The coarse Baum–Connes conjecture for spaces which admit a ... · Using comparison theorems in Riemannian geometry and the negative curvature property, it is not difﬁcult to verify

Digital Object Identifier (DOI) 10.1007/s002229900032Invent. math. 139, 201–240 (2000)

Springer-Verlag 2000

The coarse Baum–Connes conjecture forspaces which admit a uniform embeddinginto Hilbert space?

Guoliang Yu

Department of Mathematics, University of Colorado, Boulder, CO 80309–0395, USA(e-mail:[email protected] )

Oblatum 3-VIII-1997 & 23-VI-1999 / Published online: 22 September 1999

1 Introduction

Let Γ be a metric space; letH be a separable and infinite-dimensionalHilbert space. A mapf : Γ → H is said to be a uniform embedding [10]if there exist non-decreasing functionsρ1 andρ2 from R+ = [0,∞) to Rsuch that(1) ρ1(d(x, y)) ≤ ‖ f(x)− f(y)‖ ≤ ρ2(d(x, y)) for all x, y ∈ Γ;(2) limr→+∞ ρi (r) = +∞ for i = 1,2.

The main purpose of this paper is to prove the following result:

Theorem 1.1. Let Γ be a discrete metric space with bounded geometry. IfΓ admits a uniform embedding into Hilbert space, then the coarse Baum–Connes conjecture holds forΓ.

Recall that a discrete metric spaceΓ is said to have bounded geometryif ∀r > 0, ∃N(r) > 0 such that the number of elements inB(x, r) is at mostN(r) for all x ∈ Γ, whereB(x, r) = {y ∈ Γ : d(y, x) ≤ r }. Every finitelygenerated group, as a metric space with a word-length metric, has boundedgeometry.

Corollary 1.2. Let Γ be a finitely generated group. IfΓ, as a metric spacewith a word-length metric, admits a uniform embedding into Hilbert space,and its classifying spaceBΓ has the homotopy type of a finiteCW complex,then the strong Novikov conjecture holds forΓ, i.e. the index map fromK∗(BΓ) to K∗(C∗r (Γ)) is injective.

Corollary 1.2 follows from Theorem 1.1 and the descent principle [23].By index theory, the strong Novikov conjecture implies the Novikov conjec-ture on the homotopy invariance of higher signatures (cf. [8] for an excellent

? Partially supported by the National Science Foundation

202 G. Yu

survey of the Novikov conjecture). The class of finitely generated groupswhich admit a uniform embedding into Hilbert space contains a subclass ofgroups, groups with property A, which includes Gromov’s word hyperbolicgroups, discrete subgroups of connected Lie groups and amenable groups,and is closed under semi-direct product (cf. Theorem 2.2 and Proposi-tion 2.6). In general, it is an open question if every finitely generated group(or every discrete metric space with bounded geometry) admits a uniformembedding into Hilbert space ([10], page 218, [11], page 67).

The following result follows from Theorem 1.1, Proposition 4.33 in [22],and the Lichnerowicz argument.

Corollary 1.3. Let M be a complete Riemannian manifold with boundedgeometry. IfM is uniformly contractible and admits a uniform embeddinginto Hilbert space, thenM can not have uniformly positive scalar curvature.

In general, Gromov conjectures that a uniformly contractible completeRiemannian manifold can not have uniformly positive scalar curvature [12].Recall that a Riemannian manifold is said to have bounded geometry if ithas bounded sectional curvature and positive injectivity radius, and a Rie-mannian manifold is said to be uniformly contractible if for everyr > 0,there existsR ≥ r such that every ball with radiusr can be contracted toa point in a ball with radiusR.Riemannian manifolds with property A admita uniform embedding into Hilbert space (cf. Theorem 2.7).

This paper is organized as follows. In Sect. 2, we introduce a geometricproperty (called property A) which guarantees the existence of a uniformembedding into Hilbert space. We also show that the class of groups withproperty A is closed under semi-direct product. In Sect. 3, we briefly recallthe coarse Baum–Connes conjecture. In Sect. 4, we recall the localizationalgebra and discuss its application to the coarse Baum–Connes conjecture.The coarse Baum–Connes conjecture holds if and only if the evaluationmap from the K-theory of localization algebra to the K-theory of Roealgebra is an isomorphism. In Sect. 5, we introduce certain twisted Roealgebras and twisted localization algebras for bounded geometry spaceswhich admit a uniform embedding into Hilbert space. In Sect. 6, we provethat the evaluation map from the K-theory of twisted localization algebrato the K-theory of twisted Roe algebra is an isomorphism for boundedgeometry spaces which admit a uniform embedding into Hilbert space.This establishes “the twisted coarse Baum–Connes conjecture” for boundedgeometry spaces which admit a uniform embedding into Hilbert space. InSect. 7, we establish a geometric analogue of Higson–Kasparov–Trout’sinfinite-dimensional Bott periodicity for bounded geometry spaces whichadmit a uniform embedding into Hilbert space. This geometric analogueof infinite-dimensional Bott periodicity is then used to reduce the coarseBaum–Connes conjecture to the twisted coarse Baum–Connes conjecturefor bounded geometry spaces which admit a uniform embedding into Hilbertspace.

The coarse Baum–Connes conjecture 203

This work is inspired by Gromov’s deep questions concerning uniformembedding into Hilbert space ([10], [11]) and by the remarkable work ofHigson and Kasparov on the Baum–Connes conjecture [14].

I thank the referees, Vincent Lafforgue and Guihua Gong for valuablesuggestions which led to this more readable revised version.

2 Uniform embeddings into Hilbert space and property A

In this section, we shall introduce the concept of property A for metricspaces. We prove that metric spaces with property A admit a uniformembedding into Hilbert space. This result is inspired by the Bekka–Cherix–Valette theorem which states that every amenable group admits a properand isometric action on Hilbert space [2]. The class of finitely generatedgroups with property A, as metric spaces with word-length metrics, includesword hyperbolic groups, discrete subgroups of connected Lie groups andamenable groups, and is closed under semi-direct product.

Definition 2.1. A discrete metric spaceΓ is said to have property A if forany r > 0, ε > 0, there exist a family of finite subsets{Aγ }γ∈Γ of Γ × N(N is the set of all natural numbers) such that

(1) (γ,1) ∈ Aγ for all γ ∈ Γ;

(2)#(Aγ−Aγ ′ )+#(Aγ ′−Aγ )

#(Aγ∩Aγ ′ )< ε for all γ andγ ′ ∈ Γ satisfyingd(γ, γ ′) ≤ r ,

where, for each finite setA, #A is the number of elements inA;(3) ∃R > 0 such that if(x,m) ∈ Aγ , (y,n) ∈ Aγ for someγ ∈ Γ, then

d(x, y) ≤ R.

Notice that property A is invariant under quasi-isometry. In the case ofa finitely generated group, property A does not depend on the choice of theword-length metric.

The following result is inspired by the Bekka–Cherix–Valette theoremwhich states that every amenable group admits a proper and isometric actionon Hilbert space [2].

Theorem 2.2. If a discrete metric spaceΓ has property A, thenΓ admitsa uniform embedding into Hilbert space.

Proof. Let

H =∞⊕

k=1

`2(Γ×N).

By the definition of property A, there exist a family of finite subsets{A(k)γ }γ∈Γsuch that

(1) (γ,1) ∈ A(k)γ for all γ ∈ Γ;

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(2) ∃Rk > 0 such that if(x,m) ∈ A(k)γ , (y,n) ∈ A(k)γ for somek andγ ∈ Γ,thend(x, y) ≤ Rk;

(3) ∥∥∥∥∥ χA(k)γ(#A(k)γ

)1/2 −χA(k)

γ ′(#A(k)γ ′

)1/2

∥∥∥∥∥l2(Γ×N)

<1

2k

for all γ, γ ′ ∈ Γ satisfyingd(γ, γ ′) ≤ k andk ∈ N, whereχA(k)γis the

characteristic function ofA(k)γ .

Fix γ0 ∈ Γ. Define f : Γ→ H by:

f(γ) =∞⊕

k=1

(χA(k)γ(

#A(k)γ)1/2 −

χA(k)γ0(#A(k)γ0

)1/2

).

One can easily check thatf is a uniform embedding. ut

Example 2.3. Let Γ be a finitely generated amenable group.∀ r > 0,0.5> ε > 0, there exists a finite subsetF of Γ such that

#(Fg− F)+ #(F − Fg)

#F< ε/10 if d(g,1) ≤ r,

whered is the word-length metric. SetAγ = {(x,1) ∈ Γ×N : x ∈ γF}. Itis easy to see that{Aγ }γ∈Γ satisfies the conditions of property A.

Hence amenable groups admit a uniform embedding into Hilbert space.This is also a consequence of the Bekka–Cherix–Valette Theorem men-tioned before (cf. [2]) since groups acting properly and isometrically onHilbert space admit a uniform embedding into Hilbert space. However, byExample 2.5 and Proposition 2.6 below, the class of groups admitting a uni-form embedding into Hilbert space is much larger than the class of groupsacting properly and isometrically on Hilbert space since infinite property Tgroups can not act properly and isometrically on Hilbert space.

Example 2.4.Let F2 be the free group of two generators. LetT be the treeassociated toF2, where the set of all vertices ofT is F2. EndowT withthe simplicial metric. Fix a geodesic rayγ0 on T. For eachg ∈ F2, thereexists a unique geodesic rayγg on T starting fromg such thatγg ∩ γ0 isa non-empty geodesic ray. For each natural numberN, define

Ag = {(x,1) ∈ F2 ×N, x ∈ γg,d(x, g) ≤ N}for all g ∈ F2.


Givenr > 0, ε > 0, it is not difficult to see that there exists a largeNsuch that{Ag}g∈F2 satisfies the conditions of property A.

Example 2.5. Let M be a compact Riemannian manifold with negativesectional curvature, letΓ = π1(M), the fundamental group ofM. Fixa point x0 ∈ M, the universal cover ofM, and a geodesic ray (with unitspeed)γ0 on M starting fromx0. For eachg ∈ Γ, let γg be the geodesic ray(with unit speed) onM starting fromgx0 such thatγg is asymptotic toγ0(cf. Proposition 1.7.3 in “Geometry of Nonpositively Curved Manifolds”by Patrick B. Eberlein, Chicago Lecture Notes in Mathematics, 1996). Foreach natural numberN, we define

Bg = {x ∈ M,d(x, γg(t)) < 1 for somet ∈ [0, N]}.Let F be a fundamental domain inM. For eachδ0 > 0, δ1 > 0, there existsa natural numberk such that if volume(g′F ∩ Bg) ≥ δ0 for someg, g′ ∈ Γ,then there exists an integer`g′,g satisfying∣∣∣∣volume(g′F ∩ Bg)− `g′,g

k

∣∣∣∣ < δ1.

We define

Ag = {(g′,n) : volume(g′F ∩ Bg) ≥ δ0, 1≤ n ≤ `g′,g} ⊆ Γ×N.Using comparison theorems in Riemannian geometry and the negative

curvature property, it is not difficult to verify that ifN is large enough,δ0 andδ1 are small enough, then{Ag}g∈Γ satisfies the conditions of prop-erty A.

More generally, one can show that word hyperbolic groups have prop-erty A by similar argument using constructions from [25].

Proposition 2.6. Let Γ1 and Γ2 be two finitely generated groups withproperty A (as metric spaces with word-length metrics). IfΓ1 acts onΓ2 byautomorphisms, then the semi-direct productΓ2o Γ1 has property A.

Proof. Given r > 0, ε > 0. For eachr1 > 0, ε1 > 0, r2 > 0, ε2 > 0,let {A(1)γ }γ∈Γ1 and {A(2)γ }γ∈Γ2 be respectively as in the definition of prop-erty A for Γ1 with respect tor1 andε1, and forΓ2 with respect tor2 andε2.Set

f(x · y, (γ1 · γ2, (m,n))) = χA(1)x(γ1,m)χA(2)

γ−11 xyx−1γ1

(γ2,n),

wherem,n ∈ N, x ∈ Γ1, y ∈ Γ2, x · y ∈ Γ2 o Γ1, γ1 ∈ Γ1, γ2 ∈ Γ2,γ1 ·γ2 ∈ Γ2oΓ1, and, for each setA, χA is the characteristic function ofA.

