( G m ) , A ( H m )) m ∈ I ) O r → C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) O r is an isomorphism. By taking limit on r , we have that the canonical quotient map λ : C ∗ u,max (( P d ( G m ) , A ( H m )) m ∈ I ) → C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) is an isomorphism. In this section, we shall prove Theorem 3.12 using a geometric analogue of the infinite-dimensional Bottperiodicity introduced by the third author in [54]. The geometric analogue of infinite-dimensional Bottperiodicity is used to reduce the coarse Baum–Connes conjecture to the twisted coarse Baum–Connesconjecture. β and β L In this subsection, we shall define the Bott map β : S b ⊗ C ∗ u (( P d ( G m )) m ∈ I ) → C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) and the localized Bott map β L : S b ⊗ C ∗ u,L (( P d ( G m )) m ∈ I ) → C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) . For x ∈ X md . Let E be a finite-dimensional Euclidean space of H m . Define a ∗ -homomorphism β E ( x ) : S → A ( E ) by β E ( x )( g ) = g ( X b ⊗ b ⊗ C E,x ) for all g ∈ S , where X is the operator of multiplication by x on R viewed as a degree one and unboundedmultiplier of S , C E,x is the Clifford algebra-valued function on E denoted by C E,x ( v ) = v − f m ( π ( x )) ∈ E ⊂ Cliff ( E ) for all v ∈ E and g ( X b ⊗ b ⊗ C E,x ) is defined by functional calculus. By definition of thealgebra A ( H m ) , we have a ∗ -homomorphism β ( x ) : S → A ( H m ) . For each t ∈ [1 , ∞ ) , we define a map β : S b ⊗ C ∗ u [( P d ( G )) m ∈ I ] → C ∗ u,alg (( P d ( G ) , A ( H m )) m ∈ I ) (cid:0) β t (cid:0) g b ⊗ T (cid:1)(cid:1) ( m ) ( x, y ) = β ( x )( g t ) b ⊗ T ( m ) ( x, y ) for all g ∈ S , T = ( T ( m ) ) m ∈ C u [( P d ( G )) m ∈ I ] , where g t ( s ) = g ( t − s ) for all t ≥ , s ∈ R , and β ( x ) : S →A ( H m ) , is the ∗ -homomorphism associated to the inclusion of the zero-dimensional affine space into H m by mapping to f m ( π ( x )) , where A ( H m ) is defined in Section 4.2. Lemma 5.1.
The Bott map β extends to an asymptotic morphism β : S b ⊗ C ∗ u (( P d ( G )) m ∈ I ) → C ∗ u (( P d ( G ) , A ( H m )) m ∈ I ) . Proof.
Let E = ⊕ m E m as in Section 4.2, where E m = ℓ ( X md ) ⊗ A ( H m ) ⊗ K . For each m and each g ∈ S ,we define a module homomorphism V mg : E m → E m by V mg X x ∈ X md a x [ x ] = X x ∈ X md ( β ( x )( g ) ⊗ a x [ x ] for all g ∈ S and all P x ∈ X md a x [ x ] ∈ E m . Then taking direct sum gives us a module homomorphism V g = M m V mg : ⊕ m E m → M m E m . Note that β t ( g b ⊗ ( T ( m ) ) m ) = V g t ◦ (1 b ⊗ ( T ( m ) ) m ) for all g ∈ S and ( T ( m ) ) m ∈ C ∗ (( P d ( G m )) m ∈ I ) , where b ⊗ ( T ( m ) ) m ∈ I is the adjointable homomorphism on E = L m E m defined by b ⊗ ( T ( m ) ) m ∈ I : M m X x ∈ X md a x [ x ] M m X y ∈ X md X x ∈ X md (1 b ⊗ T ( m ) ( y, x )) a x [ y ] for all P x ∈ X md a x [ x ] ∈ L m E m .Since k ( T ( m ) ) m k = sup m k T ( m ) k E m , we have that k β t ( g b ⊗ ( T ( m ) ) m ) k ≤ k g k · k ( T ( m ) ) m k . (1)As a result, for each t ≥ , the linear map β t : S b ⊗ C u [( P d ( G m )) m ∈ I ] → C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) extends to a bounded leaner map β t : S b ⊗ C ∗ u (( P d ( G m )) m ∈ I ) → C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) where S b ⊗ alg C ∗ u (( P d ( G )) m ∈ I ) is the algebraic tensor product. Since k f m ( π ( x )) − f m ( π ( y )) k ≤ ρ ( d ( π ( x ) , π ( y ))) for all x, y ∈ E m by the uniformly coarse embeddability of the sequence of quotient groups ( Q m ) m , inaddition with Inequality (1) above, we have that k β t (( g b ⊗ T ) · ( g b ⊗ T )) − β t (( g b ⊗ T ) · β t (( g b ⊗ T ) k → as t → ∞ g i ∈ S and T i ∈ C ∗ u (( P d ( G m )) m ∈ I ) , i = 1 , .So we have a ∗ -homomorphism β : S b ⊗ alg C ∗ u (( P d ( G )) m ∈ I ) → C b ([1 , ∞ ) , C ∗ u (( P d ( G ) , A ( H m )) m ∈ I )) C ([1 , ∞ ) , C ∗ u (( P d ( G ) , A ( H m )) m ∈ I )) by g b ⊗ T [ β t ( g b ⊗ T )] for all g ∈ S and all T ∈ C ∗ u (( P d ( G ) , A ( H m )) m ∈ I ) , where C b ([1 , ∞ ) , C ∗ u (( P d ( G ) , A ( H m )) m ∈ I )) is the C ∗ -algebran all norm-continuous and bounded functions from [1 , ∞ ) to C ∗ u (( P d ( G ) , A ( H m )) m ∈ I ) and C ([1 , ∞ ) , C ∗ u (( P d ( G ) , A ( H m )) m ∈ I )) is the C ∗ -subalgebra of C b ([1 , ∞ ) , C ∗ u (( P d ( G ) , A ( H m )) m ∈ I )) consist-ing of functions vanishing at infinity.Accordingly, we obtain a ∗ -homomorphism between C ∗ -algebras from the maximal tensor product S b ⊗ max C ∗ (( P d ( G )) m ∈ I ) to the C ∗ -algebra C b ([1 , ∞ ) , C ∗ u (( P d ( G ) , A ( H m )) m ∈ I )) C ([1 , ∞ ) , C ∗ u (( P d ( G ) , A ( H m )) m ∈ I )) . By nuclearity of S , we have that S b ⊗ C ∗ u (( P d ( G )) m ∈ I ) ∼ = S b ⊗ max C ∗ u (( P d ( G )) m ∈ I ) . Then the ∗ -homomorphism above gives rise to an asymptotic morphism β t : S b ⊗ C ∗ u (( P d ( G )) m ∈ I ) → C ∗ u (( P d ( G ) , A ( H m )) m ∈ I ) , for t ∈ [0 , ∞ ) .Similarly, we can define asymptotic morphism between the localization algebras β L,t : S b ⊗ C ∗ u,L (( P d ( G m )) m ∈ I ) → C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) . In addition, the asymptotic morphisms ( β t ) t ∈ [0 , ∞ ) and ( β L,t ) t ∈ [1 , ∞ ) induce homomorphisms on K -theory β ∗ : K ∗ ( S b ⊗ C ∗ u (( P d ( G m )) m ∈ I )) → K ∗ ( C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I )) and β L, ∗ : K ∗ ( S b ⊗ C ∗ u,L (( P d ( G m )) m ∈ I )) → K ∗ ( C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I )) . α and α L In this subsection, we shall define the Dirac maps α and α L . We will firstly recall the definition of theBott–Dirac operator on an infinite dimensional Euclidean space, then we extend the construction of Diracmaps to the geometric analogues on Roe algebras and localization algebras.Let H be an infinite-dimensional separable Hilbert space. Let V be the countable infinite-dimensionalEuclidean dense subspace of H . If V a ⊂ V be a finite-dimensional affine subspace, we use L ( V a , Cliff ( V a )) to denote the Hilbert space of square integrable functions from V a to Cliff ( V a ) . If V a and V b are finitedimensional affine subspaces with V a ⊂ V b , then there exists a decomposition V b = V ba ⊕ V a . Then L ( V b , Cliff ( V b )) ∼ = L ( V ba , Cliff ( V ba )) b ⊗ L ( V a , Cliff ( V a )) , L ( V ba , Cliff ( V ba )) is the the Hilbert space associated with V ba . We define a unit vector ξ ba ∈ L ( V ba , Cliff ( V