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EXPANDERS ARE COUNTEREXAMPLES TO THE ℓp COARSE

BAUM-CONNES CONJECTURE

YEONG CHYUAN CHUNG AND PIOTR W. NOWAK

ABSTRACT. We consider an ℓp coarse Baum-Connes assembly map for 1<

p <∞, and show that it is not surjective for expanders arising from residu-ally finite hyperbolic groups.

CONTENTS

1. Introduction 1Acknowledgements 32. The ℓp coarse Baum-Connes assembly map 42.1. Roe algebras and localization algebras 42.2. The assembly map and its image 62.3. Equivariant assembly maps 133. p-operator norm localization and a lifting homomorphism 174. Kazhdan projections in the ℓp Roe algebra 205. Main result 216. Remarks and open questions 23References 25

1. INTRODUCTION

The coarse Baum-Connes conjecture, first formulated in [49], is a coarsegeometric analog of the original Baum-Connes conjecture for groups [2], andit posits that a certain coarse assembly map or index map is an isomorphismbetween a topological object involving the K -homology of Rips complexes of abounded geometry metric space and the K -theory of a certain C∗-algebra as-sociated to the metric space, namely the Roe algebra, which encodes the largescale geometry of the space. One can think of the conjecture as providing analgorithm for computing higher indices of generalized elliptic operators onnon-compact spaces. The significance of this conjecture lies in its applica-tions in geometry and topology, which includes the Novikov conjecture whenthe metric space is a finitely generated group equipped with the word met-ric, and also the problem of existence of positive scalar curvature metrics, as

Date: August 24, 2021.1

http://arxiv.org/abs/1811.10457v2

2 YEONG CHYUAN CHUNG AND PIOTR W. NOWAK

well as Gromov’s zero-in-the-spectrum conjecture, when the space is a Rie-mannian manifold. In fact, injectivity of the coarse assembly map (commonlyreferred to as the coarse Novikov conjecture) is sufficient for some of theseapplications.

The conjecture has been proven in a number of cases. Yu showed that itholds for metric spaces that coarsely embed into Hilbert space [60], general-izing his earlier work showing that it holds for spaces with finite asymptoticdimension [59]. In [59], there is also a counterexample to the coarse Novikovconjecture when the condition of bounded geometry is omitted. More recentpositive results on the coarse Baum-Connes conjecture include [15–18] byFukaya and Oguni.

On the other hand, Higson [27] showed that the coarse assembly map isnot surjective for certain Margulis-type expanders. Then in [28], Higson-Lafforgue-Skandalis showed that for any expander, either the coarse assem-bly map fails to be surjective or the Baum-Connes assembly map with certaincoefficients for an associated groupoid fails to be injective, and that the formeralways occurs for certain Margulis-type expanders. Although expanders pro-vide counterexamples to surjectivity of the coarse assembly map, it is knownthat the map (or the version with maximal Roe algebras) is injective for cer-tain classes of expanders [5,23,24,41].

While most of the results mentioned in the previous paragraph only ap-ply to Margulis-type expanders, Willett and Yu in [57] considered spaces ofgraphs with large girth and showed that the coarse assembly map is injectivefor such spaces while it is not surjective if the space is a weak expander. Forthe maximal version of the coarse assembly map, they showed that it is anisomorphism for such spaces if there is a uniform bound on the vertex de-grees of the graphs. They also discussed how their methods can be modifiedto yield the same results for a version of the coarse assembly map for uniformRoe algebras formulated by Špakula [55].

In this paper, our goal is to consider an ℓp analog of the coarse Baum-Connes assembly map for 1 < p <∞, and show that it fails to be surjectivefor certain expanders by adapting the arguments in [57]. It should be notedthat techniques used in the C∗-algebraic setting often do not transfer to theℓp setting in a straightforward manner.

Although the ℓp analog of the coarse Baum-Connes conjecture has no knowngeometric or topological applications when p 6= 2, in light of interest in Lp (orℓp) operator algebras in recent years (e.g. [3, 7–9, 14, 19–22, 26, 37, 45–47]),the study of assembly maps involving Lp operator algebras contributes toour general understanding of the K -theory of some of these algebras. We alsonote that other assembly maps involving Lp operator algebras have recentlybeen considered in [7, 14], as well as in unpublished work of Kasparov-Yu.In similar spirit, the Bost conjecture [34, 43, 44, 54] asks whether the Baum-Connes-type assembly map into the K -theory of the Banach algebra L1(G) isan isomorphism for a locally compact group G.

EXPANDERS ARE COUNTEREXAMPLES TO THE ℓp COARSE BAUM-CONNES CONJECTURE 3

The description of the assembly map that we use is equivalent to that in[57] when p = 2 in the sense that one is an isomorphism if and only if theother is. This equivalence was established in [58], where Yu introduced local-ization algebras and showed that a local index map from K -homology to theK -theory of the localization algebra is an isomorphism for finite-dimensionalsimplicial complexes. Qiao and Roe [48] later showed that this isomorphismholds for general locally compact metric spaces. By considering analogs ofRoe algebras and localization algebras represented on appropriate ℓp spaces,we obtain an ℓp analog of the coarse Baum-Connes assembly map of the form

µ : limR→∞

K∗(Bp

L(PR (X )))→ K∗(Bp(X )),

and the ℓp coarse Baum-Connes conjecture is the statement that this map isan isomorphism. Following the ideas in [48] and [25, Appendix B], we alsoprovide an alternative form of the assembly map (see Theorem 2.15).

Zhang and Zhou showed in [61, Proposition 5.20] that the left-hand side ofthe ℓp coarse Baum-Connes assembly map can be naturally identified withthat of the original coarse Baum-Connes assembly map. Therefore, givena metric space X with bounded geometry, if the ℓp coarse Baum-Connesconjecture for X holds for a range of values of p, then the K -theory groupsK∗(Bp(X )) are independent of p in that range. This is true when X has finiteasymptotic dimension and p ∈ (1,∞) by [61, Theorem 4.6]. Also, Shan andWang showed in [53] that the ℓp coarse Baum-Connes assembly map is injec-tive for p ∈ (1,∞) when X coarsely embeds into a simply connected completeRiemannian manifold of non-positive sectional curvature.

We end the introduction with the statement of our main theorem:

Theorem 1.1. Let p ∈ (1,∞). Let G be a residually finite hyperbolic group.

Let N1 ⊇ N2 ⊇ ·· · be a sequence of normal subgroups of finite index such that⋂i Ni = {e}. Assume that the box space äG =

⊔i G/Ni is an expander, i.e., that

G has property τ with respect to the family {Ni}. If q ∈Bp(äG) is the Kazhdan

projection associated to äG, then [q] ∈ K0(Bp(äG)) is not in the image of the

ℓp coarse Baum-Connes assembly map.

As examples, our result applies to finitely generated free groups and SL2(Z).One can see from the proof of our main result that the hyperbolicity as-

sumption can in fact be replaced by the following set of conditions:

(1) G has the p-operator norm localization property.(2) The ℓp Baum-Connes assembly map for each Ni is injective.(3) The classifying space for proper G-actions has finite homotopy type,

i.e., there is a model Z of a locally finite CW complex with universalproper G-action such that Z/G is a compact CW complex.

Acknowledgements. We would like to thank Rufus Willett for his illumi-nating comments.


This project has received funding from the European Research Council(ERC) under the European Union’s Horizon 2020 research and innovationprogramme (grant agreement no. 677120-INDEX).

2. THE ℓp COARSE BAUM-CONNES ASSEMBLY MAP

In this section, we define the ℓp coarse Baum-Connes assembly map andprovide an alternative formulation of it. We also consider the equivariantversion of the assembly map.

2.1. Roe algebras and localization algebras. We begin by introducingthe ℓp Roe algebra and ℓp localization algebra, whose K -theory groups areused to define the ℓp coarse Baum-Connes assembly map.

Definition 2.1. Let (X , d) be a metric space.

(1) We say that X is uniformly discrete if there exists δ > 0 such that

d(x, y)≥ δ for all distinct points x, y ∈ X.

(2) If X is uniformly discrete, we say that it has bounded geometry if for

all R > 0 there exists NR ∈N such that all balls of radius R in X have

cardinality at most NR .