Let h be a bijective map fromN to N× N such thath(1) = (1,1). Foreachγ = x · y ∈ Γ2o Γ1, we define

Aγ = {(γ1 · γ2,n) ∈ (Γ2o Γ1)× N : f(x · y, (γ1 · γ2,h(n))) 6= 0}.

206 G. Yu

Now it is straightforward to verify that ifri andεi (i = 1,2) are chosenappropriately, then{Aγ }γ∈Γ2oΓ1 satisfies the conditions of property A.ut

In a recent paper [17], Higson and Roe proved that a finitely generatedgroup with a word-length metric has property A if and only if the groupacts amenably on some compact Hausdorff space. It follows that discretesubgroups of connected Lie groups have property A [17].

A general metric spaceX is said to have property A if there existsa discrete subspaceΓ of X such that (1) there existsc > 0 for whichd(x,Γ) ≤ c for all x ∈ X; (2) Γ has property A.

The following result follows from Theorem 2.2.

Theorem 2.7. If a metric spaceX has property A, thenX admits a uniformembedding into Hilbert space.

In general, it is an open question if every discrete metric space withbounded geometry admits a uniform embedding into Hilbert space [10].However it is not difficult to show that every separable metric space admitsa uniform embedding into a separable Banach space.

3 The coarse Baum–Connes conjecture

In this section, we shall briefly recall the coarse Baum–Connes conjectureand its applications.

Let M be a proper metric space (a metric space is called proper if everyclosed ball is compact). LetHM be a separable Hilbert space equippedwith a faithful and non-degenerate∗-representation ofC0(M) whose rangecontains no nonzero compact operator, whereC0(M) is the algebra of allcomplex-valued continuous functions onM which vanish at infinity.

Definition 3.1. (1) The support of a bounded linear operatorT : HM → HMis the complement of the set of points(m,m′) ∈ M × M for which thereexistg andg′ in C0(M) such that

g′Tg= 0, g(m) 6= 0 and g′(m′) 6= 0;(2) A bounded operatorT : HM → HM has finite propagation if

sup{d(m,m′) : (m,m′) ∈ supp(T )} <∞;(3) A bounded operatorT : HM → HM is locally compact if the operatorsgT andTg are compact for allg ∈ C0(M).

Definition 3.2. The Roe algebraC∗(M) is the operator norm closure of the∗-algebra of all locally compact, finite propagation operators acting onHM.

C∗(M) is independent of the choice ofHM (up to∗-isomorphism). Inparticular, we can chooseHM to bel2(Z)⊗H0, whereZ is a countable densesubset ofM, H0 is a separable and infinite-dimensional Hilbert space, and


C0(M) acts onHM by: g(h1 ⊗ h2) = gh1 ⊗ h2 for all g ∈ C0(M),h1 ∈l2(Z),h2 ∈ H0 (g acts onl2(Z) by pointwise multiplication). Such a choiceof HM will be useful later on in this paper.

Let Γ be a locally finite discrete metric space (a metric space is calledlocally finite if every ball contains finitely many elements).

Definition 3.3. For eachd ≥ 0, the Rips complexPd(Γ) is the sim-plicial polyhedron where the set of all vertices isΓ, and a finite subset{γ0, . . . , γn} ⊆ Γ spans a simplex iffd(γi , γ j ) ≤ d for all 0≤ i, j ≤ n.

EndowPd(Γ) with the spherical metric. Recall that the spherical metricis the maximal metric whose restriction to each simplex{∑n

i=0 tiγi : ti ≥ 0,∑ni=0 ti = 1} is the metric obtained by identifying the simplex withSn+

through the map:

n∑i=0

tiγi → t0√∑n

i=0 t2i

, · · · , tn√∑ni=0 t2

i

,whereSn+ = {(x0, · · · , xn) ∈ Rn+1 : xi ≥ 0,

∑ni=0 x2

i = 1} is endowedwith the standard Riemannian metric. The distance of a pair of points indifferent connected components ofPd(Γ) is defined to be infinity. Use ofthe spherical metric is necessary in Sect. 4 to avoid certain pathologicalphenomena whend goes to infinity (for the simplicial metric, the distancebetween the barycenters of two faces of ann-simplex goes to zero asn goesto infinity).

Conjecture 3.4 (The Coarse Baum–Connes Conjecture).If Γ is a discretemetric space with bounded geometry, then the index map fromlimd→∞ K∗(Pd(Γ)) to limd→∞ K∗(C∗(Pd(Γ))) is an isomorphism, whereK∗(Pd(Γ)) = KK∗(C0(Pd(Γ)),C) is the locally finiteK -homology groupof Pd(Γ).

It should be pointed out that limd→∞ K∗(C∗(Pd(Γ))) is isomorphic toK∗(C∗(Γ)).

We remark that Conjecture 3.4 is false if the bounded geometry condi-tion is dropped [28]. In the case of a finitely generated group, the descentprinciple states that the coarse Baum–Connes conjecture for the group asa metric space with a word-length metric implies the strong Novikov con-jecture if the group has a finiteCW complex as its classifying space [23].If Γ is a discrete subspace of a complete Riemannian manifoldM suchthat there existsc > 0 satisfyingd(x,Γ) ≤ c for all x ∈ M, then thecoarse Baum–Connes conjecture forΓ provides a way of computing higherindices of elliptic operators onM. In particular, by the Lichnerowicz ar-gument, the coarse Baum–Connes conjecture implies Gromov’s conjecturestating that a uniformly contractible complete Riemannian manifold can nothave uniformly positive scalar curvature [22]. It also implies the zero-in-thespectrum conjecture tating that the spectrum of the Laplacian acting on the

208 G. Yu

space of allL2-forms on a uniformly contractible complete Riemannianmanifold contains zero [22].

4 The localization algebra

The localization algebra introduced in [27] plays an important role in theproof of our main result. For the convenience of the readers, we shallbriefly recall its definition and its relation with K-homology. The conceptof localization algebra is motivated by the heat kernel approach to indextheory of elliptic operators.

Let M be a proper metric space.

Definition 4.1. The localization algebraC∗L(M) is the norm-closure of thealgebra of all uniformly bounded and uniformly norm-continuous functionsf : [0,∞)→ C∗(M) such that

sup{d(m,m′) : (m,m′) ∈ supp( f(t))} → 0

ast →∞.

There exists a local index map (cf. [27]):

indL : K∗(M)→ K∗(C∗L(M)).

The evaluation homomorphisme from C∗L(M) to C∗(M) is defined by:

e( f ) = f(0).

If Γ is a locally finite discrete metric space, we have the following commut-ing diagram:

limd→∞

K∗(Pd(Γ))

indL ↙ ↘ ind

limd→∞

K∗(C∗L(Pd(Γ)))e∗−→ lim

d→∞K∗(C∗(Pd(Γ))).

Theorem 4.2 ([27]). If Γ has bounded geometry, then the local index map

indL : K∗(Pd(Γ))→ K∗(C∗L(Pd(Γ))),

is an isomorphism for everyd ≥ 0.

In general, the local index map indL : K∗(X)→ K∗(C∗L(X)), is an iso-morphism for every finite-dimensional simplicial complexX endowed withthe spherical metric. For the convenience of readers, we give an overviewof the proof of this fact given in [27]. LetX1 and X2 be two simplicialsubcomplexes ofX such thatX1 andX2 are endowed with metrics inheritedfrom the metric ofX. The local nature of the localization algebra can be


used to prove a Mayer–Vietoris sequence for the K-groups of the localiza-tion algebras forX1∪X2, X1, X2 andX1∩X2 (cf. Proposition 3.11 of [27]),and certain (strong) Lipschitz homotopy invariance of the K-theory of thelocalization algebra (cf. Lemma 6.5 in this paper, Proposition 3.7 of [27]and also page 332 of [28] for a notational correction). Now the fact that thelocal index map indL : K∗(X)→ K∗(C∗L(X)), is an isomorphism followsfrom an induction argument on the dimension of skeletons ofX using theMayer–Vietoris sequence and the (strong) Lipschitz homotopy invariancefor K-theory of localization algebras (cf. page 315–316 of [27]).

Theorem 4.2 implies that in order to prove the coarse Baum–Connesconjecture, it is enough to show that

e∗ : limd→∞

K∗(C∗L(Pd(Γ)))→ limd→∞

K∗(C∗(Pd(Γ)))

is an isomorphism.

5 Twisted Roe algebras and twisted localization algebras

In this section, we shall introduce certain twisted Roe algebras and twistedlocalization algebras for bounded geometry spaces which admit a uniformembedding into Hilbert space. These algebras play important roles in theproof of our main result. I thank one of the referees and Vincent Lafforguefor suggesting the current more friendly definition of twisted Roe algebras.

As suggested by one of the referees, we shall first introduce the twistedRoe algebras and the twisted localization algebras in the finite-dimensionalcase to motivate the infinite-dimensional construction. The readers are en-couraged to go through the paper in the finite-dimensional case first sincethe finite-dimensional case is considerably simpler.

Let H be a finite-dimensional Euclidean space. Denote byC(H ) theZ/2-gradedC∗-algebra of continuous functions fromH into the complexifiedClifford algebra of H which vanish at infinity. LetS = C0(R), gradedaccording to even and odd functions. We defineA(H ) to be the gradedtensor product ofS with C(H ).

LetΓ be a discrete metric space with bounded geometry. Letf : Γ→ H,be a uniform embedding. For eachd ≥ 0, we extendf to Pd(Γ) by:

f(x) =∑

cγ f(γ),

for all x =∑ cγ γ ∈ Pd(Γ), whereγ ∈ Γ, cγ > 0 and∑

cγ = 1.Choose a countable dense subsetΓd of Pd(Γ) for eachd > 0 such that

Γd1 ⊆ Γd2 if d2 ≥ d1.Let C∗alg(Pd(Γ),A) be the set of all functionsT onΓd × Γd such that

(1) T(x, y) ∈ A(H )⊗K for all x, y ∈ Γd, where K is the algebra ofcompact operators;

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(2) ∃M > 0 andL > 0 such that‖T(x, y)‖ ≤ M for all x, y ∈ Γd, and foreachy ∈ Γd, #{x : T(x, y) 6= 0} ≤ L, #{x : T(y, x) 6= 0} ≤ L;

(3) ∃r1 > 0 andr2 > 0 such that(a) if d(x, y) > r1, thenT(x, y) = 0;(b) support(T(x, y)) ⊆ B( f(x), r2) for all x, y ∈ Γd, where

support(T(x, y)) is defined to be the support{(s,h) ∈ R × H :(T(x, y))(s,h) 6= 0} of the function T(x, y) : R × H →Cliff (H )⊗K, (Cliff (H ) is the complexified Clifford algebra ofH),f is the uniform embedding, andB( f(x), r2) = {(s,h) ∈ R× H :s2+ ‖h− f(x)‖2 < r 2

2};(4) ∃c > 0 such thatDY(T(x, y)) exists inA(H )⊗K , and‖DY(T(x, y))‖≤ c for all x, y ∈ Γd and Y = (s,h) ∈ R × H satisfying‖Y‖ =√

s2+ ‖h‖2 ≤ 1, whereDY(T(x, y)) is the derivative of the functionT(x, y) : R× H → Cliff (H )⊗K , in the direction ofY.

We define a product structure onC∗alg(Pd(Γ),A) by:

(T1T2)(x, y) =∑z∈Γd

T1(x, z)T2(z, y).

Let

E ={∑

x∈Γd

ax[x] : ax ∈ A(H )⊗K,∑x∈Γd

a∗xax converges in norm

}.