ba )) by ξ ba ( v ba ) = π − dim ( V ba ) / exp ( − k v ba k ) for all v ba ∈ V ba . Regarding L ( V a , Cliff ( V a )) as a subspace of L ( V b , Cliff ( V b )) via the isometry ξ ξ b ⊗ ξ for V a ⊂ V b , we define H = lim −→ L ( V a , Cliff ( V a )) . For each finite dimensional affine subspace V a ⊂ V , define s a to be the space of Schwartz functionsfrom V a to Cliff ( V a ) . Let s = lim −→ s a be the algebraic direct limit of the Schwartz subspaces s a ⊂ V a .If V a ⊂ V ⊂ H is a finite-dimensional affine subspace, then the Dirac operator D V a is an unboundedoperator on L ( V a , Cliff ( V a )) with domain s a , defined by D a ξ = n X i =1 ( − degξ ∂ξ∂x i v i where { v , · · · , v n } is an orthonormal basis for V a and { x , · · · , x n } are the dual coordinates to { v , · · · , v n } .The Clifford operator C V a ,v of V a at v ∈ V a is an unbounded operator on L ( V a , Cliff ( V a )) defined by ( C V a ,v ξ )( v ) = ( v − v ) · ξ ( v ) for any ξ ∈ L ( V a , Cliff ( V a )) and v ∈ V a , where the multiplication is the Clifford multiplication of v − v ∈ V a ∈ Cliff ( V a ) and ξ ( v ) ∈ Cliff ( V a ) . The domain of the Clifford operator is the space of Schwartz functions s a . When V a is a linear subspace and v = 0 ∈ V a , we denote C V a = C V a , .Given an algebraic decomposition V = V ⊕ V ⊕ V ⊕ · · · where each V i is a finite-dimensional linear subspace of V . For each n ∈ N and each t ∈ [1 , ∞ ) , we definean unbounded, selfadjoint operator B n,t on H by B n,t = t D + t D + · · · + t n − D n − + t n ( D n + C n ) + t n +1 ( D n +1 + C n +1 ) + · · · where t i = 1 + i/t , C i and D i are the Clifford operator and the Dirac operator on L ( V i , Cliff ( V i )) ,respectively.Let K be the algebra of compact operators on the Hilbert space H and S b ⊗K the graded tensor productwhere K is endowed with the trivial grading. Let W n = V ⊕ V ⊕ · · · ⊕ V n − , then we have an asymptoticmorphism ( α n,t ) t ∈ [1 , ∞ ) : A ( W n ) → S b ⊗K by α n,t : f b ⊗ h f ( X b ⊗ b ⊗ B n,t ) · (1 b ⊗ M h t ) f ∈ S and h ∈ C ( W n ) where h t ( v ) = h ( v/t ) and M h t is the operator on H of left multiplication by h t . By Definition 2.8, Lemma 2.9 and Proposition 4.2 in [30], this map is well-defined. In addition, wehave the following asymptotically commutative diagram: A ( W n ) β Wn,Wn +1 (cid:15) (cid:15) α n,t / / S b ⊗K = (cid:15) (cid:15) A ( W n +1 ) α n +1 ,t / / S b ⊗K where β W n ,W n +1 : A ( W n ) → A ( W n +1 ) is the homomorphism induced by the inclusion from W n to W n +1 .Accordingly, we have an asymptotic morphism α t : A ( H ) → S b ⊗K , for all t ∈ [1 , ∞ ) . Moreover, it has be proved in [30] that the composition S β t −→ A ( H ) α t −→ S b ⊗K is asymptotically equivalent to the homomorphism S → S b ⊗K defined by f f b ⊗ p, for all f ∈ S , where p is a rank-one projection. As a result, the asymptotic morphism of the composition α t ◦ β t induces the identity map the the K -theory of S .For each m , recall f m : Q m → H m is the coarse embedding, and choose V m ⊂ H m to be the infinite-dimensional dense Euclidean subspace of H m defined by V m = span { f m ( x ) ∈ H m : x ∈ Q m } as in Section 4. Recall that for each m ∈ I and x ∈ P d ( G m ) , W n ( x ) ⊂ H m is the subspace W n ( x ) = span (cid:8) f m ( π ( y )) : y ∈ G m , d ( π ( x ) , π ( y )) ≤ n (cid:9) . Let V n ( x ) = W n +1 ( x ) ⊖ W n ( x ) if n ≥ , V ( x ) = W ( x ) , where x ∈ P d ( G m ) and W n ( x ) is as in Section 4.For each x ∈ P d ( G m ) , we have the algebraic decomposition: V m = V ( x ) ⊕ V ( x ) ⊕ · · · ⊕ V n ( x ) ⊕ · · · . For each m ∈ I , we define a Hilbert space H m by choosing a linear dense subspace V m of H m .For each n ∈ N , we define an unbounded operator B mn,t on H m as follows: B mn,t ( x ) = t D m + t D + · · · + t n − D mn − + t n ( D mn + C mn ) + t n +1 ( D mn +1 + C mn +1 ) + · · · where t j = 1 + t − j , D mn and C mn are respectively the Dirac operator and Clifford operator associated to V n ( x ) . The operator B mn,t ( x ) plays the role of the Dirac operator in the infinite-dimensional case. Notethat B mn,t ( x ) is an unbounded and essentially selfadjoint operator.44or each non-negative integer n , m ∈ I and x ∈ X md , we have an asymptotic morphism α n,t ( x ) : A ( W n ( x )) b ⊗ K → S b ⊗K m b ⊗ K. by α n,t ( x ) (cid:0) ( g b ⊗ h ) b ⊗ k (cid:1) = g t ( X b ⊗ b ⊗ B mn,t ( x ))(1 b ⊗ M h t ) b ⊗ k for every g ∈ S , h ∈ C ( W n ( x )) , k ∈ K and t ≥ , where g t ( s ) = g ( t − s ) for all t ≥ and s ∈ R , h t ( v ) = h ( t − v ) for all t ≥ and v ∈ W n ( x ) and M h t acts on H m by point-wise multiplication.In order to define the geometric Dirac maps, we need to introduce new C ∗ -algebras. Note that each ele-ment in S b ⊗K m b ⊗ K can be viewed as a K m b ⊗ K -valued function on R . We define C ∗ u,alg ( P d ( G m ) , S b ⊗K m )) m ∈ I ) to be the algebra of all tuples T = ( T ( m ) ) m ∈ I such that(1) for each m , T ( m ) is a bounded function from X md × X md to S b ⊗K m b ⊗ K , where K m is the algebra of allcompact operators acting on H m and K is the algebra of compact operators on the infinite-dimensionalseparable Hilbert space;(2) there exists L > such that for each y ∈ X md and each m , ♯ { x : T ( m ) ( x, y ) = 0 } ≤ L , and ♯ { y : T ( m ) ( x, y ) = 0 } ≤ L ; (3) there exists r > , T ( m ) ( x, y ) = 0 for all m and x, y ∈ X md with d ( x, y ) > r .(4) for each m ∈ I and each bounded subset B ⊂ P d ( G m ) , the set n ( x, y ) ∈ ( B × B ) ∩ ( X md × X md ) : T ( m ) ( x, y ) = 0 o is finite;(5) there exists r > such that Supp ( T ( m ) ( x, y )) ⊂ [ − r , r ] for all m and x, y ∈ X md .(6) there exists c > , (cid:13)(cid:13)(cid:13)(cid:13) ddt T ( m ) ( x, y ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ c where T ( m ) ( x, y ) ∈ S ⊗ K m b ⊗ K can be viewed a K m b ⊗ K -valued function on R for all m and all x, y ∈ X md .The algebraic structure on C ∗ u,alg ( P d ( G m ) , S b ⊗K ⊗ K ) m ) is defined by: ( T T ) ( m ) ( x, y ) = X z ∈ X d T ( m )1 ( x, z ) T ( m )2 ( z, y ) . and ( T ∗ ) ( m ) ( x, y ) = (cid:16) T ( m )1 ( y, x ) (cid:17) ∗ for all x, y ∈ X md and T , T ∈ C ∗ u,alg (( P d ( G m ) , S b ⊗K m )) m ∈ I ) .Let E m = ℓ ( X md ) b ⊗H m b ⊗ H , where K is the algebra of all compact operators on H . Then C ∗ u,alg ( P d ( G m ) , K m ) m ∈ I ) acts on E = ⊕ m E m by T ( δ x b ⊗ h b ⊗ h ) = X y ∈ X d δ y b ⊗ T ( m ) ( y, x )( h b ⊗ h ) m ∈ I , x ∈ X md , h ∈ H m , h ∈ H , where δ x and δ y are the Dirac functions at x and y , respectively.The completion of C ∗ u,alg ( P d ( G m ) , S b ⊗K m ) m ∈ I ) under the operator norm on E is denoted by C ∗ u (( P d ( G m ) , S b ⊗K m )) m ∈ I ) . Since K m b ⊗ K is isomorphic to K for all m . For each T ∈ C ∗ a,alg (( P d ( G m ) , S b ⊗K m )) m ∈ I ) , we can viewed itas a function from R to C u [( P d ( G m )) m ∈ I ] by ( T ( t )) ( m ) ( x, y ) = T ( m ) ( x, y )( t ) . Since the support of each T ( m ) ( x, y )( t ) is contained in some interval [ − r , r ] where r does not depend on x, y, m , so this function is of compactly supported. Moreover, the sequence of functions (cid:8) T ( m ) ( x, y )( t ) (cid:9) x,y,m has uniformly bounded derivative, so the functions t T ( m ) ( x, y )( t ) are equicontinuous on t . It fol-lows that the function t T ( t ) is continuous on t . As a result, we can viewed T as an element in S b ⊗ C ∗ u (( P d ( G m )) m ∈ I ) . It is easy to verify that C ∗ u (( P d ( G m ) , S ⊗ K m )) m ∈ I ) ∼ = S b ⊗ C ∗ u (( P d ( G m )) m ∈ I ) . Denoted by C ∗ u,L,alg (( P d ( G m ) , S b ⊗K m ) m ∈ I ) the algebraic uniform localization algebra consisting of alluniformly bounded and uniformly continuous functions g = ( g ( m ) ) m ∈ I : [0 , ∞ ) → C ∗ u,alg (( P d ( G m ) , S b ⊗K m ) m ∈ I ) such that(1) sup m propagation ( g ( m ) ) s )) → , as s → ∞ ;(2) there exists r > such that Supp ( g ( m ) s ( x, y )) ⊂ [ − r , r ] for all m ∈ I , x, y ∈ X md and s ∈ [0 , ∞ ) ;(3) there exists c > , (cid:13)(cid:13)(cid:13)(cid:13) ddt g ( m ) s ( x, y ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ c where each g ( m ) s ∈ S ⊗ K m b ⊗ K is viewed a K m b ⊗ K -valued function on R for all m , all x, y ∈ X md andall s ∈ [0 , ∞ ) .The uniform localization algebra, denoted by C ∗ u,L (( P d ( G m ) , S b ⊗K m ) m ∈ I ) , is the completion of the ∗ -algebra C ∗ u,L,alg (( P d ( G m ) , S b ⊗K m ) m ∈ I ) under the norm k g k = sup s k g s k for all g ∈ C ∗ u,L,alg (( P d ( G m ) , S b ⊗K m ) m ∈ I ) . Note that C ∗ u,L (( P d ( G m ) , S b ⊗K m ) m ∈ I ) ∼ = S b ⊗ C ∗ u,L (( P d ( G m )) m ∈ I ) . Definition 5.2.
For each d > and each t ∈ [1 , ∞ ) , we define a map α t : C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) → C ∗ u (( P d ( G m ) , S b ⊗K m ) m ∈ I ) by ( α t ( T )) ( m ) ( x, y ) = ( α N,t ( x ))( T m ( x, y )) for all T = ( T ( m ) ) m ∈ C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) and t ≥ , where N is a positive integer such that forevery pair ( x, y ) ∈ X md × X md , there exists T ( m )1 ( x, y ) ∈ A ( W N ( x )) b ⊗ K satisfying ( β N ( x ))( T ( m )1 ( x, y )) = T ( m ) ( x, y ) .
46y Lemma 7.2, Lemma 7..3, Lemma 7.4 and Lemma 7.4 in [54], we know that the asymptotic morphism ( α t ) t ∈ [0 , ∞ ) is well-defined. Similarly, we can define the maps between the localization algebras. Definition 5.3.
For each d > and each t ∈ [1 , ∞ ) , we define a map α L,t : C ∗ u,L,alg (( P d ( G m ) , A ( H m )) m ∈ I ) → C ∗ u,L (( P d ( G m ) , S b ⊗K m ) m ∈ I ) by ( α L,t ( T )) ( m ) ( s ) = ( α t ( x ))( T m ( s )) for all ( T ( s )) s ∈ [0 , ∞ ) = ( T ( m ) ( s )) m ∈ C ∗ u,L,alg (( P d ( G m ) , A ( H m )) m ∈ I ) and t ≥ . Lemma 5.4.
