Expanders are counterexamples to the coarse p -Baum-Connes conjecture
aa r X i v : . [ m a t h . K T ] N ov Expanders are counterexamples to the coarse p -Baum-Connes conjecture Yeong Chyuan Chung and Piotr W. NowakNovember 27, 2018
Abstract
We show that certain expanders are counterexamples to the coarse p -Baum-Connes conjecture. Contents p -operator norm localization, and surjectivity ofthe assembly map 8 φ L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 p -operator norm localization and the map φ . . . . . . . . . . . . 103.3 Kazhdan projections in B p ( X ) . . . . . . . . . . . . . . . . . . . . . 133.4 Traces and commutativity of the diagram . . . . . . . . . . . . . . 14 The coarse Baum-Connes conjecture, first formulated in [36], is a coarse ge-ometric analog of the original Baum-Connes conjecture for groups [1], and itposits that a certain coarse assembly map or index map is an isomorphismbetween a topological object involving the K -homology of Rips complexes of a1ounded 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, aswell 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 [44], gener-alizing his earlier work showing that it holds for spaces with finite asymp-totic dimension [43]. In [43], there is also a counterexample to the coarseNovikov conjecture when the condition of bounded geometry is omitted. Morerecent positive results on the coarse Baum-Connes conjecture include [9–12]by Fukaya and Oguni.On the other hand, Higson [19] showed that the coarse assembly mapis not surjective for certain Margulis-type expanders. Then in [20], 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 [3, 16, 17, 28].While most of the results mentioned in the previous paragraph only ap-ply to Margulis-type expanders, Willett and Yu in [41] 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 [39].In this paper, our goal is to consider an L p 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 [41]. It should be noted2hat techniques used in the C ∗ -algebraic setting often do not transfer to the L p setting in a straightforward manner.Although the L p analog of the coarse Baum-Connes conjecture has noknown geometric or topological applications when p
2, in light of interestin L p operator algebras in recent years (e.g. [4, 5, 8, 13–15, 18, 27, 31–34]), thestudy of assembly maps involving L p operator algebras contributes to ourgeneral understanding of the K -theory of some of these algebras. We alsonote that other assembly maps involving L p operator algebras have recentlybeen considered in [4, 8], as well as in unpublished work of Kasparov-Yu. Insimilar spirit, the Bost conjecture [24, 29, 30, 38] asks whether the Baum-Connes-type assembly map into the K -theory of the Banach algebra L ( G ) isan isomorphism for a locally compact group G .The description of the assembly map that we use is equivalent to that in[41] when p = K -homology to the K -theory of the localization algebra is an isomorphism for finite-dimensionalsimplicial complexes. Qiao and Roe [35] later showed that this isomorphismholds for general locally compact metric spaces. By considering L p analogs ofRoe algebras and localization algebras, we obtain an L p analog of the coarseBaum-Connes assembly map of the form µ : lim R →∞ K ∗ ( B pL ( P R ( X ))) → K ∗ ( B p ( X )),and the coarse p -Baum-Connes conjecture is the statement that this map isan isomorphism. Our main theorem is stated as follows: Theorem 1.
Let p ∈ (1, ∞ ) . Let G be a residually finite group with property (T)and the p-operator norm localization property. Let N ⊇ N ⊇ · · · be a sequenceof normal subgroups of finite index such that T i N i = { e } , and let ä G = F i G / N i be the box space. If q ∈ B p ( ä G ) is the Kazhdan projection associated to ä G,then [ q ] ∈ K ( B p ( ä G )) is not in the image of the coarse p-Baum-Connes as-sembly map. Acknowledgements
We would like to thank Rufus Willett for his illuminating 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).3
Preliminaries
Definition 2.
Let ( X , d ) be a metric space. We say that X is proper if allclosed balls in X are compact, and we say that X has bounded geometry if forall R > there exists N R ∈ N such that all balls of radius R have cardinalityat most N R .A net in X is a discrete subset Y ⊆ X such that there exists r > with theproperties that d ( x , y ) ≥ r for all x , y ∈ Y , and for any x ∈ X there is y ∈ Y withd ( x , y ) < r. We now associate certain L p operator algebras called Roe algebras toproper metric spaces. These algebras encode the large scale geometry of themetric space, and the K -theory of the Roe algebra serves as the target of thecoarse p -Baum-Connes assembly map. Definition 3.