E is a Hilbert module overA(H )⊗K :⟨∑x∈Γd

ax[x],∑x∈Γd

bx[x]⟩=∑x∈Γd

a∗xbx,

(∑x∈Γd

ax[x])

a =∑x∈Γd

axa[x]

for all a ∈ A(H )⊗K .C∗alg(Pd(Γ),A) acts onE by:

T

(∑x∈Γd

ax[x])=∑y∈Γd

(∑x∈Γd

T(y, x)ax

)[y],

whereT ∈ C∗alg(Pd(Γ),A),∑

ax[x] ∈ E. One can easily verify thatT isa module homomorphism which has an adjoint module homomorphism.

Definition 5.1 (The finite-dimensional case).C∗(Pd(Γ),A) is the opera-tor norm closure ofC∗alg(Pd(Γ),A) in B(E), theC∗-algebra of all module


homomorphisms fromE to E for which there is an adjoint module homo-morphism.

Let C∗L,alg(Pd(Γ),A) be the set of all uniformly norm-continuous anduniformly bounded functionsg : R+ = [0,∞) → C∗alg(Pd(Γ),A) suchthat

(1) ∃ a bounded functionr(t) : R+ → R+ such that limt→∞ r(t) = 0and if d(x, y) > r(t), then(g(t))(x, y) = 0;

(2) ∃R> 0 such that support((g(t))(x, y)) ⊆ B( f(x), R) for all t ∈ R+,x, y ∈ Γd, where f is the uniform embedding, andB( f(x), R) = {(s,h) ∈R× H : s2+ ‖h− f(x)‖2 < R2};

(3) ∃C > 0 such that‖DY((g(t))(x, y))‖ ≤ C for all t ∈ R+, x, y ∈ ΓdandY ∈ R× H satisfying‖Y‖ ≤ 1.

Definition 5.2 (The finite-dimensional case).C∗L(Pd(Γ),A) is the normclosure ofC∗L,alg(Pd(Γ),A), whereC∗L,alg(Pd(Γ),A) is endowed with thenorm:

‖g‖ = supt∈R+‖g(t)‖C∗(Pd(Γ),A).

Next we shall introduce the twisted Roe algebras and twisted localiza-tion algebras in the infinite-dimensional case. We shall first recall an algebraassociated to an infinite-dimensional Euclidean space introduced by Hig-son, Kasparov and Trout [15]. LetV be a countably infinite-dimensionalEuclidean space. Denote byVa,Vb, so on, the finite-dimensional, affinesubspaces ofV. Denote byV0

a the finite-dimensional linear subspace ofVconsisting of differences of elements inVa. Let C(Va) be theZ/2-gradedC∗-algebra of continuous functions fromVa into the complexified Cliffordalgebra ofV0

a which vanish at infinity. LetA(Va) be the graded tensorproduct ofS with C(Va).

If Va ⊂ Vb, we have a decomposition:

Vb = V0ba+ Va,

whereV0ba is the orthogonal complement ofV0

a in V0b . For eachvb ∈ Vb, we

have a corresponding decomposition:vb = vba+ va, wherevba ∈ V0ba and

va ∈ Va. Every functionh on Va can be extended to a functionh on Vb bythe formula:h(vb) = h(va).

Definition 5.3. (1) If Va ⊂ Vb, denote byCba the Clifford algebra-valuedfunction onVb which mapsvb to vba ∈ V0

ba ⊂ Cliff (V0b ). Define a homo-

morphismβba : A(Va)→ A(Vb), by

βba(g⊗h) = g(X⊗1+ 1⊗Cba)(1⊗h)

for all g ∈ S andh ∈ C(Va), where X is the function of multiplicationby x onR, considered as a degree one and unbounded multiplier ofS, andg(X⊗1+ 1⊗Cba) is defined by functional calculus.

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(2) We define aC∗-algebraA(V) by:

A(V) = lim−→A(Va),

where the direct limit is over the directed set of all finite-dimensional affinesubspacesVa ⊂ V, using the homomorphismβba in (1).

Let H be a separable and infinite-dimensional Hilbert space. LetΓ bea discrete metric space with bounded geometry. Assume thatΓ admitsa uniform embeddingf : Γ→ H. For eachn ∈ N, x ∈ Γ, defineWn(x) tobe the finite-dimensional Euclidean subspace ofH spanned by{ f(y) ∈ H :d(y, x) ≤ n2}. Let W(x) =⋃n∈N Wn(x).

We have the following:(1) Wn(x) ⊆ Wn+1(x) for all n ∈ N, x ∈ Γ;(2) for eachr > 0, there existsN > 0 such thatWn(x) ⊂ Wn+1(y) for

all n ≥ N, x andy in Γ satisfyingd(x, y) ≤ r .(3) ∀ n ∈ N, ∃ dn > 0 such that dimWn(x) ≤ dn for all x ∈ Γ;(4) there exists an Euclidean subspaceV of H such thatW(x) = V for

all x ∈ Γ.Note that (3) follows from the bounded geometry property ofΓ. We

remark that(2) will be useful in the proof of Lemma 7.2.For everyx ∈ Pd(Γ), write x = ∑

γ∈Γcγ>0

cγ γ . Define Wn(x) to be the

Euclidean subspace ofH spanned byWn(γ) for all γ such thatcγ > 0. Weextend f to Pd(Γ) by:

f(x) =∑

cγ f(γ).

Without loss of generality we assume thatV is dense inH (otherwisewe can takeH to be the norm closure ofV).

Throughout the rest of this paper,R+ × H is endowed with the weakesttopology for which the projection toH is weakly continuous and the functiont2+‖h‖2 is continuous((t,h) ∈ R+ × H ) (cf. the proof of Proposition 4.2in [14]). Note that, for allx ∈ H andr > 0, B(x, r) = {(t,h) ∈ R+ × H :t2+||h−x||2 < r 2} is an open subset ofR+×H. For each finite-dimensionalaffine subspaceVa of V, the center ofA(Va) containsC0(R+ × Va). IfVa ⊂ Vb, thenβba takesC0(R+ × Va) into C0(R+ × Vb), whereβba is asin Definition 5.3. TheC∗-subalgebra lim−→C0(R+ × Va) is ∗-isomorphic to

C0(R+ × H ), the algebra of all continuous functions onR+ × H whichvanish at infinity, where the direct limit is over the directed set of all finite-dimensional affine subspacesVa ⊂ V using the homomorphismβba, andR+ × H is the locally compact space with the topology given as above.Hence the center ofA(V) containsC0(R+×H ). The support of an elementa ∈ A(V) is defined to be the complement (inR+×H) of all (t,h) for whichthere existsg ∈ C0(R+×H ) such thatag= 0 andg(t,h) 6= 0.C0(R+×H )acts onA(V)⊗K by: g(a⊗k) = ga⊗k for all g ∈ C0(R+×H ), a ∈ A(V),


k ∈ K . Hence we can define the support of an element inA(V)⊗K ina similar way.

Let C∗alg(Pd(Γ),A) be the set of all functionsT onΓd × Γd such that

(1) ∃an integerN such thatT(x, y) ∈ (βN(x))(A(WN(x))⊗K) ⊆ A(V)⊗Kfor all x, y ∈ Γd, whereβN(x) : A(WN(x))⊗K → A(V)⊗K , is the∗-homomorphism associated to the inclusion ofWN(x) into V as inDefinition 5.3, andK is the algebra of compact operators;

(2) ∃M > 0 andL > 0 such that‖T(x, y)‖ ≤ M for all x, y ∈ Γd, and foreachy ∈ Γd, #{x : T(x, y) 6= 0} ≤ L, #{x : T(y, x) 6= 0} ≤ L;

(3) ∃r1 > 0 andr2 > 0 such that(a) if d(x, y) > r1, thenT(x, y) = 0;(b) support(T(x, y)) ⊆ B( f(x), r2) for all x, y ∈ Γd, where f is the

uniform embedding, andB( f(x), r2) = {(s,h) ∈ R+ × H : s2 +‖h− f(x)‖2 < r 2

2};(4) ∃c > 0 such that DY(T1(x, y)) exists in A(WN(x))⊗K , and‖DY(T1(x, y))‖ ≤ c for all x, y ∈ Γd andY = (s,h) ∈ R×WN(x) satis-fying ‖Y‖ = √s2+ ‖h‖2 ≤ 1, where(βN(x))(T1(x, y)) = T(x, y), andDY(T1(x, y)) is the derivative of the functionT1(x, y) : R×WN(x)→Cliff (WN(x))⊗K , in the direction ofY.

It will become clear in the proof of Lemma 7.2 why we require con-dition (4) in the above definition. Note also that condition (4) is preservedunder the∗-homomorphismβN′N(x) : A(WN(x))⊗K → A(WN′(x))⊗K ,associated to the inclusion ofWN(x) into WN′(x) for any N′ > N.

We define a product structure onC∗alg(Pd(Γ),A) by:

(T1T2)(x, y) =∑z∈Γd

T1(x, z)T2(z, y).

Let

E ={∑

x∈Γd

ax[x] : ax ∈ A(V)⊗K,∑x∈Γd

a∗xax converges in norm

}.

E is a Hilbert module overA(V)⊗K :⟨∑x∈Γd

ax[x],∑x∈Γd

bx[x]⟩=∑x∈Γd

a∗xbx,

(∑x∈Γd

ax[x])

a =∑x∈Γd

axa[x]

for all a ∈ A(V)⊗K .

214 G. Yu

C∗alg(Pd(Γ),A) acts onE by:

T

(∑x∈Γd

ax[x])=∑y∈Γd

(∑x∈Γd

T(y, x)ax

)[y],

whereT ∈ C∗alg(Pd(Γ),A),∑

ax[x] ∈ E. One can easily verify thatT isa module homomorphism which has an adjoint module homomorphism.

Definition 5.4. C∗(Pd(Γ),A) is the operator norm closure ofC∗alg(Pd(Γ),A)in B(E), the C∗-algebra of all module homomorphisms fromE to E forwhich there is an adjoint module homomorphism.

Let C∗L,alg(Pd(Γ),A) be the set of all uniformly norm-continuous anduniformly bounded functionsg : R+ → C∗alg(Pd(Γ),A) such that

(1) ∃N such that(g(t))(x, y) ∈ (βN(x))(A(WN(x))⊗K) ⊆ A(V)⊗Kfor all t ∈ R+, x, y ∈ Γd;

(2) ∃ a bounded functionr(t) : R+ → R+ such that limt→∞ r(t) = 0and if d(x, y) > r(t), then(g(t))(x, y) = 0;

(3) ∃R> 0 such that support((g(t))(x, y)) ⊆ B( f(x), R) for all t ∈ R+,x, y ∈ Γd, where f is the uniform embedding, andB( f(x), R) = {(s,h) ∈R+ × H : s2+ ‖h− f(x)‖2 < R2};

(4) ∃C > 0 such that‖DY((g1(t))(x, y))‖ ≤ C for all t ∈ R+, x, y ∈ ΓdandY ∈ R ×WN(x) satisfying‖Y‖ ≤ 1, where(βN(x))((g1(t))(x, y)) =(g(t))(x, y).

Definition 5.5. C∗L(Pd(Γ),A) is the norm closure ofC∗L,alg(Pd(Γ),A),whereC∗L,alg(Pd(Γ),A) is endowed with the norm:

‖g‖ = supt∈R+‖g(t)‖C∗(Pd(Γ),A).

6 K-Theory of twisted Roe algebras and twisted localization algebras

In this section, we shall study the K-theory of twisted Roe algebras andtwisted localization algebras. In particular, we show that the evaluationmap from the K-theory of twisted localization algebra is isomorphic to theK-theory of twisted Roe algebra for bounded geometry spaces which ad-mit a uniform embedding into Hilbert space. This establishes “the twistedcoarse Baum–Connes conjecture” for bounded geometry spaces which ad-mit a uniform embedding into Hilbert space.

Definition 6.1. (1) The support of an elementT in C∗alg(Pd(Γ),A) is definedto be

{(x, y,u) ∈ Γd × Γd × (R+ × H ) : u ∈ support(T(x, y))},whereu = (t,h) ∈ R+ × H;


(2) The support of an elementg in C∗L,alg(Pd(Γ),A) is defined to be⋃t∈R+

support(g(t)).