The maps ( α t ) t ∈ [1 , ∞ ) and ( α L,t ) t ∈ [1 , ∞ ) extends to an asymptotic morphism ( α t ) t ∈ [1 , ∞ ) : C ∗ u,max (( P d ( G m ) , A ( H m )) m ∈ I ) → C ∗ u (( P d ( G m ) , S b ⊗K m ) m ∈ I ) , and ( α L,t ) t ∈ [1 , ∞ ) : C ∗ u,max (( P d ( G m ) , A ( H m )) m ∈ I ) → C ∗ u (( P d ( G m ) , S b ⊗K m ) m ∈ I ) . Proof.
Given any
T, S ∈ C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) . By the definition of W n ( x ) , there exists N > independent of m such that W n ( x ) ⊂ W n +1 ( y ) for all n ≥ N , m ∈ I and x, y ∈ X md satisfying d ( x, y ) ≤ propagation ( T ) . By the definition of twisted Roe algebra (Definition 4.6), there exists N suchthat for each n ≥ N , and for every pair ( x, y ) ∈ X md × X md there exists T ( m )1 ( x, y ) ∈ A ( W n ( x )) ˆ ⊗ K satisfying ( β n ( x ))( T ( m )1 ( x, y )) = T ( m ) ( x, y ) and ( β n ( x ))( S ( m )1 ( x, y )) = S ( m ) ( x, y ) . It follows from Lemma7.3, Lemma 7.4 and 7.5 in [54] that k α n,t ( x )( T m ( x, y ) S m ( x, y )) − α n,y ( x )( T m ( x, y )) α n,t ( x )( T m ( x, y )) k → , and ( α n,t ( T ( m )1 ( x, y ))) ∗ − α n,t (( T ( m )1 ( x, y )) ∗ ) → uniformly as t → ∞ . By Lemma 4.15, we have that k α t ( ST ) − α t ( S ) α t ( T ) k → , and k α t ( a ) ∗ − α t ( a ∗ ) k → as t → ∞ . As a result, we have an asymptotic morphism ( α t ) t ∈ [1 , ∞ ) : C ∗ u,max (( P d ( G m ) , A ( H m )) m ∈ I ) → S b ⊗ C ∗ u (( P d ( G m )) m ∈ I ) . Similarly, we have an asymptotic morphism between localization algebras ( α L,t ) t ∈ [1 , ∞ ) : C ∗ u,max (( P d ( G m ) , A ( H m )) m ∈ I ) → S b ⊗ C ∗ u (( P d ( G m )) m ∈ I ) . By Theorem 4.4, we have the isomorphisms λ : C ∗ u,max (( P d ( G m ) , A ( H m )) m ∈ I ) → C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) and λ : C ∗ u,L,max (( P d ( G m ) , A ( H m )) m ∈ I ) → C ∗ u,L ( P d ( G m ) , A ( H m )) m ∈ I ) . λ , still denoted by ( α t ) t ∈ [0 , ∞ ) : C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) → C ∗ u (( P d ( G m ) , S b ⊗K m ) m ∈ I ) = S b ⊗ C ∗ u (( P d ( G m )) m ∈ I ) and ( α L,t ) t ∈ [0 , ∞ ) : C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) → C ∗ u,L (( P d ( G m ) , S b ⊗K m ) m ∈ I ) = S b ⊗ C ∗ u,L (( P d ( G m )) m ∈ I ) . Accordingly, we have the homomorphisms α ∗ : K ∗ ( C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I ) → K ∗ ( S b ⊗ C ∗ u (( P d ( G m )) m ∈ I ) , and α L, ∗ : K ∗ ( C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I ) → K ∗ ( S b ⊗ C ∗ u,L (( P d ( G m )) m ∈ I ) induced by the asymptotic morphisms ( α t ) [1 , ∞ ) and ( α L,t ) [1 , ∞ ) on K -theory. Note that the asymptotic morphisms α , α L , β and β L induce the following commutative diagram on K -theory: K ∗ ( S b ⊗ C ∗ u,L (( P d ( G m )) m ∈ I )) e ∗ / / ( β L ) ∗ (cid:15) (cid:15) K ∗ ( S b ⊗ C ∗ u (( P d ( G m )) m ∈ I ) β ∗ (cid:15) (cid:15) K ∗ ( C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I )) ( α L ) ∗ (cid:15) (cid:15) e ∗ ∼ = / / K ∗ ( C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I )) α ∗ (cid:15) (cid:15) K ∗ ( S b ⊗ C ∗ u,L (( P d ( G m )) m ∈ I ) e ∗ / / K ∗ ( S b ⊗ C ∗ u ( P d ( G m )) m ∈ I )) In this subsection, we shall prove a geometric analogue of the infinite-dimensional Bott periodicityintroduced by Higson, Kasparov and Trout [30]. This geometric analogue was essentially proved by thethird author [54].