(3) We say that X is proper if all closed balls in X are compact.

(4) A net in X is a discrete subset Y ⊆ X such that there exists r > 0 with

the properties that d(x, y)≥ r for all x, y ∈Y , and for any x ∈ X there is

y ∈Y with d(x, y)< r.

(5) If X is proper, we say that it has bounded geometry if it contains a net

with bounded geometry.

We now associate certain ℓp operator algebras called Roe algebras to propermetric spaces. These algebras encode the large scale geometry of the metricspace, and the K -theory of the Roe algebra serves as the target of the ℓp

coarse Baum-Connes assembly map.

Definition 2.2. Let X be a proper metric space, and fix a countable dense sub-

set Z ⊆ X. Let T be a bounded operator on ℓp(Z,ℓp), and write T = (Tx,y)x,y∈Z

so that each Tx,y is a bounded operator on ℓp. T is said to be locally compact

if

• each Tx,y is a compact operator on ℓp;

• for every bounded subset B ⊆ X, the set{(x, y) ∈ (B×B)∩ (Z×Z) : Tx,y 6= 0

}

is finite.

The propagation of T is defined to be

prop(T)= inf{S > 0 : Tx,y = 0 for all x, y∈ Z with d(x, y)> S

}.

The algebraic Roe algebra of X, denoted Cp[X ], is the subalgebra of B(ℓp(Z,ℓp))

consisting of all finite propagation, locally compact operators. The ℓp Roe al-

gebra of X, denoted Bp(X ), is the closure of Cp[X ] in B(ℓp(Z,ℓp)).


If X is uniformly discrete, we define Cpu[X ] to be the subalgebra of B(ℓp(X ))

consisting of all finite propagation operators, and we define the ℓp uniform

Roe algebra of X, denoted Bpu(X ), to be the closure of C

pu[X ] in B(ℓp(X )).

One can check that just like in the p = 2 case, up to non-canonical isomor-phism, Bp(X ) does not depend on the choice of dense subspace Z, while upto canonical isomorphism, its K -theory does not depend on the choice of Z.Moreover, Bp(X ) is a coarse invariant, as noted in [8].

Remark 2.3. In the definition above, one may consider bounded operators onℓp(Z,E) for a fixed separable infinite-dimensional Lp space E. Recall thatwhen p ∈ [1,∞)\{2}, the separable infinite-dimensional Lp spaces are classi-fied as follows:

• Up to isometric isomorphism: ℓp, Lp[0,1], Lp[0,1]⊕pℓpn, Lp[0,1]⊕pℓ

p,where ℓ

pn denotes C

n with the ℓp norm.• Up to non-isometric isomorphism: ℓp, Lp[0,1].

Different choices of E may result in non-isomorphic Banach algebras. Forexample, if X is a point, then using ℓp in the definition results in Bp(X ) =K (ℓp), while using Lp[0,1] in the definition results in Bp(X ) = K (Lp[0,1]),and these two Banach algebras are non-isomorphic.

The domain of the ℓp coarse Baum-Connes assembly map involves the K -theory of localization algebras of Rips complexes of the metric space. We willnow define these notions and formulate the ℓp coarse Baum-Connes conjec-ture.

Definition 2.4. Let X be a proper metric space, and let Cp[X ] be its algebraic

Roe algebra. Let Cp

L[X ] be the algebra of bounded, uniformly continuous func-

tions f : [0,∞) →Cp[X ] such that prop( f (t))→ 0 as t →∞. Equip C

p

L[X ] with

the norm

|| f || := supt∈[0,∞)

|| f (t)||Bp(X ).

The completion of Cp

L[X ] under this norm, denoted by B

p

L(X ), is the ℓp local-

ization algebra of X.

Definition 2.5. Let (X , d) be a bounded geometry metric space, and let R > 0.

The Rips complex of X at scale R, denoted PR(X ), is the simplicial complex

with vertex set X and such that a finite set {x1, . . . , xn} ⊆ X spans a simplex if

and only if d(xi, x j)≤ R for all i, j =1, . . . , n.

Equip PR(X ) with the spherical metric defined by identifying each n-simplex

with the part of the n-sphere in the positive orthant, and equipping PR (X ) with

the associated length metric.

For any R > 0, there is a homomorphism

iR : K∗(Bp(PR (X )))→ K∗(Bp(X )),


(cf. [58, Lemma 2.8] for the p = 2 case) and the ℓp coarse Baum-Connesassembly map

µ : limR→∞

K∗(Bp

L(PR (X )))→ K∗(Bp(X ))

is defined to be the limit of the composition

K∗(Bp

L(PR(X )))

e∗→ K∗(Bp(PR (X )))

iR→ K∗(Bp(X )),

where e : Bp

L(PR (X ))→ Bp(PR(X )) is the evaluation-at-zero map.

The ℓp coarse Baum-Connes conjecture for a proper metric space X withbounded geometry is the statement that µ is an isomorphism.

It was shown in [61, Proposition 5.20] that the left-hand side of the ℓp

coarse Baum-Connes assembly map can be naturally identified with that ofthe original coarse Baum-Connes assembly map.

2.2. The assembly map and its image. Our main goal in this subsectionis to show that classes in the image of the ℓp coarse Baum-Connes assem-bly map can be represented by elements with finite propagation (at least forK0). We do this by considering a larger algebra generated by pseudolocal op-erators. With a little more work, we are also able to provide an alternativedescription of the assembly map, which may be of independent interest, andthe idea is based on the p = 2 case in [48] and [25, Appendix B].

For a proper metric space X , and a countable dense subset Z ⊆ X , theBanach space ℓp(Z,ℓp) is equipped with a natural multiplication action ofC0(X ) by restriction to Z.

Definition 2.6. Let X be a proper metric space, and fix a countable dense

subset Z ⊆ X. A bounded operator T on ℓp(Z,ℓp) is said to be pseudolocal if

the commutator [ f ,T] is a compact operator for every f ∈C0(X ).Let Dp[X ] be the algebra of all finite propagation, pseudolocal operators on

ℓp(Z,ℓp), and define D p(X ) to be the closure of Dp(X ) in B(ℓp(Z),ℓp).Let D

p

L[X ] be the algebra of all bounded, uniformly continuous functions

f : [0,∞) → Dp[X ] such that prop( f (t)) → 0 as t →∞, and define D

p

L(X ) to be

the completion of Dp

L[X ] under the supremum norm.

Note that D p(X ) and Dp

L(X ) are closed ideals in Bp(X ) and B

p

L(X ) respec-

tively for a proper metric space X with bounded geometry.If T is pseudolocal, then f T g is a compact operator for every pair of func-

tions f , g ∈C0(X ) with disjoint supports. This is because f T g = f [T, g]. Con-versely, we have the following version of Kasparov’s lemma (cf. [33, Proposi-tion 3.4]) whose proof follows closely that of [31, Lemma 5.4.7]:

Lemma 2.7. Let X be a proper metric space, let Z be a countable dense subset

of X, and let T ∈ B(ℓp(Z,ℓp)). Suppose that f T g is a compact operator for

every pair of functions f , g ∈C0(X ) with disjoint compact supports. Then T is

pseudolocal.


Proof. Note that under the stated hypotheses, f T g will be a compact oper-ator for every pair of bounded Borel functions f , g on X with disjoint com-pact supports (acting as multiplication operators on ℓp(Z,ℓp)). This is be-cause by Urysohn’s lemma there exist f ′, g′ ∈ C0(X ) with disjoint compactsupports such that f f ′ = f and gg′ = g. Moreover, it suffices to show that iff ∈C0(X ,R), then [ f ,T] may be approximated in norm by compact operators.

Let ε > 0 and f ∈ C0(X ,R). Cover the range of f by finitely many non-overlapping half-open intervals U1, . . . ,Un, each of diameter less than ε/||T||,such that Ui intersects U j if and only if |i− j| ≤ 1. Let f1, . . . , fn be the char-acteristic functions of the Borel sets f −1(U1), . . ., f −1(Un). Observe that

(1) if |i− j| > 1, then f iT f j is a compact operator, and(2) if f = f (x1) f1 +·· ·+ f (xn) fn, where xi ∈ f −1(Ui), then || f − f || < ε/||T||.