Let O be an open subset ofR+ × H. DefineC∗alg(Pd(Γ),A)O to be thesubalgebra ofC∗alg(Pd(Γ),A) consisting of all elements whose supports arecontained inΓd × Γd × O. DefineC∗(Pd(Γ),A)O to be the norm closureof C∗alg(Pd(Γ),A)O. We can similarly defineC∗L(Pd(Γ),A)O.

Lemma 6.2. Let O and O′ be open subsets ofR+ × H. If O ⊆ O′,thenC∗(Pd(Γ),A)O andC∗L(Pd(Γ),A)O are respectively closed, two sidedideals ofC∗(Pd(Γ),A)O′ andC∗L(Pd(Γ),A)O′.

The proof of Lemma 6.2 is straightforward and is therefore omitted.

Lemma 6.3. Let B(x, r) = {(t,h) ∈ R+ × H : t2 + ‖h − x‖2 < r 2}for each r > 0 and x ∈ H; let Xi, j and X′i, j be subsets ofΓ for all

1 ≤ i ≤ i0 and 1 ≤ j ≤ j0. If Or, j = ∩i0i=1(∪γ∈Xi, j B( f(γ), r)) and

O′r, j = ∩i0i=1(∪γ∈X′i, j B( f(γ), r)) for each1 ≤ j ≤ j0 ( f is the uniform

embedding),Or = ∪ j0j=1Or, j and O′r = ∪ j0

j=1O′r, j , then for eachr0 > 0 wehave

limr<r0,r→r0

C∗(Pd(Γ),A)Or + limr<r0,r→r0

C∗(Pd(Γ),A)O′r(1)

= limr<r0,r→r0

C∗(Pd(Γ),A)Or∪O′r ,

limr<r0,r→r0

C∗L(Pd(Γ),A)Or + limr<r0,r→r0

C∗L(Pd(Γ),A)O′r

= limr<r0,r→r0

C∗L(Pd(Γ),A)Or∪O′r ;lim

r<r0,r→r0(C∗(Pd(Γ),A)Or ∩ C∗(Pd(Γ),A)O′r )(2)

= limr<r0,r→r0

C∗(Pd(Γ),A)Or∩O′r ,

limr<r0,r→r0

(C∗L(Pd(Γ),A)Or ∩ C∗L(Pd(Γ),A)O′r )

= limr<r0,r→r0

C∗L(Pd(Γ),A)Or∩O′r .

Proof. We shall only prove the first identity. The rest of the identities canbe proved in a similar way. It is enough to show that

limr<r0,r→r0

C∗alg(Pd(Γ),A)Or∪O′r ⊆ limr<r0,r→r0

C∗alg(Pd(Γ),A)Or

+ limr<r0,r→r0

C∗(Pd(Γ),A)O′r .

216 G. Yu

Given T ∈ C∗alg(Pd(Γ),A)Or∪O′r for somer0 > r > 0, by the uniformembedding property and property (3) in the definition ofC∗alg(Pd(Γ),A),there existsR> 0 such that

support(T(x, y)) ⊆(∪ j0

j=1 ∩i0i=1

(∪γ∈Xi, j ,d(γ,x)≤RB( f(γ), r)))

∪(∪ j0

j=1 ∩i0i=1

(∪γ∈X′i, j ,d(γ,x)≤RB( f(γ), r)

))for all x, y ∈ Γd.

For eachk ∈ N, let cr,k be an even function inS such that (1) 0≤cr,k ≤ 1, support(cr,k) ⊆ (−r − 1

k, r + 1k), cr,k|(−r− 1

2k ,r+ 12k )= 1; (2) cr,k is

differentiable and its derivative function is continuous. For eachγ ∈ Γ, letgr,k,γ = (β(γ))(cr,k), whereβ(γ) is the∗ homomorphism:S = A(0) →A(V), associated to the inclusion of the zero-dimensional affine space 0into V by mapping 0 tof(γ), where f is the uniform embedding. Note thatgr,k,γ ∈ C0(R+ × H ).

For eachx ∈ Γd, k ∈ N, let hk,x andh′k,x be functions inC0(R+ × H )defined by:

hk,x =j0∑

j=1

Πi0i=1

∑γ∈Xi, j ,d(γ,x)≤R

gr,k,γ

,h′k,x =

j0∑j=1

Πi0i=1

∑γ∈X′i, j ,d(γ,x)≤R

gr,k,γ

.Choosek ∈ N such thatr + 1

k < r0. We definegx andg′x in C0(R+ × H )by:

gx = h2k,x

hk,x + h′k,x,

g′x =h′2k,x

hk,x + h′k,x.

DefineT1 andT2 in C∗(Pd(Γ),A) by:

T1(x, y) = gxT(x, y),

T2(x, y) = g′xT(x, y).

We have

T = T1+ T2.


By the properties ofgx andg′x, and the bounded geometry property ofΓ, itis not difficult to verify that

T1 ∈ limr<r0,r→r0

C∗alg (Pd(Γ),A)Or,

T2 ∈ limr<r0,r→r0

C∗alg (Pd(Γ),A)O′r .

utLet e be the evaluation homomorphism fromC∗L(Pd(Γ),A) to

C∗(Pd(Γ),A) defined by:

e(g) = g(0).

Lemma 6.4. If O is the union of a family of open subsets{Oi }i∈J inR+×Hsuch that

(1) Oi ∩ Oj = ∅ if i 6= j ;(2) ∃r > 0, γi ∈ Γ such that Oi ⊆ B( f(γi), r) for all i , where

B( f(γi), r) = {(t,h) ∈ R+ × H : t2 + ‖h − f(γi )‖2 < r 2}, and f isthe uniform embedding,

then

e∗ : limd→∞

K∗(C∗L(Pd(Γ),A)O)→ limd→∞

K∗(C∗(Pd(Γ),A)O),

is an isomorphism.

We need some preparations before we can prove Lemma 6.4. Weshall first introduce twoC∗-algebras which will be useful in the proofof Lemma 6.4.

Let {Oi }i∈J and{γi}i∈J be as in Lemma 6.4, let{Xi }i∈J be a family ofclosed subspaces ofPd(Γ) for somed such that

(1) γi ∈ Xi for everyi ∈ J;(2) {Xi }i∈J is uniformly bounded in the sense that there existsr0 > 0 for

which diameter(Xi ) ≤ r0 for everyi ∈ J.LetA(V)Oi be theC∗-subalgebra ofA(V) generated by elements whose

supports are contained inOi . We defineA∗(Xi : i ∈ J) to be theC∗-subalgebra of

{⊕i∈Jbi : bi ∈ A(V)Oi ⊗C∗(Xi ), supi∈J||bi || <∞}generated by elements⊕i∈Jbi for which there existN> 0, c0 > 0 such that,for eachi , there isb′i ∈ A(WN(γi ))⊗C∗(Xi ) for which (βN(γi ))(b′i ) = bi ,DY(b′i ) exists in A(WN(γi ))⊗C∗(Xi ) and ‖DY(b′i )‖ ≤ c0 for all Y =(s,h) ∈ R × WN(γi ) satisfying‖Y‖ ≤ 1, whereβN(γi ) is the∗-homo-morphism:

A(WN(γi ))⊗C∗(Xi )→ A(V)⊗C∗(Xi ),

induced by the inclusion ofWN(γi ) into V.

218 G. Yu

Similarly we defineA∗L(Xi : i ∈ J) to be theC∗-subalgebra of

{⊕i∈Jbi : bi ∈ A(V)Oi ⊗C∗L(Xi ), supi∈J||bi || <∞}generated by elements⊕i∈Jbi such that

(1) the function

⊕i∈Jbi (t) : R+ → ⊕i∈J(A(V)Oi ⊗C∗(Xi )),

is uniformly norm-continuous int;(2) there exists a bounded functionc(t) onR+ for which

limt→∞ c(t) = 0 and sup{d(x, y) : (x, y) ∈ supp(bi (t))} ≤ c(t)

for all i ∈ J, where supp(bi (t)) is defined to be the complement (inXi ×Xi )of all (x, y) ∈ Xi × Xi for which there exist functionsh1 andh2 in C(Xi ),the algebra of all complex-valued continuous functions onXi , such thath1(x) 6= 0, h2(y) 6= 0, and

(a1⊗h1)(bi (t))(a2⊗h2) = 0

for all a1,a2 ∈ A(V)Oi ;(3) there existN > 0, c0 > 0 such that, for eachi , there isb′i ∈

A(WN(γi ))⊗C∗L(Xi ) for which (βN(γi ))(b′i ) = bi , DY(b′i ) exists inA(WN(γi ))⊗C∗L(Xi ) and‖DY(b′i )‖ ≤ c0 for all Y = (s,h) ∈ R×WN(γi )satisfying‖Y‖ ≤ 1.

Next we shall recall the concepts of Lipschitz maps and strong Lipschitzhomotopy equivalence [27].

Let {Yi }i∈J be another family of closed subspaces ofPd(Γ) for somedsuch that

(1) γi ∈ Yi for everyi ∈ J;(2) {Yi }i∈J is uniformly bounded.A map

g :⊔i∈J

Xi →⊔i∈J

Yi

is said to be Lipschitz (⊔

i∈J Xi and⊔

i∈J Yi , are respectively the disjointunions of{Xi }i∈J and{Yi }i∈J) if

(1) g(x) ∈ Yi for all x ∈ Xi ;(2) there exists a constantc such that

d(g(x), g(y)) ≤ cd(x, y)

for all x andy in Xi , wherec is independent ofi , andi is any element inJ.Letgi (i = 1,2) be Lipschitz maps from

⊔i∈J Xi to

⊔i∈J Yi . g1 is said to

be strongly Lipschitz homotopy equivalent tog2 if there exists a continuousmap

F : [0,1] ×(⊔

i∈J

Xi

)→⊔i∈J

Yi


such that(1) F(0, x) = g1(x), F(1, x) = g2(x) for all x ∈⊔i∈J Xi ;(2) there exists a constantc for which

d(F(t, x), F(t, y)) ≤ cd(x, y)

for all x andy in Xi , t ∈ [0,1], wherei is any element inJ;(3) F is equicontinuous int, i.e. for anyε > 0, there existsδ > 0 such

that

d(F(t1, x), F(t2, x)) < ε

for all x ∈⊔i∈J Xi if |t1− t2| < δ.{Xi }i∈J is said to be strongly Lipschitz homotopy equivalent to{Yi}i∈J

if there exist Lipschitz mapsg1 :⊔i∈J Xi →⊔i∈J Yi , andg2 :⊔i∈J Yi →⊔

i∈J Xi , such thatg1g2 andg2g1 are respectively strongly Lipschitz homo-topy equivalent to identity maps.

Define A∗L,0(Xi : i ∈ J) to be theC∗-subalgebra ofA∗L(Xi : i ∈ J)consisting of elements⊕i∈Jbi (t) satisfyingbi (0) = 0 for all i ∈ J. When ne-cessary (for example in the proof of the next lemma), we useA∗L,0(Xi , HXi :i ∈ J) to denoteA∗L,0(Xi : i ∈ J), where HXi is the separable Hilbertspace (equipped with a faithful and nondegenerate∗-representation ofC(Xi )whose range contains no non-zero compact operator) in the definition ofC∗(Xi ) (cf. Definition 3.1). Note thatA∗L,0(Xi , HXi : i ∈ J) is independentof the choice of{HXi }i∈J (up to∗-isomorphism).

The following result and its proof are essentially from [27].

Lemma 6.5. If {Xi }i∈J is strongly Lipschitz homotopy equivalent to{Yi }i∈J,thenK∗(A∗L,0(Xi : i ∈ J)) is isomorphic toK∗(A∗L,0(Yi : i ∈ J)).