Proposition 5.5.
For each d > , the compositions α ∗ ◦ β ∗ and ( α L ) ∗ ◦ ( β L ) ∗ are the identity maps.Proof. Following the arguments in Lemma 5.1, we can define the asymptotic morphism γ : S b ⊗ C ∗ u (( P d ( G m )) m ∈ I ) → S b ⊗ C ∗ u (( P d ( G m ) , K m ) m ∈ I ) by γ t ( g b ⊗ T ) ( m ) ( x, y ) = g t ( X b ⊗ b ⊗ B m ,t ( x )) b ⊗ T ( m ) ( x, y )) for each m , ( x, y ) ∈ X md × X md , where g ∈ C ( R ) is a continuously differentiable function with compactsupport and T = ( T ( m ) ) m ∈ C u [( P d ( G m )) m ∈ I ] . Let g ∈ S = C ( R ) be a continuously differentiablefunction with compact support, and T = ( T ( m ) ) m ∈ C u ( P d ( G m )) m ∈ I ] . It is obvious that β t ( g b ⊗ T ) ∈ C ∗ u,alg (( P d ( G m ) , A ( H m )) m ∈ I ) for all t ∈ [1 , ∞ ) . Accordingly, the asymptotic morphism ( γ t ) t ∈ [1 , ∞ ) is welldefined. 48or each m and x ∈ X md t ≥ , we define η ( x ) : A ( W ( x )) b ⊗ K → H m b ⊗ K by ( η t ( x ))(( g b ⊗ h ) b ⊗ k ) = g t ( X b ⊗ b ⊗ B m ,t ( x )) M h t b ⊗ k for all g ∈ S , h ∈ C ( W ,t ) , k ∈ K . Let γ ′ be the asymptotic morphism γ ′ : S b ⊗ C ∗ u (( P d ( G m )) m ∈ I ) → S b ⊗ C ∗ u (( P d ( G m ) , K m ) m ∈ I ) by γ ′ t ( g b ⊗ T ) ( m ) ( x, y ) = ( η t ( x ))( β W ( x ) ( x )( g )) b ⊗ T ( m ) ( x, y )) for each g ∈ C ( R ) a continuously differentiable function with compact support, T = ( T ( m ) ) m ∈ C u [( P d ( G m )) m ∈ I ] , t ≥ , m ∈ I , ( x, y ) ∈ X md × X md , where β W ( x ) ( x ) : S → A ( W ( x )) is the ∗ -homomorphism induced by the map g g ( X b ⊗ b ⊗ C W ( x ) ,x ) . It follows from Proposition 4.2 in [30]that γ is asymptotic equivalent to γ ′ . Therefore, γ ∗ = γ ′∗ .For every m ∈ I , t ≥ and x ∈ X md , let U mx : H m → H m be the unitary operator acting on the Hilbertspace H m = lim V a ⊂ H m L ( V a , Cllif C ( V a )) defined by ( U x ξ )( v ) = ξ ( v − f m ( π ( x ))) , for all ξ ∈ H m and all v ∈ V a . Then we have U − x g t ( X b ⊗ b ⊗ B m ,t ( x )) U x = ( η t ( x ))( β W ( x ) ( x )( g )) for all m ∈ I and x ∈ X md , g ∈ S .For each s ∈ [0 , , let R ( s ) = cos ( π s ) sin ( π s ) − sin ( π s ) cos ( π s ) ! . For each m ∈ I , x ∈ X md , s ∈ [0 , and t ≥ , we define a unitary operator U x,s acting on ℓ ( X md ) b ⊗H m b ⊗ H ⊕ ℓ ( X md ) b ⊗H m b ⊗ H by: U x,s = R ( s ) b ⊗ U x b ⊗ ! R ( s ) − , For each s ∈ [0 , , we define an asymptotic morphism γ ( s ) : S b ⊗ C ∗ U (( P d ( G m )) m ∈ I ) → S b ⊗ C ∗ U (( P d ( G m )) m ∈ I ) b ⊗ M ( C ) by ( γ t ( s )( g b ⊗ T )) ( m ) ( x, y ) = U mx,s γ ′ t ( g b ⊗ T ( m ) )( x, y ) 00 0 ! U − x,s , for each g ∈ S , T = ( T ( m ) ) m ∈ C u [( P d ( G m )) m ∈ I ] , t ≥ , s ∈ [0 , and ( x, y ) ∈ X