Now ||[ f ,T]− [ f ,T]|| < 2ε, and since∑n

i=1 f i = 1, we have

f T −T f =∑

i, j

f (xi) f iT f j − f iT f (x j) f j

=∑

|i− j|>1( f (xi)− f (x j)) f iT f j +

∑

|i− j|=1( f (xi)− f (x j)) f iT f j.

The first sum is a compact operator so we just need to estimate the norm ofthe second sum.

In the case where i = j+1, we get a sum of operators of the form

( f (x j+1)− f (x j)) f j+1T f j : ℓp(Z∩ f −1(U j),ℓp)→ ℓp(Z∩ f −1(U j+1),ℓp).

The norm of this sum is the maximum of the norms of its summands, whichis less than 2ε. The case where i = j−1 is similar, so

||∑

|i− j|=1( f (xi)− f (x j)) f iT f j|| < 4ε.

Hence [ f ,T] differs in norm from a compact operator by 6ε, and since ε> 0 isarbitrary, it follows that [ f ,T] is a norm limit of compact operators. �

For the rest of this subsection, we will assume that X is a proper metricspace with bounded geometry.

Proposition 2.8. The algebra Dp

L(X ) has trivial K-theory. Hence the bound-

ary map ∂ : K∗+1(D p

L(X )/Bp

L(X ))→ K∗(Bp

L(X )) is an isomorphism.

Proof. Let Z be a countable dense subset of X , and let E denote the ℓp directsum of countably infinitely many copies of ℓp(Z,ℓp). Let D

p

L(X ;E) denote the

algebra obtained in the same way as Dp

L(X ) but by considering operators on

E instead of operators on ℓp(Z,ℓp). Note that the “top left corner” inclusionD

p

L(X )→ D

p

L(X ;E) induces an isomorphism on K -theory.


For each n ∈N and a ∈ Dp

L(X ), let an(t) = a(t+n). Consider the homomor-

phisms µ,µ+1,µ0 : Dp

L(X )→ D

p

L(X ;E) defined by

µ(a)(t)= a(t)⊕a1(t)⊕a2(t)⊕·· · ,

µ+1(a)(t)= 0⊕a1(t)⊕a2(t)⊕·· · ,

µ0(a)(t)= a(t)⊕0⊕0⊕·· · .

Lemma 2.7 is used to show that these homomorphisms indeed take valuesin D

p

L(X ;E) (cf. [48, Lemma 3.7]). Using uniform continuity of a and also

conjugation by an isometry, note that µ and µ+1 induce the same map on K -theory. Moreover, µ0

∗ +µ+1∗ = (µ0 +µ+1)∗ = µ∗. Hence µ0

∗ = 0. But since µ0

induces an isomorphism on K -theory, we must have K∗(D p

L(X ))= 0. �

Now we want to show that classes in K1(D p

L(X )/Bp

L(X )) have finite prop-

agation representatives whose inverses also have finite propagation. Fromthis, we see that classes in K0(Bp

L(X )) have finite propagation representa-

tives. To simplify notation, given a Banach algebra A, we will write TA todenote the algebra of all bounded, uniformly continuous functions from [0,∞)to A, equipped with the supremum norm.

Note that TD p(X ) contains TBp(X ) as a closed ideal, while Dp

L(X ) contains

Bp

L(X ) as a closed ideal. Thus we have the following commutative diagram

of short exact sequences, where the upper vertical maps are induced by eval-uation at zero, while the lower vertical maps are induced by forgetting thecondition that propagation tends to zero.

0 −−→ Bp(X ) −−→ D p(X ) −−→ D p(X )/Bp(X ) −−→ 0x

xx

0 −−→ TBp(X ) −−→ TD p(X ) −−→ TD p(X )/TBp(X ) −−→ 0x

xx

0 −−→ Bp

L(X ) −−→ D

p

L(X ) −−→ D

p

L(X )/Bp

L(X ) −−→ 0

The proof of the next lemma involves truncating operators using a par-tition of unity in a way adapted to the ℓp norm and the dual ℓq norm (cf.[56, Lemma 6.3]) for p ∈ (1,∞).

Lemma 2.9. There is a contractive linear map Φ :TD p(X )→ Dp

L(X ) such that

(1) Φ(a)(t) depends only on a(t),(2) prop(Φ(a)(t))≤ prop(a(t)) for all a ∈TD p(X ),(3) Φ(a)(t)−a(t)∈Bp(X ) for all a ∈TD p(X ).

Moreover, Φ induces a homomorphism

Φ :TD p(X )/TBp(X )→ Dp

L(X )/Bp

L(X ).

Proof. For each positive integer n, let {Un,i}i be a locally finite open cover of X

such that each Un,i has diameter less than 1/n, and let {φn,i}i be a continuous


partition of unity subordinate to {Un,i}i. Let Z be a countable dense subset ofX , and let EX = ℓp(Z,ℓp). Define a linear map Φn : B(EX )→ B(EX ) by

Φn(T)=∞∑

i=1φ

1/qn,i Tφ

1/pn,i ,

where 1p+ 1

q= 1. Note that prop(Φn(T)) ≤ min{prop(T),1/n}. Moreover, if

T ∈ D p(X ), then T −Φn(T) =∑

i φ1/qn,i (φ1/p

n,i T −Tφ1/pn,i ) ∈ Bp(X ). If v ∈ EX and

w ∈ E∗X

have finite support, then

|⟨Φn(T)v,w⟩| ≤

∞∑

i=1|⟨Tφ

1/pn,i v,φ1/q

n,i w⟩|

≤ ||T||

∞∑

i=1||φ

1/pn,i v||p||φ

1/qn,i w||q

≤ ||T||

(∞∑

i=1||φ

1/pn,i v||

pp

)1/p (∞∑

i=1||φ

1/qn,i w||

qq

)1/q

= ||T||

(∞∑

i=1

∑

z∈Z

φn,i(z)||v(z)||p)1/p (

∞∑

i=1

∑

z∈Z

φn,i(z)||w(z)||q)1/q

= ||T||

(∑

z∈Z

∞∑

i=1φn,i(z)||v(z)||p

)1/p (∑

z∈Z

∞∑

i=1φn,i(z)||w(z)||q

)1/q

= ||T||

(∑

z∈Z

||v(z)||p)1/p (

∑

z∈Z

||w(z)||q)1/q

= ||T||||v||p||w||q ,

and it follows that ||Φn(T)|| ≤ ||T|| for each n.Now define Φ by linearly interpolating between the maps Φn, so that when

t ∈ [n, n+1] and a ∈TD p(X ),

Φ(a)(t)= (n+1− t)Φn+1(a(t))+ (t−n)Φn+2(a(t)).

It is straightforward to check that Φ has the required properties. �

Remark 2.10. The formula at the end of the proof results in

prop(Φ(a)(0))= prop(Φ1(a(0)))≤ 1.

We could also have used the formula

Φ′(a)(t)= (n+1− t)Φn+k(a(t))+ (t−n)Φn+k+1(a(t))

for any k ≥1, obtaining a map Φ′ with the desired properties but with

prop(Φ′(a)(0))= prop(Φk(a(0)))≤ 1/k.

Proposition 2.11. The homomorphism

ι : Dp

L(X )/Bp

L(X )→TD p(X )/TBp(X )

induced by inclusion is an isomorphism (with inverse Φ).


Proof. Given a ∈TD p(X ), we have Φ(a) ∈ Dp

L(X ). Write [a] and [Φ(a)] for the

images of a and Φ(a) in the respective quotients. Now t 7→ a(t)−Φ(a)(t) is inTBp(X ). Hence ι([Φ(a)])= [a], i.e., ι is surjective.