Proof. For any Lipschitz mapg from⊔

i∈J Xi to⊔

i∈J Yi , we shall firstrecall the construction of a homomorphism fromK∗(A∗L,0(Xi : i ∈ J)) toK∗(A∗L,0(Yi : i ∈ J)) associated tog [27].

Let {εk}∞k=1 be a sequence of positive numbers such that limk→∞ εk = 0.For eachk, there exists an isometry

Uk,i : HXi → HYi

such that

supp(Uk,i ) ⊆ {(x, y) ∈ Xi × Yi : d(g(x), y) < εk},where HXi is the separable Hilbert space (equipped with a faithful andnondegenerate∗-representation ofC(Xi ) whose range contains no non-zero compact operator) in the definition ofC∗(Xi ) (cf. Definition 3.1),and supp(Uk,i ) is the complement (inXi × Yi ) of the set of all points(x, y) ∈ Xi ×Yi for which there existg1 ∈ C(Xi ) andg2 ∈ C(Yi ) such thatg2Uk,i g1 = 0, g1(x) 6= 0 andg2(y) 6= 0.

220 G. Yu

Let

Uk = ⊕i∈JUk,i .

Define a family of isometries

Vg(t) : (⊕i∈J HXi )⊕ (⊕i∈J HXi )→ (⊕i∈J HYi )⊕ (⊕i∈J HYi )

(t ∈ [0,∞)) by:

Vg(t) = R(t − k+ 1)

(Uk 00 Uk+1

)R∗(t − k+ 1)

for all natural numbersk andk− 1≤ t < k, where

R(t) =(

cos(π2 t) sin(π2 t)− sin(π2 t) cos(π2 t)

).

We define a∗-homomorphism

Ad(Vg) : A∗L,0(Xi : i ∈ J)+ → A∗L,0(Yi : i ∈ J)+ ⊗ M2(C)

by:

Ad(Vg)(b+ cI ) = Vg(t)

(b(t) 0

0 0

)V∗g (t)+ cI,

whereb∈ A∗L,0(Xi : i ∈ J), c∈C, A∗L,0(Xi : i ∈ J)+ and A∗L,0(Yi : i ∈ J)+are respectively obtained fromA∗L,0(Xi : i ∈ J) and A∗L,0(Yi : i ∈ J) byadjoining the identity.

Note thatAd(Vg) is well defined althoughVg(t) is not norm-continuousat t = k. Ad(Vg) induces a homomorphism

Ad(Vg)∗ : K∗(A∗L,0(Xi : i ∈ J))→ K∗(A∗L,0(Yi : i ∈ J)).

It is not difficult to verify thatAd(Vg)∗ is independent of the choices of{εk}∞k=1 andVg.

By assumption there exist Lipschitz maps

g1 :⊔i∈J

Xi →⊔i∈J

Yi ,

and

g2 :⊔i∈J

Yi →⊔i∈J

Xi ,

such thatg1g2 and g2g1 are respectively strongly Lipschitz homotopyequivalent to identity maps. We have

Ad(Vg1)∗Ad(Vg2)∗ = Ad(Vg1g2)∗,


Ad(Vg2)∗Ad(Vg1)∗ = Ad(Vg2g1)∗.

Hence it is enough to show that

Ad(Vg1g2)∗ = Id∗, Ad(Vg2g1)∗ = Id∗.

Next we shall prove the second identity (the first identity can be proved inthe same way).

Let g = g2g1. There is a strong Lipschitz homotopyF such thatF(0, x) = g(x) andF(1, x) = x for all x ∈⊔i∈J Xi . Choose{tk,l }k≥0,l≥0 ⊆[0,1] such that

(1) t0,l = 0, tk,l+1 ≤ tk,l , tk+1,l ≥ tk,l ;(2) there exists a sequenceNl → ∞ such thattk,l = 1 for all k ≥ Nl ,

andNl+1 ≥ Nl for all l ;(3) there exists a sequence of positive numbersε′l → 0 such that

d(F(tk+1,l , x), F(tk,l , x)) < ε′l

and

d(F(tk,l+1, x), F(tk,l , x)) < ε′lfor all non-negative integerk andl , and allx ∈⊔i∈J Xi .

For example, we can take

tk,l ={

kl+1 if k ≤ l + 1;1 otherwise.

Let Uk,l,i be an isometry fromHXi to HXi such that

supp(Uk,l,i ) ⊂ {(x, y) ∈ Xi × Xi : d(F(tk,l , x), y) < εl}and

Uk,l,i = I

if F(tk,l , x) = x for all x ∈ Xi . Let

Vk,l = ⊕i∈JUk,l,i .

Define a family of isometries

Vk(t) : (⊕i∈J HXi )⊕ (⊕i∈J HXi )→ (⊕i∈J HXi )⊕ (⊕i∈J HXi )

(t ∈ [0,∞)) by:

Vk(t) = R(t − l + 1)

(Vk,l−1 0

0 Vk,l

)R∗(t − l + 1)

for all natural numbersl andl − 1≤ t < l . Vk induces a homomorphism

Ad(Vk) : A∗L,0(Xi : i ∈ J)+ → A∗L,0(Xi : i ∈ J)+ ⊗ M2(C)

222 G. Yu

by

Ad(Vk)(b+ cI ) = Vk(t)

(b(t) 0

0 0

)V∗k (t)+ cI

for all b ∈ A∗L,0(Xi : i ∈ J) andc ∈ C.Note thatA∗L,0(Xi : i ∈ J) is stable. Hence any element inK1(A∗L,0(Xi :

i ∈ J)) can be represented by a unitaryu in A∗L,0(Xi : i ∈ J)+. We define

a= ⊕k≥0(Ad(Vk)(u))

(u−1 00 I

),

b= ⊕k≥0(Ad(Vk+1)(u))

(u−1 00 I

),

c= I ⊕k≥1 (Ad(Vk)(u))

(u−1 00 I

).

By the definition ofVk, it is not difficult to verify thata, b and c areinvertible elements inA∗L,0(Xi ,⊕k≥0HXi : i ∈ J)+ ⊗ M2(C). Again by thedefinitions ofVk andVk+1, a is equivalent tob in K1(A∗L,0(Xi ,⊕k≥0HXi :i ∈ J)). Note thatb is equivalent toc in K1(A∗L,0(Xi ,⊕k≥0HXi : i ∈ J)).Hence(Ad(Vg)(u))(u−1 ⊕ I ) ⊕k≥1 I = ac−1 is equivalent to⊕k≥0I inK1(A∗L,0(Xi ,⊕k≥0HXi : i ∈ J)). This implies thatAd(Vg)(u) ⊕k≥1 I isequivalent tou⊕k≥1 I in K1(A∗L,0(Xi ,⊕k≥0HXi : i ∈ J)). But

j∗ : K1(A∗L,0(Xi : i ∈ J))→ K1(A

∗L,0(Xi ,⊕k≥0HXi : i ∈ J))

is an isomorphism, where

j : A∗L,0(Xi : i ∈ J)→ A∗L,0(Xi ,⊕k≥0HXi : i ∈ J)

is the homomorphism defined by:

⊕i∈Jbi →⊕i∈J(bi ⊕k≥1 0).

ThereforeAd(Vg)(u) is equivalent tou in K1(A∗L,0(Xi : i ∈ J)). The caseof K0 can be dealt with in a similar way using a suspension argument.ut

Let e be the evaluation homomorphism fromA∗L(Xi : i ∈ J) to A∗(Xi :i ∈ J) defined by:

⊕i∈Jbi (t)→⊕i∈Jbi (0).

The following result is essentially from [27].


Lemma 6.6. Let {γi }i∈J be as in Lemma 6.4. If{4i }i∈J is a family ofsimplices inPd(Γ) for somed such thatγi ∈ 4i for all i ∈ J, then

e∗ : K∗(A∗L(4i : i ∈ J))→ K∗(A∗(4i : i ∈ J))

is an isomorphism.

Proof. We have a short exact sequence:

0→ A∗L,0(4i : i ∈ J)→ A∗L(4i : i ∈ J)→ A∗(4i : i ∈ J)→ 0.

Hence it is enough to show

K∗(A∗L,0(4i : i ∈ J)) = 0.

Note that{4i }i∈J is strongly Lipschitz homotopy equivalent to{{γi}}i∈J.Hence by Lemma 6.5 it is enough to show

K∗(A∗L,0({γi } : i ∈ J)) = 0.

We shall prove

K1(A∗L,0({γi} : i ∈ J)) = 0.

TheK0-case can be dealt with in a similar way using a suspension argument.Note thatA∗L,0({γi } : i ∈ J) is stable. Hence any element inK1(A∗L,0({γi } :

i ∈ J)) can be represented by a unitaryu in A∗L,0({γi } : i ∈ J)+. For eachs ∈ [0,∞), we define

us(t) ={

I if 0 ≤ t ≤ s;u(t − s) if s≤ t <∞.

Consider

w(s) = (⊕∞k=0uk ⊕ I )(I ⊕∞k=1 u−1k−s⊕ I ),

wheres ∈ [0,1]. Notice thatw(s) is an element inA∗L,0({γi },⊕∞k=0H{γi } ⊕H{γi } : i ∈ J)+ for eachs ∈ [0,1]. We have

w(0) = u⊕∞k=1 I ⊕ I,

w(1) = (⊕∞k=0uk ⊕ I )(I ⊕∞k=1 u−1k−1⊕ I ).

w(1) is obviously equivalent to⊕∞k=0I ⊕ I in K1(A∗L,0({γi },⊕∞k=0H{γi } ⊕H{γi } : i ∈ J)). This, together with the above two identities, implies thatu⊕∞k=1 I ⊕ I is equivalent to⊕∞k=0I ⊕ I in K1(A∗L,0({γi },⊕∞k=0H{γi } ⊕H{γi } :i ∈ J)). But

j∗ : K1(A∗L,0({γi} : i ∈ J))→ K1(A

∗L,0({γi},⊕∞k=0H{γi } ⊕ H{γi } : i ∈ J))

224 G. Yu

is an isomorphism, where

j : A∗L,0({γi} : i ∈ J)→ A∗L,0({γi },⊕∞k=0H{γi } ⊕ H{γi } : i ∈ J)

is the homomorphism defined by:

⊕i∈Jbi →⊕i∈J(bi ⊕k≥1 0⊕ 0).

Henceu is equivalent to the zero element inK1(A∗L,0({γi } : i ∈ J)). utProof of Lemma 6.4.Let A(V)O be theC∗-subalgebra ofA(V) generatedby elements whose supports are contained inO. The support of an element∑

ax[x] in E is defined to be

{(x,u) ∈ Γd × (R+ × H ) : u ∈ support(ax)},whereE is as in Definition 5.4. LetEO be the closure of the set of all elementsin E whose supports are contained inΓd × O. EO is a Hilbert module overA(V)O⊗K . C∗(Pd(Γ),A)O has a faithful representation onEO. We havea decomposition:

EO =⊕i∈J

EOi .

By the uniform embedding property and property (3) in the definition ofC∗alg(Pd(Γ),A), each elementa ∈ C∗alg(Pd(Γ),A)O has a correspondingdecomposition:

a=⊕i∈J

ai

such that there existsR> 0 for which eachai is supported on

{(x, y,u) : u ∈ Oi , x ∈ Γd, y ∈ Γd,d(x, γi ) ≤ R,d(y, γi ) ≤ R}.Let

Oi (R) = {(x,u) : u ∈ Oi , x ∈ Γd,d(x, γi ) ≤ R}.DefineEOi (R) to be the closure of the set of all elements inE whose supportsare contained inOi (R). ai lives in the image of the injective homomorphismfrom B(EOi (R)) to B(EOi ):

ψi : b→(

b 00 0

),

with respect to the decomposition

EOi = EOi (R) ⊕ E′Oi (R)

for some Hilbert submoduleE′Oi (R)of EOi (such Hilbert submodule exists

in this case).