md × X md , where thealgebra of all complex -by- matrices M ( C ) is endowed with the trivial grading.49ince f m ( π ( x )) ∈ W ( x ) , for each m ∈ I and each x ∈ X md , we can verify that γ t (0) − α t ( β t ( g b ⊗ T )) 00 0 ! → in norm as t → ∞ for every compactly supported function g ∈ S with continuous derivative and T ∈ C u [( P d ( G m )) m ∈ I ] . It follows that γ (0) ∗ = α ∗ ◦ β ∗ . On the other hand, γ (1) = γ ′ t
00 0 ! . We have that γ (1) ∗ = γ ′∗ , As a result, we have that α ∗ ◦ β ∗ = γ ∗ . Replacing B m ,t ( x ) with s − B ,t ( x ) in thedefinition of γ , we obtain a homotopy between γ and the ∗ -homomorphism g b ⊗ ( T ( m ) ) m → g b ⊗ ( P ( m ) b ⊗ T ( m ) ) ,where P ( m ) is the rank-one projection onto the one-dimensional kernel of B m ,t ( e ) , where e ∈ G m is theidentity element of G m . Since f m ( π ( e )) = 0 ∈ H m , P m does not depend on x . It follows that γ ∗ is theidentity homomorphism. Therefore, α ∗ ◦ β ∗ is the identity. Proof of Theorem 1.3:
Consider the following commutative diagram: lim d →∞ K ∗ ( S b ⊗ C ∗ u,L (( P d ( G m )) m ∈ I )) e ∗ / / ( β L ) ∗ (cid:15) (cid:15) lim d →∞ K ∗ ( S b ⊗ C ∗ u (( P d ( G m )) m ∈ I ) β ∗ (cid:15) (cid:15) lim d →∞ K ∗ ( C ∗ u,L (( P d ( G m ) , A ( H m )) m ∈ I )) ( α L ) ∗ (cid:15) (cid:15) e ∗ ∼ = / / lim d →∞ K ∗ ( C ∗ u (( P d ( G m ) , A ( H m )) m ∈ I )) α ∗ (cid:15) (cid:15) lim d →∞ K ∗ ( S b ⊗ C ∗ u,L (( P d ( G m )) m ∈ I ) e ∗ / / lim d →∞ K ∗ ( S b ⊗ C ∗ u ( P d ( G m )) m ∈ I )) For any element x ∈ lim d →∞ K ∗ ( S b ⊗ C ∗ u,L (( P d ( G m )) m ∈ I )) with e ∗ ( x ) = 0 , we have that e ∗ ◦ ( β L ) ∗ ( x ) = α ∗ ◦ e ∗ ( x ) = 0 . Since the map e ∗ is an isomorphism by Theorem 4.9, we have that ( β L ) ∗ ( x ) = 0 . In addition, id = ( α L ) ∗ ◦ ( β L ) ∗ by Theorem 5.5, we have that x = ( α L ) ∗ ◦ ( β L ) ∗ ( x ) = 0 . Thus, the map e ∗ : lim d →∞ K ∗ ( S b ⊗ C ∗ u,L (( P d ( G m )) m ∈ I )) → lim d →∞ K ∗ ( S b ⊗ C ∗ u (( P d ( G m )) m ∈ I )) is injective.For any element y ∈ lim d →∞ K ∗ ( S b ⊗ C ∗ u (( P d ( G m )) m ∈ I )) , since the middle horizontal map is an isomor-phism, we can find an element z ∈ lim d →∞ K ∗ ( C ∗ u,L ( P d ( G m ) , A ( H m )) m ∈ I )) , such that e ∗ ( z ) = β ∗ ( y ) .
50y Theorem 4.9 and commutativity of the above diagram, we have that y = α ∗ ◦ β ∗ ( y ) = α ∗ ◦ e ∗ ( z ) = e ∗ ◦ ( β L ) ∗ ( z ) . Thus, the map e ∗ : lim −→ d →∞ K ∗ ( S b ⊗ C ∗ u,L (( P d ( G m )) m ∈ I )) → lim d →∞ K ∗ ( S b ⊗ C ∗ u (( P d ( G m )) m ∈ I )) is surjective. Acknowledgement
The authors would like to acknowledge Rufus Willett for useful conversations.
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