To show that ι is injective, let b ∈ Dp

L(X ) be such that b(t) ∈ Bp(X ) for

each t, i.e., ι([b]) = 0. Let (bn) be a sequence in Dp

L(X ) that converges to b

in Dp

L(X ). Since b(t) belongs to Bp(X ), so does Φ(b)(t) by (3) in the previous

lemma. Thus Φ(b) ∈ Bp

L(X ). On the other hand, b−Φ(b) is the norm limit of

the sequence (bn −Φ(bn)). Since each bn −Φ(bn) belongs to Bp

L(X ), so does

b−Φ(b). Hence b ∈Bp

L(X ), i.e., [b]= 0. �

Corollary 2.12. Every class in Dp

L(X )/Bp

L(X ) can be represented by an ele-

ment with finite propagation. Thus every class in K1(D p

L(X )/Bp

L(X )) can be

represented by an element with finite propagation and whose inverse also has

finite propagation.

From an explicit formula for the K -theory index map (see [10, Definition1.46 and Equation (1.43)]), we get the following:

Corollary 2.13. Every class in K0(Bp

L(X )) can be represented as a formal

difference [p]− [q], where p and q are idempotents with finite propagation in

matrix algebras over the unitization âBp

L(X ).

Hence, every class in the image of the ℓp coarse Baum-Connes assembly

map can be represented as a formal difference [p′]− [q′], where p′ and q′ are

idempotents with finite propagation in matrix algebras over the unitizationâBp(X ).

The remainder of this subsection is about an alternative description of theℓp coarse Baum-Connes assembly map.

Proposition 2.14. The homomorphisms

TBp(X )→ Bp(X ),

TD p(X )→ D p(X ),

TD p(X )/TBp(X )→ D p(X )/Bp(X ),

induced by evaluation at zero induce isomorphisms on K-theory.

Proof. Using the six-term exact sequence and the five lemma, it suffices toshow that the first two homomorphisms induce isomorphisms on K -theory.Since these homomorphisms are surjective, it suffices to show that their ker-nels

T0 A(X )= {a ∈TA(X ) : a(0)= 0}

have trivial K -theory, where A(X ) is either D p(X ) or Bp(X ).Given a ∈ A(X ), extend a to a function on R by defining a(t) = 0 for all

t < 0. For each n ∈ N, let an(t) = a(t− n). (Note that unlike in the proof ofProposition 2.8, we are now translating to the right.) Denote by A(X ;E)

EXPANDERS ARE COUNTEREXAMPLES TO THE ℓp COARSE BAUM-CONNES CONJECTURE11

the algebra obtained in the same way as A(X ) but by considering opera-tors on E =

⊕nℓ

p(Z,ℓp) instead of ℓp(Z,ℓp). Consider the homomorphismsµ,µ+1,µ0 :T0 A(X )→T0A(X ;E) defined by

µ(a)(t)= a(t)⊕a1(t)⊕a2(t)⊕·· · ,

µ+1(a)(t)= 0⊕a1(t)⊕a2(t)⊕·· · ,

µ0(a)(t)= a(t)⊕0⊕0⊕·· · .

Note that for any fixed t, all but finitely many of the terms an(t) are zero, so µ

and µ+1 indeed take values in T0 A(X ;E). A similar argument as in the proofof Proposition 2.8 shows that µ0

∗ = 0, and since µ0 induces an isomorphism onK -theory, we must have K∗(T0 A(X ))= 0. �

Combining the isomorphisms in Propositions 2.8, 2.11, and 2.14, we getthe following commutative diagram:

K∗+1(D p(X )/Bp(X ))∂

−−−−→ K∗(Bp(X ))

∼=

xe0

xe0

K∗+1(D p

L(X )/Bp

L(X ))

∼=−−−−→

∂K∗(Bp

L(X ))

In particular, ∂ : K∗+1(D p(X )/Bp(X )) → K∗(Bp(X )) is an isomorphism if andonly if e0 : K∗(Bp

L(X ))→ K∗(Bp(X )) is an isomorphism.

Applying this fact to each Rips complex and taking limits, we get

Theorem 2.15. The ℓp coarse Baum-Connes conjecture holds for X if and

only if the limit (as R →∞) of the compositions

K∗+1(D p(PR (X ))/Bp(PR (X )))∂→ K∗(Bp(PR (X )))

iR→ K∗(Bp(X ))

is an isomorphism.

To end this subsection, we note that failure of the ℓp coarse Baum-Connesconjecture can be detected by certain obstruction groups (cf. [6, 59, 60] whenp = 2).

Definition 2.16. For a proper metric space X with bounded geometry, define

Bp

L,0(X )= { f ∈Bp

L(X ) : f (0)= 0},

Dp

L,0(X )= { f ∈ Dp

L(X ) : f (0)= 0}.

Note that Bp

L,0(X ) is a closed ideal in Dp

L,0(X ). Moreover, we have the fol-lowing commutative diagram with exact rows and columns (cf. [6, Section


5]):0 0 0y

yy

0 −−→ Bp

L,0(X ) −−→ Dp

L,0(X ) −−→ Dp

L,0(X )/Bp

L,0(X ) −−→ 0y

yy

0 −−→ Bp

L(X ) −−→ D

p

L(X ) −−→ D

p

L(X )/Bp

L(X ) −−→ 0

ye0

ye0

ye0

0 −−→ Bp(X ) −−→ D p(X ) −−→ D p(X )/Bp(X ) −−→ 0y

yy

0 0 0Applying the K -theory functor and applying the earlier results, we then getthe following commutative diagram with exact rows and columns:

......

......

yy

yy

· · · −→K∗+1(D

p

L,0(X )

Bp

L,0(X )) −→ K∗(Bp

L,0(X )) −→K∗(D p

L,0(X )) −→K∗(D

p

L,0(X )

Bp

L,0(X )) −→···

yy

yy

· · · −→ K∗+1(D

p

L(X )

Bp

L(X )

)∼=−→ K∗(Bp

L(X )) −→ 0 −→ K∗(

Dp

L(X )

Bp

L(X )

) −→···

∼=

yy

y ∼=

y

· · · −→ K∗+1( D p(X )Bp (X ) ) −→ K∗(Bp(X )) −→ K∗(D p(X )) −→ K∗( D p(X )

Bp(X ) ) −→···y

yy

y...

......

...

It follows that K∗(D p

L,0(X )/Bp

L,0(X ))= 0 so

K∗(Bp

L,0(X ))∼= K∗(D p

L,0(X ))∼= K∗+1(D p(X )).

Corollary 2.17. The ℓp coarse Baum-Connes conjecture holds for X if and

only if one of the following vanishes:

• limR→∞ K∗(Bp

L,0(PR (X ))),

• limR→∞ K∗(D p

L,0(PR(X ))),• limR→∞ K∗(D p(PR (X ))).

In other words, any of the above can serve as an obstruction group for theℓp coarse Baum-Connes assembly map.


2.3. Equivariant assembly maps. We will also require equivariant ver-sions of the Roe algebra, localization algebra, and assembly map. This sub-section contains all the necessary information about these.

Definition 2.18. Let X be a proper metric space, and let Γ be a countable dis-

crete group acting freely and properly on X by isometries. Fix a Γ-invariant

countable dense subset Z ⊆ X and define Cp[X ] as above. The equivariant

algebraic Roe algebra of X, denoted Cp[X ]Γ, is the subalgebra of Cp[X ] con-

sisting of matrices (Tx,y)x,y∈Z satisfying Tgx,gy = Tx,y for all g ∈Γ and x, y∈ Z.

The equivariant ℓp Roe algebra of X, denoted Bp(X )Γ, is the closure of Cp[X ]Γ

in B(ℓp(Z,ℓp)).The equivariant ℓp localization algebra B

p

L(X )Γ is defined by considering

the algebra of bounded, uniformly continuous functions f : [0,∞) → Cp[X ]Γ

such that prop( f (t))→ 0 as t →∞, and completing in the norm

|| f || := supt∈[0,∞)

|| f (t)||Bp(X )Γ .

If Γ is a discrete group, then we may represent the group ring CΓ on ℓp(Γ)by left translation. Its completion, which we denote by B

pr (Γ), is the reduced

Lp group algebra of Γ, also known as the algebra of p-pseudofunctions on Γ

in the literature. When p = 2, it is the reduced group C∗-algebra.Just as in the p = 2 case, the equivariant ℓp Roe algebra of X as defined

above is related to the reduced Lp group algebra of Γ. Before making thisprecise, we recall some facts about Lp tensor products, details of which canbe found in [12, Chapter 7].