Note that

EOi (R) = (A(V)Oi ⊗K)⊗`2({x ∈ Γd : d(x, γi ) ≤ R}).HenceB(EOi (R)) can be identified with

A(V)Oi ⊗K⊗B(l2({x ∈ Γd : d(x, γi ) ≤ R})).Using this identification it is not difficult to verify that the map:

a→⊕i∈J

ai ,

gives a∗-isomorphism fromC∗(Pd(Γ),A)O to theC∗-algebra

limR→∞

A∗({x ∈ Pd(Γ) : d(x, γi ) ≤ R} : i ∈ J).

Similarly C∗L(Pd(Γ),A)O is ∗-isomorphic to theC∗-algebra

limR→∞

A∗L({x ∈ Pd(Γ) : d(x, γi ) ≤ R} : i ∈ J).

For each natural numberk, let4(k)i be the simplex with vertices{γ ∈ Γ :d(γ, γi ) ≤ k}. By the above discussions, we know that limd→∞C∗(Pd(Γ),A)O and limd→∞C∗L(Pd(Γ),A)O are respectively∗-isomorphic tolimk→∞ A∗(4(k)i : i ∈ J) and limk→∞ A∗L (4(k)i : i ∈ J). This, togetherwith Lemma 6.6, implies Lemma 6.4. utLemma 6.7. Givenr > 0, let B(x, r)={(t,h)∈R+×H : t2+‖h− x‖2<r 2}for eachx ∈ H. If Γ has bounded geometry, then there exists an integerl0

such that whenever⋂`

k=1 B( f(γk), r) 6= ∅ for distinct elementsγk in Γ, wehave` ≤ l0, where f is the uniform embedding.

Proof.⋂`

k=1 B( f(γk), r) 6= ∅ implies that

|| f(γk)− f(γ1)|| ≤ 2r

for all 1 ≤ k ≤ l . Hence, by the uniform embedding property, there existsR > 0 such thatd(γk, γ1) ≤ R for all 1 ≤ k ≤ l , whereR depends onlyon r . This, together with the bounded geometry property ofΓ, impliesLemma 6.7. ut

Now we are ready to prove “the twisted coarse Baum–Connes conjec-ture” for bounded geometry spaces which admit a uniform embedding intoHilbert space. This result plays a key role in the proof of the main theoremof this paper.

Theorem 6.8. If Γ has bounded geometry, then

e∗ : limd→∞ K∗(C∗L(Pd(Γ),A))→ lim

d→∞ K∗(C∗(Pd(Γ),A)),

is an isomorphism.

226 G. Yu

Proof. Let B(x, r) = {(t,h) ∈ R+ × H : t2 + ‖h − x‖2 < r 2} for allr > 0, x ∈ H. We defineOr by:

Or =⋃γ∈Γ

B( f(γ), r),

where f is the uniform embedding.We have

C∗L(Pd(Γ),A) = limr→∞C∗L(Pd(Γ),A)Or ,

C∗(Pd(Γ),A) = limr→∞C∗(Pd(Γ),A)Or .

Hence it is enough to show that

e∗ : limd→∞ K∗( lim

r<r0,r→r0C∗L(Pd(Γ),A)Or )

→ limd→∞

K∗( limr<r0,r→r0

C∗(Pd(Γ),A)Or ),

is an isomorphism for everyr0 > 0. By Lemma 6.7, for eachr0 > 0, thereexist finitely many subsets{Jk}k0

k=1 of Γ such thatOr = ⋃k0k=1 Or,k for all

r < r0, where eachOr,k is the disjoint union of{B( f(γ), r)}γ∈Jk for allr < r0. Now Theorem 6.8 follows from Lemmas 6.4, 6.2, 6.3, and a Mayer–Vietoris sequence argument (cf. the K-theory Mayer–Vietoris sequence inSect. 3 of [18]). ut

7 Proof of the main theorem

In this section, we shall prove the main theorem of this paper. The proof ofthe main theorem is considerably simpler for the finite-dimensional case.For this reason, the readers are encouraged to go through the proof for thefinite-dimensional case first. Note that the main theorem is not trivial in thefinite-dimensional case.

We shall first prove a geometric analogue of the infinite-dimensionalBott periodicity introduced by Higson, Kasparov and Trout in [15]. Thegeometric analogue of infinite-dimensional Bott periodicity is then used toreduce the coarse Baum–Connes conjecture to the twisted coarse Baum–Connes conjecture for bounded geometry spaces which admit a uniformembedding into Hilbert space.

To prove the Bott periodicity in our context, we need to recall the def-initions of certain elliptic operators and construct their associated asymp-totic morphisms. For convenience of the readers, we shall introduce theseconstructions in the finite-dimensional case before we discuss their infinite-dimensional counterparts.

Let H be a finite-dimensional Euclidean space. Denote byH the Hilbertspace of square integrable functions fromH into Cliff (H ), the complexified


Clifford algebra ofH. Let s be the Schwartz subspace ofH . We define theDirac operatorD, an unbounded operator onH with domains, to be:

Dξ =n∑

i=1

(−1)degξ ∂ξ

∂xivi ,

where{v1, . . . , vn} is an orthonormal basis forH and{x1, . . . , xn} are thedual coordinates to{v1, . . . , vn}.

Next we shall recall certain infinite-dimensional elliptic operators intro-duced by Higson, Kasparov and Trout [15], which play the role of the Diracoperator in the infinite-dimensional case.

Let H be a separable and infinite-dimensional Hilbert space. LetV be thecountably infinite-dimensional Euclidean subspace ofH defined in Sect. 5.If Va ⊂ V is a finite-dimensional affine subspace, denote byHa the Hilbertspace of square integrable functions fromVa into Cliff (V0

a ). If Va ⊂ Vb,then there exists an isomorphism:

Hb∼= Hba⊗Ha,

whereHba is the Hilbert space associated toV0ba. Let ξ0 ∈ Hba be the unit

vector defined by:

ξ0(vba) = π−nba/4 exp

(−1

2‖vba‖2

),

wherevba ∈ V0ba, nba = dim (V0

ba). We considerHa as included inHb viathe isometryξ → ξ0⊗ξ. We define

H = lim−→Ha,

where the Hilbert space direct limit is taken over the direct system of finite-dimensional affine subspaces ofV.

Let s = lim−→sa be the algebraic direct limit of the Schwartz subspaces

sa ⊂ Ha. If Va ⊂ V is a finite-dimensional affine subspace, we define theDirac operatorDa, an unbounded operator onH with domains, to be:

Daξ =n∑

i=1

(−1)degξ ∂ξ

∂xivi ,

where{v1, . . . , vn} is an orthonormal basis forV0a and{x1, . . . , xn} are the

dual coordinates to{v1, . . . , vn}. If Va is a linear subspace, then we definethe Clifford operatorCa acting ons by:

(Caξ)(vb) = vaξ(vb)

for any ξ ∈ sb andvb ∈ Vb for someVb satisfyingVa ⊆ Vb, whereva isthe vector projection ofvb onto Va, andvaξ(vb) is the Clifford product of

228 G. Yu

va and ξ(vb). Note thatCa is a well-defined operator ons. Let Vn(x) =Wn+1(x) Wn(x) if n ≥ 1, V0(x) = W1(x), wherex ∈ Pd(Γ), andWn(x)is as in Sect. 5. We have the algebraic decomposition:

V = V0(x)⊕ V1(x)⊕ · · · ⊕ Vn(x)⊕ · · · .For eachn, define an unbounded operatorBn,t(x) on H associated to theabove decomposition by:

Bn,t(x) = t0D0+ t1D1+ · · · + tn−1Dn−1 + tn(Dn + Cn)

+ tn+1(Dn+1+ Cn+1)+ · · ·wheret j = 1+ t−1 j , Dn andCn are respectively the Dirac operator andClifford operator associated toVn(x).

We remark thatBn,t(x) plays the role of the Dirac operator in the infinite-dimensional case. By Lemma 5.8 in [15],Bn,t(x) is essentially selfadjoint.

We need to introduce anotherC∗-algebra before we can construct anasymptotic morphism associated to the Dirac operatorD in the finite-dimensional case and the Higson–Kasparov–Trout operatorBn,t in theinfinite-dimensional case.

We defineC∗alg(Pd(Γ), K) to be the algebra of all functionsT onΓd×Γdsuch that

(1) T(x, y) ∈ K(H)⊗K for all x, y ∈ Γd, whereH is respectively asin the definitions of the Dirac operatorD in the finite-dimensional case andthe Higson–Kasparov–Trout operatorBn,t in the infinite-dimensional case,andK(H) is the algebra of all compact operators acting onH;

(2)∃M > 0 andL > 0 such that‖T(x, y)‖ ≤ M for all x, y ∈ Γd, and foreachy ∈ Γd, #{x ∈ Γd : T(x, y) 6= 0} ≤ L, #{x ∈ Γd : T(y, x) 6= 0} ≤ L;

(3) ∃r > 0 such that ifd(x, y) > r , thenT(x, y) = 0.The product structure onC∗alg(Pd(Γ), K) is defined by:

(T1 · T2)(x, y) =∑z∈Γd

T1(x, z)T2(z, y).

Let E = `2(Γd)⊗H⊗H0, whereH0 is a separable and infinite-dimen-sional Hilbert space with a faithful∗-representation ofK .

C∗alg(Pd(Γ), K) acts onE by:

T(δx⊗h⊗h0) =∑y∈Γd

δy⊗T(y, x)(h⊗h0)

for all x ∈ Γd,h ∈ H,h0 ∈ H0, whereδx andδy are respectively Diracfunctions atx andy.

Definition 7.1. C∗(Pd(Γ), K) is defined to be the operator norm closure ofC∗alg(Pd(Γ), K) with respect to the above∗-representation.


Now we are ready to construct an asymptotic morphismα fromC∗(Pd(Γ),A) to C∗(Pd(Γ), K) associated to the Dirac operator in thefinite-dimensional case and the Higson–Kasparov–Trout operator in theinfinite-dimensional case. We remark that the asymptotic morphismα isadapted from [14].

If H is a finite-dimensional Euclidean space, we define

θt : A(H )⊗K → K(H)⊗K,

by:

θt((g⊗h)⊗k) = gt(D)π(ht)⊗k

for everyg ∈ S, h ∈ C(H ), k ∈ K , andt ≥ 1, wheregt(s) = g(t−1s) forall t ≥ 1 ands ∈ R, ht(v) = h(t−1v) for all t ≥ 1 andv ∈ H, D is theDirac operator as above, andπ(ht) acts onH by pointwise multiplication.

We define

αt : C∗alg(Pd(Γ),A)→ C∗(Pd(Γ), K),

by:

(αt(T ))(x, y) = θt(T(x, y)),

for everyT ∈ C∗alg(Pd(Γ),A), all (x, y) ∈ Γd × Γd andt ≥ 1.If H is a separable and infinite-dimensional Hilbert space, letV be the

countably infinite-dimensional Euclidean subspace ofH as in Sect. 5. Foreachx ∈ Pd(Γ), let Bn,t(x) be the infinite-dimensional elliptic operator ofHigson–Kasparov–Trout defined in this section, letV = V0(x) ⊕ V1(x) ⊕· · · ⊕ Vn(x)⊕ · · · andWn(x) = V0(x)⊕ V1(x)⊕ · · · ⊕ Vn−1(x) be as in thedefinition of Bn,t(x).

For every non-negative integern andx ∈ Γd, we define

θnt (x) : A(Wn(x))⊗K → K(H)⊗K

by:

(θnt (x))((g⊗h)⊗k) = gt(Bn,t(x))π(ht)⊗k

for everyg ∈ S, h ∈ C(Wn(x)), k ∈ K , andt ≥ 1, wheregt(s) = g(t−1s)for all t ≥ 1 ands ∈ R, ht(v) = h(t−1v) for all t ≥ 1 andv ∈ Wn(x),π(ht) acts onH by pointwise multiplication (ifVa = Wn(x) ⊆ Vb, wedefine(π(ht))ξ = htξ for all ξ ∈ Hb, whereht is as in Definition 5.3,HaandHb are as in the definition ofH). The fact that(θn

t (x))((g⊗h)⊗k) is inK(H)⊗K follows from Lemma 5.8 in [15]. Note thatθn

t (x) is defined ona dense subalgebra ofA(Wn(x))⊗K .