For p ∈ [1,∞), there is a tensor product of Lp spaces such that we have acanonical isometric isomorphism Lp(X ,µ)⊗Lp(Y ,ν)∼= Lp(X ×Y ,µ×ν), whichidentifies, for every ξ ∈ Lp(X ,µ) and η ∈ Lp(Y ,ν), the element ξ⊗η with thefunction (x, y) 7→ ξ(x)η(y) on X ×Y . Moreover, we have the following proper-ties:

• Under the identification above, the linear span of all ξ⊗η is dense inLp(X ×Y ,µ×ν).

• ||ξ⊗η||p = ||ξ||p||η||p for all ξ ∈ Lp(X ,µ) and η ∈ Lp(Y ,ν).• The tensor product is commutative and associative.• If a ∈ B(Lp(X1,µ1),Lp(X2,µ2)) and b ∈ B(Lp(Y1,ν1),Lp(Y2,ν2)), then

there exists a unique c ∈ B(Lp(X1 ×Y1,µ1 ×ν1),Lp(X2 ×Y2,µ2 ×ν2))such that under the identification above, c(ξ⊗η) = a(ξ)⊗ b(η) for allξ ∈ Lp(X1,µ1) and η ∈ Lp(Y1,ν1). We will denote this operator by a⊗b.Moreover, ||a⊗b|| = ||a||||b||.

• The tensor product of operators is associative, bilinear, and satisfies(a1⊗b1)(a2⊗b2)= a1a2⊗b1b2.

If A ⊆ B(Lp(X ,µ)) and B ⊆ B(Lp(Y ,ν)) are norm-closed subalgebras, we thendefine A⊗B⊆ B(Lp(X ×Y ,µ×ν)) to be the closed linear span of all a⊗b witha ∈ A and b ∈B.

Regarding Mn(C) as B(ℓpn), we may view Mn(A) as Mn(C)⊗A when A is an

Lp operator algebra and the tensor product is as described above (see Remark


1.14 and Example 1.15 in [46]). Writing Mp∞ for

⋃n∈N Mn(C)

B(ℓp), we see that

Mp∞ is a closed subalgebra of B(ℓp). Let Pn be the projection onto the first

n coordinates with respect to the standard basis in ℓp. When p ∈ (1,∞), wehave limn→∞ ||a−PnaPn|| = 0 for any compact operator a ∈ K (ℓp). It follows

that Mp∞ = K (ℓp) for p ∈ (1,∞). However, when p = 1, we can only say that

limn→∞ ||a−Pna|| = 0 for a ∈K (ℓ1). In fact, there is a rank one operator on ℓ1

that is not in M1∞. We refer the reader to Proposition 1.8 and Example 1.10

in [46] for details.The standard proof in the p = 2 case allows one to show that if A is an Lp

operator algebra for some p ∈ [1,∞), then

K∗(Mp∞⊗ A)∼= K∗(A).

In particular, when p ∈ (1,∞), we have

K∗(K (ℓp)⊗ A)∼= K∗(A).

We refer to [46, Lemma 6.6] for details.The following lemma is well-known when p = 2 (cf. [57, Lemma 3.7]).

Lemma 2.19. Let Γ be a discrete group acting freely, properly, and cocom-

pactly by isometries on a proper metric space X. Let Z ⊆ X be the countable

dense Γ-invariant subset used to define C[X ]Γ. Let D ⊆ Z be a precompact

fundamental domain for the Γ-action on Z. Then for p ∈ (1,∞) there is an

isomorphism

ψD : Bp(X )Γ → Bpr (Γ)⊗K (ℓp(D,ℓp)).

Moreover, the induced isomorphism on K-theory is independent of the choice

of D.

Proof. Let K f (ℓp(D,ℓp)) be the dense subalgebra of K (ℓp(D,ℓp)) consistingof those operators (Kx,y)x,y∈D with only finitely many nonzero matrix entries.For T ∈C

p[X ]Γ and g ∈Γ, define an element T(g) ∈ K f (ℓp(D,ℓp)) by the matrixformula

T(g)x,y := Tx,gy for all x, y∈ D.

Define a homomorphism

ψD :Cp[X ]Γ →CΓ⊙K f (ℓp(D,ℓp))

by the formulaT 7→

∑

g∈Γ

λg ⊙T(g).

Note that only finitely many T(g) are nonzero since T has finite propagation.Moreover, ψD is an isomorphism.

In fact, ψD is implemented by conjugating T by the isometric isomorphism

U : ℓp(Z,ℓp)→ ℓp(Γ)⊗ℓp(D,ℓp), ξ 7→∑

g∈Γ

δg ⊗χDUgξ,

i.e., ψD(T) = UTU−1, and so ψD extends to an isometric isomorphism be-tween the completions.


If D′ ⊆ Z is another precompact fundamental domain for the Γ-action, thenψD(T) and ψD ′(T) differ by conjugation by an invertible multiplier of B

pr (Γ)⊗

K (ℓp(D′,ℓp)), which induces the identity map on K -theory (cf. [31, Lemma4.6.1]). �

If Γ is a countable discrete group acting freely and properly on X by isome-tries, then by considering the equivariant versions of the localization algebraand the Roe algebra, we have the ℓp equivariant assembly map

limR→∞

K∗(Bp

L(PR(X ))Γ)→ K∗(Bp(X )Γ)∼= K∗(Bp

r (Γ)),

which is (a model for) the Baum-Connes assembly map for Γ when p = 2 [52].As discussed in [7, Section 6.2], the domain of the ℓp equivariant assembly

map can be identified with that of the classical Baum-Connes assembly map,similar to how it is done for the coarse assembly map in [61, Section 5].

Using involutive versions of the ℓp Roe algebra and localization algebra,we can further establish a relationship between the ℓp equivariant assemblymap and the classical Baum-Connes assembly map for a given group. Weshall use some terminology from [61, Section 5] for this purpose.

Given p ∈ (1,∞), and countable discrete measure spaces Z and Z′, theBanach spaces ℓp(Z) and ℓp(Z′) have canonical Schauder bases (e i)i∈Z and(e′

i)i∈Z′ respectively. Any bounded linear operator T ∈ B(ℓp(Z),ℓp(Z′)) may

be viewed as an infinite matrix with respect to the Schauder bases. We maythen consider the conjugate transpose matrix T∗.

We say that T ∈ B(ℓp(Z),ℓp(Z′)) is a dual-operator if T∗ is a bounded oper-ator from ℓp(Z′) to ℓp(Z). In this case, T may be regarded as a bounded linearoperator from ℓq(Z) to ℓq(Z′) with 1/p+1/q = 1. We say that T is a compactdual-operator if T and T∗ are compact operators from ℓp(Z) to ℓp(Z′), andfrom ℓp(Z′) to ℓp(Z), respectively. Given a dual-operator T, we define itsmaximal norm by ||T||max =max(||T||B(ℓp (Z),ℓp(Z′)), ||T∗||B(ℓp (Z′),ℓp(Z))).

One can now consider the following definition analogous to Definition 2.2and Definition 2.18.

Definition 2.20. Let X be a proper metric space, and fix a countable dense

subset Z ⊆ X. A dual-operator T = (Tx,y)x,y∈Z ∈ B(ℓp(Z,ℓp)) is said to be lo-

cally compact if

• each Tx,y is a compact dual-operator on ℓp;

• for every bounded subset B ⊆ X, the set

{(x, y)∈ (B×B)∩ (Z×Z) : Tx,y 6= 0}

is finite.

The dual ℓp Roe algebra of X, denoted by Bp,∗(X ), is the maximal norm clo-

sure of the algebra of all locally compact dual-operators on ℓp(Z,ℓp) with

finite propagation.

Let Γ be a countable discrete group acting freely and properly on X by

isometries. The equivariant dual ℓp Roe algebra, denoted by Bp,∗(X )Γ, is the

maximal norm closure of the algebra of all locally compact dual-operators on


ℓp(Z,ℓp) with finite propagation and satisfying Tgx,gy = Tx,y for all g ∈ Γ and

x, y∈ Z.

The completion of the group ring CΓ in the maximal norm

|| f ||max =max(|| f ||B(ℓp(Γ)), || f∗||B(ℓp (Γ))),

where f ∗(g) = f (g−1), is the involutive version of the reduced Lp group al-gebra of Γ, and is denoted by Bp,∗(Γ). One obtains the involutive version ofLemma 2.19 with essentially the same proof.

The dual ℓp localization algebra Bp,∗L

(X ) and the equivariant dual ℓp lo-calization algebra B

p,∗L

(X )Γ are defined in the obvious manner analogous toDefinition 2.4 and Definition 2.18.

One immediately sees that there are inclusion homomorphisms

Bp,∗(X )Γ → Bp(X )Γ,

Bp,∗L

(X )Γ → Bp

L(X )Γ.

Moreover, using the Riesz-Thorin interpolation theorem, one obtains contrac-tive homomorphisms

Bp,∗(X )Γ → C∗(X )Γ = B2(X )Γ,

Bp,∗L

(X )Γ → C∗L(X )Γ =B2

L(X )Γ.

When X is a finite-dimensional simplicial complex, one can show that thehomomorphisms between the localization algebras induce isomorphisms onK -theory as outlined in [7, Section 6.2] (cf. [61, Propositions 5.18 and 5.19]for the non-equivariant case).

Proposition 2.21. If the Baum-Connes assembly map for Γ is injective, and

the inclusion Bp,∗r (Γ) → B

pr (Γ) induces an injection on K-theory, then the ℓp

Baum-Connes assembly map for Γ is injective.

Proof. The result follows immediately from the following commutative dia-gram of assembly maps:

limR→∞ K∗(Bp

L(PR (X ))Γ)

e∗−−−−→ K∗(Bp(X )Γ)∼= K∗(Bp

r (Γ))

∼=

xx

limR→∞ K∗(Bp,∗L

(PR (X ))Γ)e∗

−−−−→ K∗(Bp,∗(X )Γ)∼= K∗(Bp,∗r (Γ))

∼=

yy

limR→∞ K∗(C∗L

(PR(X ))Γ)e∗

−−−−→ K∗(C∗(X )Γ)∼= K∗(C∗r (Γ))

The bottom horizontal arrow is the Baum-Connes assembly map for Γ. Thetop horizontal arrow is the ℓp Baum-Connes assembly map for Γ. The uppervertical arrow on the right is induced by inclusion, and the lower verticalarrow on the right is induced by complex interpolation. �


In [37], Liao and Yu introduced a Banach version of the rapid decay prop-erty for groups, and considered the question of when the inclusion B

p,∗r (Γ) →

Bpr (Γ) induces an isomorphism on K -theory.

Proposition 2.22. [37, Corollary 4.9] Let q0 ∈ [1,∞], q ∈ [1, q0], and 1/p +

1/q = 1. If Γ has property (RD)q0 , then the inclusion Bp,∗r (Γ) → B

pr (Γ) induces

an isomorphism on K-theory.

By [37, Theorem 4.4], the result applies to all groups with property RD.Moreover, groups with property RD in Lafforgue’s class C

′ satisfy the Baum-Connes conjecture [34, Corollaire 0.0.4].

Corollary 2.23. The ℓp Baum-Connes assembly map is injective for groups

with property RD in Lafforgue’s class C′.

Lafforgue’s class C′ includes groups acting properly and isometrically on

a strongly bolic, weakly geodesic, uniformly locally finite metric space, whichby [40] includes hyperbolic groups (and their subgroups). Hyperbolic groupsalso have property RD [11]. Thus the ℓp Baum-Connes assembly map isinjective for hyperbolic groups.

Next, we note that the results in the previous subsection can be adaptedto the equivariant case (cf. [25, Appendix B] when p = 2) and we shall omitthe details. We also have the following lemma relating the equivariant andnon-equivariant algebras, which when p = 2 gives the K -homology inductionisomorphism.

Lemma 2.24. Let Y be a compact metric space, and π : Y → Y a regular

(or Galois) cover with deck transformation group Γ. Then D p(Y )Γ/Bp(Y )Γ is

isomorphic to D p(Y )/Bp(Y ).

The proof is the same as [31, Proof of Lemma 12.5.4] so we omit it. Us-ing Corollary 2.12, the proof works in essentially the same way to give anisomorphism

Dp

L(Y )/Bp

L(Y )

∼=→ D

p

L(Y )Γ/Bp

L(Y )Γ.

By Proposition 2.8 and its equivariant analog, we then get an isomorphism

K∗(Bp

L(Y ))

∼=→ K∗(Bp

L(Y )Γ).

3. p-OPERATOR NORM LOCALIZATION AND A LIFTING HOMOMORPHISM

In this section, we introduce the p-operator norm localization property,which was defined for p = 2 in [5]. It enables us to get a bounded lifting ho-momorphism on the Roe algebra of the box space of a residually finite group.

Definition 3.1. Let (X ,ν) be a metric space equipped with a positive locally

finite Borel measure ν, let p ∈ [1,∞), and let E be an infinite-dimensional

Banach space. Let f :N→N be a non-decreasing function. We say that X has

the p-operator norm localization property relative to f (and E) with constant

0 < c ≤ 1 if for all r ∈ N and every T ∈ B(Lp(X ,ν;E)) with prop(T) ≤ r, there

exists a nonzero ξ ∈ Lp(X ,ν;E) satisfying


(1) diam(supp(ξ))≤ f (r),(2) ||Tξ|| ≥ c||T||||ξ||.

Definition 3.2. A metric space X is said to have the p-operator norm localiza-

tion property if there exists a constant 0< c ≤ 1 and a non-decreasing function

f :N→N such that for every positive locally finite Borel measure ν on X, (X ,ν)has the p-operator norm localization property relative to f with constant c.

Remark 3.3. (cf. [5, Proposition 2.4]) If a metric space has the p-operatornorm localization property, then it has the property with constant c for all0< c <1.

It was shown in [5, Proposition 4.1] that the metric sparsification propertyimplies the 2-operator norm localization property. For metric spaces withbounded geometry, it was shown in [4] that property A implies the metricsparsification property, and it was shown in [51] that property A is equivalentto the 2-operator norm localization property.

As noted in [56, Section 7], where a similar property called lower normlocalization was considered, the proof of [5, Proposition 4.1] can be adaptedwith the obvious modifications to yield the following.

Proposition 3.4. The metric sparsification property implies the p-operator

norm localization property for every p ∈ [1,∞).

The following corollary is obtained by combining the proposition with theresult in [51].

Corollary 3.5. If X is a metric space with bounded geometry, then the 2-

operator norm localization property implies the p-operator norm localization

property for all p ∈ [1,∞).

In particular, by [30, Lemma 4.3], if X has finite asymptotic dimension,then X has the p-operator norm localization property for all p ∈ [1,∞). Thisapplies to hyperbolic groups by [50].

Now we consider a lifting map φ defined in [57, Lemma 3.8] (and also in[5, Section 7]).

Definition 3.6. Let X be a metric space, and let π : X → X be a surjective

map. Let ε> 0. Then (X ,π) is called an ε-metric cover of X if for all x ∈ X , the

restriction of π to the open ball B(x,ε) of radius ε around x in X is an isometry

onto the open ball B(π(x),ε) of radius ε around π(x) in X.

Let G be a finitely generated, residually finite group with a sequence ofnormal subgroups of finite index N1 ⊇ N2 ⊇ ·· · such that

⋂i Ni = {e}. Let

äG =⊔

i G/Ni be the box space, i.e., the disjoint union of the finite quotientsG/Ni, endowed with a metric d such that its restriction to each G/Ni is thequotient metric, while d(G/Ni,G/N j)≥ i+ j if i 6= j.

Let T ∈Cp[äG] have propagation S, and let M be such that for all i, j ≥ M,

we have d(G/Ni,G/N j) ≥ 2S and πi : G → G/Ni is a 2S-metric cover. We maythen write T = T(0) ⊕

∏i≥M T(i), where T(0) ∈ B(ℓp(G/N1 ⊔ ·· · ⊔G/NM−1,ℓp)),


and each T(i) ∈ B(ℓp(G/Ni,ℓp)). For each i ≥ M, define an operator T(i) ∈

Cp[G]Ni by

T(i)x,y =

{T(i)πi(x),πi (y) if d(x, y)≤ S

0 otherwise,

and define φ(T) to be the image of∏

i≥M T(i) in∏

i Cp[G]Ni

⊕i C

p[G]Ni. This defines a

homomorphism

φ :Cp[äG]→∏

i Cp[G]Ni

⊕i C

p[G]Ni.