We define

αt : C∗alg(Pd(Γ),A)→ C∗(Pd(Γ), K),

230 G. Yu

by:

(αt(T))(x, y) = (θn0t (x))(T1(x, y))

for every T ∈ C∗alg(Pd(Γ),A) and t ≥ 1, wheren0 is a non-negativeinteger such that, for every pairx and y in Γd, there existsT1(x, y) ∈A(Wn0(x))⊗K satisfying(βn0(x))(T1(x, y)) = T(x, y). Note that, by theproof of Proposition 4.2 in [15],αt(T ) does not asymptotically depend onthe choice ofn0, i.e. the difference of the values ofαt(T ) for two differentchoices ofn0 goes to 0 in norm ast goes to∞.

Lemma 7.2. α extends to an asymptotic morphism fromC∗(Pd(Γ),A) toC∗(Pd(Γ), K).

We need some preparations before we can prove Lemma 7.2.Given a non-negative integerm and an algebraic decopmosition:

V = V0⊕ V1⊕ · · · ⊕ Vn ⊕ · · · ,define an unbounded operatorBt onH by

Bt = (1+ t−1m)(D0+ C0)+ (1+ t−1(m+ 1))(D1+ C1)

+ · · · (1+ t−1(m+ n))(Dn + Cn)+ · · · ,where eachVn is a finite-dimensional Euclidean subspace ofV, Dn andCnare respectively the Dirac operator and Clifford operator associated toVn.

The following result is a modified version of Lemma 5.9 in [15].

Lemma 7.3. Let B′t be the unbounded operator onH :B′t = (1+ t−1m)(D′0+ C′0)+ (1+ t−1(m+ 1))(D′1+ C′1)+ · · · (1+ t−1(m+ n))(D′n + C′n)+ · · · ,

associated to a second decomposition

V = V ′0⊕ V ′1⊕ · · · ⊕ V ′n ⊕ · · · ,whereD′n andC′n are respectively the Dirac operator and Clifford operatorassociated to the finite-dimensional Euclidean spaceV ′n. If

V0⊕ · · · ⊕ Vn ⊆ V ′0⊕ · · · ⊕ V ′n+1

and

V ′0⊕ · · · ⊕ V ′n ⊆ V0⊕ · · · ⊕ Vn+1

for all n, then for anyε > 0, R> 0, c> 0, there existst0 > 0 such that

||g(sB′t)− g(sBt)|| < ε


for all t > t0, s ∈ R,and allg ∈ C0(R) for which (1) support(g) ⊆ [−R, R],|g(t)| ≤ c for all t ∈ R; (2) the derivative functiong′ of g is continuous,and |g′(t)| ≤ c for all t ∈ R.

Proof. Let

g1(θ) = g

(2

eiθ − i− i

).

By assumption, there existsc′ > 0 such that(1) c′ depends only onR andc;(2) dg1

dθ is continuous,|(dg1dθ )(θ)| ≤ c′ for all θ ∈ [0,2π].

Claim.There exists

p(θ) =k0∑

k=−k0

ckeikθ

such that

|g1(θ)− p(θ)| < ε/10

for all θ ∈ [0,2π], where(1) k0 is a non-negative integer which depends only onc′ andε;(2) |ck| ≤ c for all −k0 ≤ k ≤ k0.

Proof of Claim.Let

ck = 1

2π

∫ 2π

0g1(θ)e

−ikθdθ,

ak = 1

2π

∫ 2π

0

(dg1

dθ

)(θ)e−ikθdθ.

We have

ak = ikck

for all k. Hence ∣∣g1(θ)− p(θ)∣∣ = ∣∣∣∣ ∑

|k|>|k0|

ak

keikθ

∣∣∣∣≤( ∑|k|>k0

1

k2

)( ∞∑k=−∞

|ak|2)≤ 2π(c′)2

∑|k|>k0

1

k2.

The above inequality implies our Claim.

232 G. Yu

Substitutingeiθ by 2x+i + i in the above Claim, we obtain∣∣∣∣∣g(x)−

(c0+

k0∑k=1

(ck2k

(1

x+ i+ i

2

)k

+c−k2k

(1

x− i− i

2

)k))∣∣∣∣∣<ε/10

for all x ∈ R, where(1) k0 depends only onR, c andε;(2) |ck| ≤ c for all −k0 ≤ k ≤ k0.If

V0⊕ · · · ⊕ Vn ⊆ V ′0⊕ · · · ⊕ V ′n

and

V ′0⊕ · · · ⊕ V ′n ⊆ V0⊕ · · · ⊕ Vn+1

for all n, Lemma 7.3 follows from the above approximation result and thefollowing inequalities

||(sB′t + i)−1− (sBt + i)−1|| ≤ t−1

and

||(sB′t − i)−1− (sBt − i)−1|| ≤ t−1,

which follow from the estimations in the proof of Lemma 5.9 in [15].The general case can be reduced to the above special case as follows. Let

{Vn}∞n=0 be the family of finite-dimensional Euclidean subspaces ofV suchthatV0⊕· · ·⊕ Vn is spanned byV0⊕V1⊕· · ·⊕Vn andV ′0⊕V ′1⊕· · ·⊕V ′nfor all n. We have

V0⊕ · · · ⊕ Vn ⊆ V0⊕ · · · ⊕ Vn

and

V0⊕ · · · ⊕ Vn ⊆ V0⊕ · · · ⊕ Vn+1

for all n;

V ′0⊕ · · · ⊕ V ′n ⊆ V0⊕ · · · ⊕ Vn

and

V0⊕ · · · ⊕ Vn ⊆ V ′0⊕ · · · ⊕ V ′n+1

for all n. Now the general case follows by applying the special case twice.ut


Let E be a finite-dimensional Euclidean space. Given a decomposition

E = E0⊕ · · · ⊕ En−1,

define an unbounded operatorD(t) on the Hilbert space of all square inte-grable functions fromE into Cliff(E) by

D(t) = D0+ (1+ t−1)D1+ · · · + (1+ t−1(n− 1))Dn−1,

whereEi is a Euclidean subspace ofE andDi is the Dirac operator associ-ated toEi for all 0≤ i < n.

Lemma 7.4. Let

E = E′0⊕ · · · ⊕ E′n−1,

be a second decomposition ofE, let D(t)′ be the unbounded operator onthe Hilbert space of all square integrable functions fromE into Cliff(E)defined by

D(t)′ = D′0+ (1+ t−1)D′1+ · · · + (1+ t−1(n− 1))D′n−1,

whereD′i is the Dirac operator associated to the Euclidean spaceE′i for all0≤ i < n. For anyε > 0, R> 0, c > 0, there existst0 > 0 such that

||g(sD(t)′)− g(sD(t))|| < ε

for all t > t0 and s ∈ R, and all g ∈ C0(R) for which (1) support(g) ⊆[−R, R], |g(t)| ≤ c for all t ∈ R; (2) the derivative functiong′ is continuous,and |g′(t)| ≤ c for all t ∈ R.

Proof. By the argument in the proof of Lemma 5.9 in [15], we have

||(sD(t)′ + i)−1− (sD(t)+ i)−1|| ≤ nt−1,

||(sD(t)′ − i)−1− (sD(t)− i)−1|| ≤ nt−1.

The above inequalities, together with the approximation argument in theproof of Lemma 7.3, imply Lemma 7.4. utLemma 7.5. Let E be a finite-dimensional Euclidean space, letD(t) be asin Lemma 7.4. For anyε > 0, R> 0, c > 0, there existst0 > 0 such that

||[gt(D(t)), π(ht)]|| < εfor all t > t0, g ∈ C0(R)andh ∈ C(E) for which (1) support(g) ⊆ [−R, R],|g(t)| ≤ c for all t ∈ R; (2) the derivative functiong′ is continuous and|g′(t)| ≤ c for all t ∈ R; (3) DYh ∈ C(E) for all Y ∈ E, and||DYh|| ≤ cfor all ||Y|| ≤ 1, where [ , ] denotes the graded commutator andπ(ht)denotes the multiplication operator byht, and DYh is the derivative of thefunction:h : E→ Cliff(E), in the direction ofY.

234 G. Yu

Proof. If g(x) = (x+ i)−1 or (x− i)−1, by the computation in the proof ofLemma 2.9 in [15], we have

||[gt(D(t)), π(ht)]|| ≤ t−2dim2(E)c,

wheredim(E) is the dimension ofE. This, together with the approximationargument in the proof of Lemma 7.3, implies Lemma 7.5. utProof of Lemma 7.2.∀ T ∈ C∗alg(Pd(Γ),A), define

‖T‖max= supψ

‖ψ(T )‖,

where the sup is taken over all∗-representationψ of C∗alg(Pd(Γ),A). LetC∗max(Pd(Γ),A) be the completion ofC∗alg(Pd(Γ),A) with respect to thenorm‖ ‖max.

For anyT ∈ C∗alg(Pd(Γ),A), there existsr > 0 such thatT(x, y) = 0if d(x, y) > r . By the definition ofWn(x), there existsN > 0 such thatWn(x) ⊂ Wn+1(y) for all n ≥ N, x andy in Γd satisfyingd(x, y) ≤ r . LetβWn+1(y),Wn(x) be the ∗-homomorphism from A(Wn(x))⊗K toA(Wn+1(y))⊗K induced by the inclusion ofWn(x) into Wn+1(y) if n ≥ Nandd(x, y) ≤ r . Let n0 be a non-negative integer such that ifn ≥ n0, thenfor every pairx andy in Γd there existsT1(x, y) ∈ A(Wn(x))⊗K satisfying(βn(x))(T1(x, y)) = T(x, y). For eachn ≥ max{n0, N}, x andy in Γd satis-fying d(x, y) ≤ r, let Bn+1,t(x, y) be the Higson–Kasparov–Trout operatorassociated to the algebraic decomposition:

V = V0(x)⊕ · · · ⊕ Vn−1(x)⊕ (Wn+1(y)Wn(x))⊕ Vn+1(y)⊕ Vn+2(y)⊕ · · · .

Defineθn+1t (x, y) : A(Wn+1(y))⊗K → K(H)⊗K, by:

(θn+1t (x, y))((g⊗h)⊗k) = gt(Bn+1,t(x, y))π(ht)⊗k

for every(g⊗h)⊗k ∈ (S⊗C(Wn+1(y)))⊗K, n ≥ max{n0, N}, x andy inΓd satisfyingd(x, y) ≤ r . By Lemma 7.3, the definition ofC∗alg(Pd(Γ),A)and the proof of Proposition 4.2 in [15], for eachn ≥ {n0, N}, we know that

(θnt (x))(T1(x, y))− (θn+1

t (x, y))(βWn+1(y),Wn(x)(T1(x, y)))

converges uniformly to 0 in norm on{(x, y) ∈ Γd × Γd : d(x, y) ≤ r } ast goes to∞. But by Lemma 7.4 and the definition ofC∗alg(Pd(Γ),A), foreachn ≥ {n0, N},(θn+1

t (x, y))(βWn+1(y),Wn(x)(T1(x, y)))− (θn+1t (y))(βWn+1(y),Wn(x)(T1(x, y)))

converges uniformly to 0 in norm on{(x, y) ∈ Γd × Γd : d(x, y) ≤ r } astgoes to∞. Hence for eachn ≥ max{n0, N}, we know that

(θnt (x))(T1(x, y))− (θn+1

t (y))(βWn+1(y),Wn(x)(T1(x, y)))


converges uniformly to 0 in norm on{(x, y) ∈ Γd × Γd : d(x, y) ≤ r }as t goes to∞. This fact, together with Lemma 7.5 and the definition ofC∗alg(Pd(Γ),A), implies thatα extends to an asymptotic morphism fromC∗alg(Pd(Γ),A) to C∗(Pd(Γ), K). Henceα extends to an asymptotic mor-phism from C∗max(Pd(Γ),A) to C∗(Pd(Γ), K). But by the uniform em-bedding property we haveC∗(Pd(Γ),A) = C∗max(Pd(Γ),A). Thereforeαextends to an asymptotic morphism fromC∗(Pd(Γ),A) to C∗(Pd(Γ), K).

utLet C∗alg(Pd(Γ)) be the algebra of functionsT onΓd × Γd such that(1) T(x, y) ∈ K for all x andy ∈ Γd;(2) ∃M > 0 and L > 0 such that‖T(x, y)‖ ≤ M for all x, y ∈ Γd,

#{x ∈ Γd : T(x, y) 6= 0} ≤ L and #{x ∈ Γd : T(y, x) 6= 0} ≤ L for ally ∈ Γd;

(3) ∃r > 0 such that ifd(x, y) > r , thenT(x, y) = 0.The product structure onC∗alg(Pd(Γ)) is defined by:

(T1T2)(x, y) =∑z∈Γd

T1(x, z)T2(z, y).