Lemma 3.7. If G has the p-operator norm localization property, then φ ex-

tends to a bounded homomorphism

φ : Bp(äG)→

∏∞i=1 Bp(|G|)Ni

⊕∞i=1 Bp(|G|)Ni

.

Proof. Suppose G has the p-operator norm localization property relative to f

with constant c. Let T ∈Cp[äG], and suppose T has propagation r. For each

sufficiently large i, there exists a nonzero ξ ∈ ℓp(G,ℓp) with diam(supp(ξ)) ≤f (r) and ||T(i)||||ξ|| ≥ ||T(i)ξ|| ≥ c||T(i)||||ξ||. Hence ||T|| ≥ ||T(i)|| ≥ c||T(i)|| forall such i, so ||φ(T)|| ≤ limsupi ||T

(i)|| ≤ 1c||T||. �

Definition 3.8. Let X be a proper metric space, and let Z be a countable dense

subset of X used to define Cp[X ]. An operator T ∈ Bp(X ) is said to be a ghost

if for all R,ε> 0, there exists a bounded set B ⊆ X such that if ξ ∈ ℓp(Z,ℓp) is

of norm one and supported in the open ball of radius R about some x ∉B, then

||Tξ|| < ε.

The proof of the following lemma is exactly the same as that of [57, Lemma5.5], and we include it for the convenience of the reader.

Lemma 3.9. Suppose that G has the p-operator norm localization property.

Let

φ : Bp(äG)→

∏∞i=1 Bp(|G|)Ni

⊕∞i=1 Bp(|G|)Ni

be the homomorphism in Lemma 3.7. Then φ(TG) = 0 for any ghost operator

TG .

Proof. Fix ε > 0. Let TG be a ghost operator, and let T ∈ Cp[äG] have prop-

agation R and be such that ||TG −T|| < ε. Let T(i) be as in the definition ofφ(T), and note that each T(i) has propagation at most R. Suppose G has thep-operator norm localization property relative to f with constant c. Then foreach i, there exists a nonzero ξi ∈ ℓp(G,ℓp) of norm one with support diame-ter at most f (R) such that

||T(i)ξi|| ≥ c||T(i)||.


On the other hand, for all sufficiently large i, there exists ξi ∈ ℓp(G/Ni,ℓp) ofnorm one such that ||T(i)ξi|| = ||T(i)ξi||. For such i, since TG is a ghost, wehave

c||T(i)|| ≤ ||T(i)ξi|| ≤ ||TG−T||+ ||TGξi|| < 2ε.

Hence

||φ(TG )|| < ε||φ||+ ||φ(T)|| ≤ ε||φ||+ limsupi

||T(i)|| < ε||φ||+2ε

c.

Since ε is arbitrary, and c is independent of ε, this completes the proof. �

Remark 3.10. One can consider the ℓp uniform Roe algebra Bpu(äG), defining

φu :Cpu[äG]→

∏iC

pu[G]Ni

⊕i C

pu[G]Ni

in a completely analogous manner using the same formula for the lifts of op-erators. If G has the p-operator norm localization property, then φu extendsto a bounded homomorphism

φu : Bpu(äG)→

∏i B

pu(|G|)Ni

⊕i B

pu(|G|)Ni

by essentially the same proof as that of Lemma 3.7. Moreover, if e is a fixedrank one idempotent operator on ℓp, then we have the following commutativediagram:

∏i B

pu (|G|)Ni

⊕i B

pu (|G|)Ni

∏i (·⊗e)

−−−−−→

∏i Bp(|G|)Ni

⊕i Bp(|G|)Ni

φu

xxφ

Bpu(äG)

·⊗e−−−−→ Bp(äG)

4. KAZHDAN PROJECTIONS IN THE ℓp ROE ALGEBRA

At this point an interlude is necessary in order to introduce and discussKazhdan projections in the setting of ℓp spaces. Our description follows thatof [13]. A representation π of G on a Banach space E induces a representationof CG on E by the formula

π( f )=∑

g∈G

f (g)πg,

for every f ∈ CG. For a faithful family F of representations of G on ℓp,consider the following norm on CG,

‖ f ‖F ,p = sup {‖π( f )‖ℓp : π ∈F } .

The completion of CG in this norm will be denoted Bp

F(G).

Recall that given an isometric representation π of a locally compact groupG on a reflexive Banach space E, the dual space E∗ is naturally equippedwith the representation πg = (π−1

g )∗. We have a canonical decomposition of πinto the trivial representation and its complement,

E = Eπ⊕π Eπ,


where Eπ is the subspace of invariant vectors of π, and Eπ = Ann((E∗)π) (cf.[1, Proposition 2.6 and Example 2.29]).

Definition 4.1. A Kazhdan projection p ∈ Bp

F(G) is an idempotent such that

π(p)∈ B(ℓp) is the projection onto (ℓp)π along (ℓp)π for every π ∈F .

Given a finite graph Γ= (V ,E), the edge boundary ∂A of a subset A ⊆ V isdefined to be the set of those edges E that have exactly one vertex in A. TheCheeger constant of Γ is then defined to be

h(Γ)= infA⊆V

#∂A

min{#A,#V \ A}.

Recall that a sequence of finite graphs {Γn} is a sequence of expanders if

infn

h(Γn)> 0.

See e.g. [32] for an overview.Let G be a residually finite group and let {Ni}i∈N be a family of finite index,

normal subgroups. Consider the box space X =⊔

i G/Ni, as before, and let πi

be the quasi-regular representation of G on ℓp(G/Ni). Fix F to denote thefamily of representations {πi}i∈N. We say that G has property τ with respectto the sequence {Ni} if the Cayley graphs G/Ni form a family of expanders,see [38]. The following is a special case of [13, Theorem 1.2].

Theorem 4.2. [13] Let G be a finitely generated group with property τ with

respect to the family {Ni}. Then for every 1 < p <∞, there exists a Kazhdan

idempotent in Bp

F(G).

Under the assumptions of the above theorem we have

Theorem 4.3. There exists a non-compact ghost idempotent Q =Q2 in Bp(X ).

Sketch of proof. Consider the projection qi =1

[G : Ni]Mi, where Mi is a square

matrix indexed by the elements G/Ni with all entries equal to 1. Then q =⊕qi belongs to Bp(X ). Indeed, by the construction in [13] one can choose a

finitely supported probability measure µ on G such that

‖µn− qi‖→ 0

uniformly in i. As µn are finite propagation operators, it suffices to takeQ to be the matrix defined by Q(x, y) = q(x, y)P, where P is some rank oneprojection on ℓp. �

5. MAIN RESULT

After recording a few more ingredients, we shall be ready for the proof ofthe main result.

For any compact metric space Y , the algebra Bp(Y ) is isomorphic to thealgebra of compact operators K (ℓp), so it admits a canonical densely definedtrace (see [42, Subsection 1.7.11]), which we denote by Tr.


While it follows from standard arguments that a bounded trace on a Ba-nach algebra A induces a map from K0(A) to R, we need to know that thesame is true for the densely defined trace above, and we now provide a justi-fication.

Let N denote the set of nuclear operators on ℓp, which is a Banach algebrawhen equipped with the nuclear norm.1 Then Tr is a bounded trace on N ,which induces a map Tr∗ : K0(N )→R.

We have bounded homomorphisms Mn → N for n ∈ N, which give homo-morphisms K0(Mn) → K0(N ). These homomorphisms then induce a homo-morphism

K0(K (ℓp))∼= lim−−→

K0(Mn)→ K0(N ),

which upon composing with Tr∗ gives a map K0(K (ℓp))→R.Let G be a residually finite group, let N1 ⊇ N2 ⊇ ·· · be a sequence of normal

subgroups of finite index such that⋂

i Ni = {e}, and let äG =⊔

i G/Ni be thebox space. Note that if n < R, then

PR(äG)= PR(n−1⊔

i=1G/Ni)⊔

⊔

i≥n

PR(G/Ni).