C∗alg(Pd(Γ)) has a∗-representation on2(Γd) ⊗ H0, whereH0 is a separ-able and infinite-dimensional Hilbert space with a faithful∗-representationof K . The operator norm completion ofC∗alg(Pd(Γ)) with respect to this∗-representation is∗-isomorphic toC∗(Pd(Γ)) whenΓ has bounded geom-etry. Similarly we can defineC∗L,alg(Pd(Γ)), and the operator norm comple-tion of C∗L,alg(Pd(Γ)) is ∗-isomorphic toC∗L(Pd(Γ)) whenΓ has boundedgeometry.

Next we shall construct an asymptotic morphismβ from S⊗C∗(Pd(Γ))to C∗(Pd(Γ),A), which plays the role of the Bott map in the classical Bottperiodicity theorem.

If H is a finite-dimensional Euclidean space, for eachx ∈ Γd, let

β(x) : S→ A(H ),

be the∗-homomorphism defined by:

(β(x))(g) = g(X⊗1+ 1⊗CH,x)

for all g ∈ S, whereX is the degree one and unbounded multiplier ofSdefined by:(Xa)(s) = sa(s) for eacha ∈ S with compact support and alls ∈ R, CH,x is the Clifford algebra-valued function onH which mapsv tov− f(x) ∈ H ⊂ Cliff (H ) for all v ∈ H ( f is the uniform embedding), andg(X⊗1+ 1⊗CH,x) is defined by functional calculus. We define

βt : S⊗C∗alg(Pd(Γ))→ C∗(Pd(Γ),A)

by:

(βt(g⊗T ))(x, y) = (β(x))(gt)⊗T(x, y)

236 G. Yu

for all g ∈ S, T ∈ C∗alg(Pd(Γ)) and t ≥ 1, wheregt(s) = g(t−1s) for allt ≥ 1 ands ∈ R.

If H is a separable and infinite-dimensional Hilbert space, we define

βt : S⊗C∗alg(Pd(Γ))→ C∗(Pd(Γ),A)

by:

(βt(g⊗T ))(x, y) = (β(x))(gt)⊗T(x, y)

for all g ∈ S, T ∈ C∗alg(Pd(Γ)) and t ≥ 1, wheregt(s) = g(t−1s) for allt ≥ 1 ands ∈ R, andβ(x) : S = A(0)→ A(V), is the∗-homomorphismassociated to the inclusion of the zero-dimensional affine space 0 intoV bymapping 0 tof(x) ( f is the uniform embedding).

Lemma 7.6. β extends to an asymptotic morphism fromS⊗C∗(Pd(Γ)) toC∗(Pd(Γ),A).

Proof. Let E be as in Definition 5.4. For everyg ∈ S, we define a boundedmodule homomorphismNg : E→ E, by:

Ng

(∑x∈Γd

ax[x])=∑x∈Γd

((β(x))(g)⊗1)ax[x]

for all∑

x∈Γdax[x] ∈ E. We have

βt(g⊗T ) = Ngt · (1⊗T )

for all g ∈ S and T ∈ C∗alg(Pd(Γ)), where 1⊗T is a bounded modulehomomorphism fromE to E defined by:

(1⊗T )

(∑x∈Γd

ax[x])=∑y∈Γd

(∑x∈Γd

(1⊗T(y, x))ax

)[y]

for all∑

x∈Γdax[x] ∈ E. It follows thatβt extends to a linear map from the

algebraic tensor productS⊗algC∗(Pd(Γ)) to C∗(Pd(Γ),A) satisfying

||βt(g⊗T )|| ≤ ||g||||T||for all g ∈ S andT ∈ C∗alg(Pd(Γ)).

Using the above fact, the definition ofC∗alg(Pd(Γ)) and the uniformembedding property, we can verify thatβ is an asymptotic morphismfrom the algebraic tensor productS⊗algC∗(Pd(Γ)) to C∗(Pd(Γ),A). Henceβ extends to an asymptotic morphism from the maximal tensor productS⊗maxC∗(Pd(Γ)) to C∗(Pd(Γ),A). This fact, together with the nuclearityof S, implies thatβ extends to an asymptotic morphism fromS⊗C∗(Pd(Γ))to C∗(Pd(Γ),A). ut


We remark that the asymptotic morphismβ is adapted from [14] and [15].Note thatα andβ induce homomorphisms:

α∗ : K∗(C∗(Pd(Γ),A))→ K∗(C∗(Pd(Γ))),

β∗ : K∗(C∗(Pd(Γ)))→ K∗(C∗(Pd(Γ),A)).

The following result is a geometric analogue of the infinite-dimensionalBott periodicity introduced by Higson, Kasparov and Trout in [15].

Proposition 7.7. α∗ ◦ β∗ equals the identity homomorphism fromK∗(C∗(Pd(Γ))) to K∗(C∗(Pd(Γ))).

Proof. Let γ be the asymptotic morphism

γ : S⊗C∗(Pd(Γ))→ C∗(Pd(Γ), K)

defined by:

(γt(g⊗T))(x, y) = gt2(B0,t(x))⊗T(x, y)

for everyg ∈ S, T ∈ C∗alg(Pd(Γ)), t ≥ 1, (x, y) ∈ Γd × Γd.For everyx ∈ Γd, we defineηt(x) : A(W1(x))⊗K → K(H)⊗K , by:

(ηt(x))((g⊗h)⊗k) = gt2(B1,t(x))π(ht2)⊗k

for everyg ∈ S, h ∈ C(W1(x)), k ∈ K , andt ≥ 1. Letγ ′ be the asymptoticmorphism

γ ′ : S⊗C∗(Pd(Γ))→ C∗(Pd(Γ), K)

defined by:

(γ ′t (g⊗T))(x, y) = (ηt(x))(βW1(x)(g)⊗T(x, y))

for every g ∈ S, T ∈ C∗alg(Pd(Γ)), t ≥ 1, (x, y) ∈ Γd × Γd, whereβW1(x) : S = A(0) → A(W1(x)), is the ∗-homomorphism associatedto the inclusion of the zero-dimensional linear space 0 intoW1(x) by map-ping 0 to 0. By the proof of Proposition 4.2 in [15], we know thatγ isasymptotically equivalent toγ ′. Hence we haveγ∗ = γ ′∗.

For everyt ≥ 1 andx ∈ Γd, let Ux(t) be the unitary operator acting onH induced by the translation operatorUx(t) on V defined by:

(Ux(t))v = v + t f(x)

for all v ∈ V, where f is the uniform embedding. Recall thatC∗(Pd(Γ), K)has a faithful∗-representation onl2(Γd)⊗H⊗H0 (cf. Definition 7.1). Forevery x ∈ Γd, s ∈ [0,1] and t ≥ 1, we define a unitary operatorUx,s(t)acting on(l2(Γd)⊗H⊗H0)⊕ (l2(Γd)⊗H⊗H0) by:

Ux,s(t) = R(s)

(1⊗Ux(t)⊗1 0

0 1

)R−1(s),

238 G. Yu

where

R(s) =(

cos(π2 s) sin(π2 s)− sin(π2 s) cos(π2 s)

).

We define a homotopy of asymptotic morphisms

γ(s) : S⊗C∗(Pd(Γ))→ C∗(Pd(Γ), K)⊗M2(C)

by:

((γt(s))(g⊗T))(x, y) = Ux,s(t)

((γ ′t (g⊗T))(x, y) 0

0 0

)U−1

x,s(t)

for everyg ∈ S, T ∈ C∗alg(Pd(Γ)), t ≥ 1, s ∈ [0,1], and(x, y) ∈ Γd × Γd,where M2(C) is the algebra of all 2× 2 matrices overC endowed withthe trivial grading. The fact thatγ(s) is an asymptotic morphism for eachs ∈ [0,1] follows from straightforward computation using the definitionsof γt(s) andC∗alg(Pd(Γ)), the uniform embedding property, and the fact thatf(x) ∈ W1(x) andUx(t) commutes withB1,t(x) for all x ∈ Γd andt ≥ 1.

Again by the fact thatf(x) ∈ W1(x) andUx(t) commutes withB1,t(x)for all x ∈ Γd andt ≥ 1, we can verify that

(γt(0))(g⊗T )−(αt(βt(g⊗T )) 0

0 0

)→ 0

in norm ast →∞ for all g ∈ S andT ∈ C∗alg(Pd(Γ)). It follows that

γ(0)∗ = α∗ ◦ β∗.We also have

γt(1) =(γ ′t 00 0

).

Henceγ(1)∗ = γ ′∗. Therefore we haveα∗ ◦β∗ = γ∗. ReplacingB0,t(x) withs−1B0,t(x) in the definition ofγ , for 0 < s ≤ 1, we obtain a homotopybetweenγ and the∗ homomorphism:g⊗T → g(0)P⊗T, whereP is theprojection onto the one-dimensional kernel ofB0,t(x) andP does not dependon x (cf. Corollary 2.15 and the proof of Lemma 5.8 in [15]). It follows thatγ∗ equals the identity homomorphism. Thereforeα∗ ◦β∗ equals the identityhomomorphism. ut

We can similarly construct asymptotic morphisms

αL : C∗L(Pd(Γ),A)→ C∗L(Pd(Γ), K),

βL : S⊗C∗L(Pd(Γ))→ C∗L(Pd(Γ),A),

where C∗L(Pd(Γ), K) is defined in a way similar to the definition ofC∗(Pd(Γ), K).


Proposition 7.8. (αL)∗ ◦ (βL)∗ equals the identity homomorphism fromK∗(C∗L(Pd(Γ)) to K∗(C∗L(Pd(Γ)).

The proof of Proposition 7.8 is similar to the proof of Proposition 7.7and is therefore omitted.

Finally we are ready to prove the main result of this paper.

Proof of Theorem 1.1.By Theorem 4.2, it is enough to show that

e∗ : limd→∞ K∗(C∗L(Pd(Γ)))→ lim

d→∞ K∗(C∗(Pd(Γ)))

is an isomorphism. Consider the following commuting diagram:

limd→∞ K∗(C∗L(Pd(Γ)))e∗−−−→ limd→∞ K∗(C∗(Pd(Γ)))

(βL )∗y β∗

ylimd→∞ K∗(C∗L(Pd(Γ),A))

e∗−−−→∼= limd→∞ K∗(C∗(Pd(Γ),A))

(αL )∗y α∗

ylimd→∞ K∗(C∗L(Pd(Γ)))

e∗−−−→ limd→∞ K∗(C∗(Pd(Γ)))

By Theorem 6.8 the middle horizontal homomorphism is an isomorphism(note that the uniform embedding condition is used here and the middlehorizontal homomorphism is an isomorphism only after the inductive limitslimd→∞ are taken). By the geometric analogue of the infinite-dimensionalBott periodicity (Propositions 7.7 and 7.8),(βL)∗ ◦ (αL)∗ andβ∗ ◦ α∗ areidentity homomorphisms even before the inductive limits limd→∞ are taken(note that the uniform embedding condition is also used here). The abovefacts, together with a diagram chasing argument, imply Theorem 1.1.ut

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