Each Ni acts properly on PR (G). Moreover, if B(e, r)∩Ni = {e}, then the actionof Ni on PR(G) is free, and π : PR(G) → PR (G)/Ni is a covering map (cf. [41,Lemma 4.2]). Since Ni has finite index in G, this covering is cocompact.Moreover, we have an isomorphism

K0(Bp

L(PR (äG)))∼= K0(Bp

L(PR (

n−1⊔

i=1G/Ni)))⊕

∏

i≥n

K0(Bp

L(PR(G/Ni))),

which is the cluster axiom for K -homology.Since d(G/Ni,G/N j)≥ i+ j if i 6= j, we also have a homomorphism

d : Bp(äG)→∏

i Bp(|G/Ni|)⊕i Bp(|G/Ni|)

∼=

∏i K (ℓp(G/Ni,ℓp))⊕i K (ℓp(G/Ni,ℓp))

,

which induces a homomorphism

d∗ : K0(Bp(äG))→ K0

( ∏i K (ℓp(G/Ni,ℓp))⊕i K (ℓp(G/Ni,ℓp))

)→

∏i K0(K (ℓp(G/Ni,ℓp)))⊕i K0(K (ℓp(G/Ni,ℓp)))

∼=

∏i Z⊕i Z

where the isomorphism at the end is induced by the trace Tr. This map canbe used to detect non-zero classes in K0(Bp(äG)).

The following proposition is a restatement of [23, Proposition 2.7] afteridentifying equivariant K -homology with the K -theory of equivariant local-ization algebras.

Proposition 5.1. If the classifying space for proper Γ-actions has finite ho-

motopy type, i.e., there is a model Z of a locally finite CW complex with uni-

versal proper Γ-action such that Z/Γ is a compact CW complex, then for any

r > 0, there is R > 0 such that the following is true: for any two elements

1When p = 2, this is exactly the algebra of trace class operators on ℓ2 with the trace norm.See [42, Subsection 1.7.11] for more details about nuclear operators on Banach spaces.


[x], [y] ∈ K∗(Bp

L(Pr(Γ))Γ

′

), where Γ′ is a subgroup of Γ with finite index, if

[x]= [y] in limr→∞ K∗(Bp

L(Pr(Γ))Γ

′

), then [x]= [y] in K∗(Bp

L(PR(Γ))Γ

′

).

By [39, Theorem 1], the proposition applies to hyperbolic groups. Now wehave all the necessary ingredients to prove our main result.

Theorem 5.2. Let p ∈ (1,∞). Let G be a residually finite hyperbolic group.

Let N1 ⊇ N2 ⊇ ·· · be a sequence of normal subgroups of finite index such that⋂i Ni = {e}. Assume that the box space äG =

⊔i G/Ni is an expander, i.e., that

G has property τ with respect to the family {Ni}. If q ∈Bp(äG) is the Kazhdan

projection associated to äG, then [q] ∈ K0(Bp(äG)) is not in the image of the

ℓp coarse Baum-Connes assembly map.

Proof. Since q =∏

i q(i) with q(i) ∈ B(ℓp(G/Ni,ℓp)) given by q(i)x,y =

1|G/Ni |

e11,where e11 is the rank one projection given by projecting onto the first coordi-nate, we have d∗[q]= [1,1,1, . . .]. In particular, [q] 6= 0 in K0(Bp(äG)).

On the other hand, since q is a ghost operator, φ(q) = 0 by Lemma 3.9 so

φ∗[q]= 0 in∏

i K0(Bp(|G|)Ni⊕

i K0(Bp(|G|)Ni.

Consider the following diagram for fixed R and n < R, in which the hori-zontal arrows are given by the respective assembly maps and the left verticalarrow is given by the induction isomorphisms.

0⊕∏

i≥n K0(Bp

L(PR (G))Ni ) −−−−→

∏i K0(Bp (|G|)Ni )⊕i K0(Bp (|G|)Ni )

0⊕∏

i≥n indi

xxφ∗

K0(Bp

L(PR (

⊔n−1i=1 G/Ni)))⊕

∏i≥n K0(Bp

L(PR(G/Ni))) −−−−→ K0(Bp(äG))

It follows from the definitions of the maps that the diagram commutes.Suppose [q] is in the image of the ℓp coarse Baum-Connes assembly map.

Then there exist r >0 and x ∈ K0(Bp

L(Pr(äG))) that maps to [q].

Set (yi)=φ∗[q]. Then there exists Mr such that yi = 0 in K0(Bp(|G|)Ni ) forall i ≥ Mr. Each Ni is hyperbolic so the ℓp Baum-Connes assembly map isinjective for each Ni by Corollary 2.23. Thus by Proposition 5.1, there existsR > 0 independent of the subgroups Ni, and there exists MR ≥ Mr such thatindi(xi)= 0 for all i ≥ MR , and thus xi = 0 for all i ≥ MR . But this contradicts[q] 6= 0. �

6. REMARKS AND OPEN QUESTIONS

In this final section, we list a few questions that we do not have the answerto and that may be of interest to the reader.

In the results above it was necessary to assume 1 < p < ∞. The mainreason for this assumption is the construction of Kazhdan projections andthe associated ghost projection in BP (X ). Indeed, the techniques used hereand originally in [13] require uniform convexity of the underlying Banachspace. Therefore the following question is natural in this context.


Question 6.1. What happens when p = 1? (How to construct Kazhdan pro-

jections?)

The above situation bears certain resemblance to the case of the Bost con-jecture, in which the right hand side of the Baum-Connes conjecture, namelythe K -theory of the reduced group C∗-algebra C∗

r (G) is replaced with the K -theory of the Banach algebra ℓ1(G).

It also seems that the argument used here would extend also to the case ofℓp(Z,E), where p > 1 and E is a uniformly convex Banach space, or a Banachspace of nontrivial type. In this case also ℓp(Z,E) is uniformly convex, orhas nontrivial type, respectively. Lafforgue [35] and Liao [36] showed that insuch cases there exist Kazhdan projections and it would therefore be possibleto construct the associated ghost projection. It is natural to state

Question 6.2. What happens if we consider operators on ℓp(Z,E) for Banach

spaces E other than ℓp in the definition of the Roe algebra?

As noted earlier, even E =Lp[0,1] results in an algebra that is not isomor-phic to the one we have used in this paper, although its K -theory may be thesame.

Our formulation of the ℓp coarse Baum-Connes assembly map is based ona straightforward modification of a model of the original coarse Baum-Connesassembly map from [58]. One can check that for each p and for a metric spaceX with bounded geometry, the functors X 7→ limR→∞ Kn(Bp

L(PR (X ))) form a

coarse homology theory in the sense of [29, Definition 2.3]. What is of inter-est, and which goes back to one of the original motivations for studying Lp

analogs of Baum-Connes type assembly maps, is to identify the left-hand sideof these assembly maps. The following question has been answered affirma-tively for finite-dimensional simplicial complexes in [61, Proposition 5.20].

Question 6.3. Is the K-theory of ℓp localization algebras associated to bounded

geometry metric spaces independent of p?

The next question is about the p-operator norm localization property. Asnoted in Corollary 3.5, a bounded geometry metric space with the 2-operatornorm localization property will have the p-operator norm localization prop-erty for all p ∈ [1,∞) but we do not know whether the converse holds.

Question 6.4. For p ∈ [1,∞) \ {2}, is the p-operator norm localization prop-

erty equivalent to the original (2-)operator norm localization property (equiv-

alently, Yu’s property A) for metric spaces with bounded geometry?

In this paper, we considered the problem of whether the ℓp coarse assem-bly map is surjective but we have not considered injectivity, whereas it wasshown in [57] for the p = 2 case that the coarse assembly map is injective forspaces of graphs with large girth.

Question 6.5. Is the ℓp coarse Baum-Connes assembly map injective for the

expanders considered in this paper?


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Yeong Chyuan Chung and Piotr W. Nowak November 27, 2018