Let X be a proper metric space, and fix a countable dense subsetZ ⊆ X . Let T be a bounded operator on ℓ p ( Z , ℓ p ) , and write T = ( T x , y ) x , y ∈ Z sothat each T x , y is a bounded operator on ℓ p . T is said to be locally compact if• each T x , y is a compact operator on ℓ p ;• for every bounded subset B ⊆ X , the set © ( x , y ) ∈ ( B × B ) ∩ ( Z × Z ) : T x , y ª is finite.The propagation of T is defined to be prop( T ) = inf © S > T x , y = for all x , y ∈ Z with d ( x , y ) > S ª . The algebraic Roe algebra of X , denoted C p [ X ] , is the subalgebra of B ( ℓ p ( Z , ℓ p )) consisting of all finite propagation, locally compact operators. The ℓ p Roe al-gebra of X , denoted B p ( X ) , is the closure of C p [ X ] in B ( ℓ p ( Z , ℓ p )) . One can check that just like in the p = B p ( 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, B p ( X ) is a coarse invariant, as noted in [5]. Remark 4.
In the definition above, one may consider bounded operators on ℓ p ( Z , E ) for a fixed separable infinite-dimensional L p space E . Recall thatwhen p ∈ [1, ∞ ) \ { } , the separable infinite-dimensional L p spaces are classi-fied as follows: 4 Up to isometric isomorphism: ℓ p , L p [0, 1], L p [0, 1] ⊕ p ℓ pn , L p [0, 1] ⊕ p ℓ p ,where ℓ pn denotes C n with the ℓ p norm.• Up to non-isometric isomorphism: ℓ p , L p [0, 1].Different choices of E may result in non-isomorphic Banach algebras. Forexample, if X is a point, then using ℓ p in the definition results in B p ( X ) = K ( ℓ p ), while using L p [0, 1] in the definition results in B p ( X ) = K ( L p [0, 1]),and these two Banach algebras are non-isomorphic. Definition 5.
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 -invariantcountable dense subset Z ⊆ X and define C p [ X ] as above. The equivariantalgebraic Roe algebra of X , denoted C p [ X ] Γ , is the subalgebra of C p [ X ] con-sisting of matrices ( T x , y ) x , y ∈ Z satisfying T gx , gy = T x , y for all g ∈ Γ and x , y ∈ Z.The equivariant ℓ p Roe algebra of X , denoted B p ( X ) Γ , is the closure of C p [ X ] Γ in B ( ℓ p ( Z , ℓ p )) . If Γ is a discrete group, then we may represent the group ring C Γ on ℓ p ( Γ )by left translation. Its completion, which we denote by B pr ( Γ ), is the reduced L p 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 = ℓ p Roe algebra of X as definedabove is related to the reduced L p group algebra of Γ . Before making thisprecise, we recall some facts about L p tensor products, details of which canbe found in [6, Chapter 7].For p ∈ [1, ∞ ), there is a tensor product of L p spaces such that we have acanonical isometric isomorphism L p ( X , µ ) ⊗ L p ( Y , ν ) ∼= L p ( X × Y , µ × ν ), whichidentifies, for every ξ ∈ L p ( X , µ ) and η ∈ L p ( Y , ν ), the element ξ ⊗ η with thefunction ( x , y ) 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 in L p ( X × Y , µ × ν ).• || ξ ⊗ η || p = || ξ || p || η || p for all ξ ∈ L p ( X , µ ) and η ∈ L p ( Y , ν ).• The tensor product is commutative and associative.• If a ∈ B ( L p ( X , µ ), L p ( X , µ )) and b ∈ B ( L p ( Y , ν ), L p ( Y , ν )), thenthere exists a unique c ∈ B ( L p ( X × Y , µ × ν ), L p ( X × Y , µ × ν ))such that under the identification above, c ( ξ ⊗ η ) = a ( ξ ) ⊗ b ( η ) for all ξ ∈ L p ( X , µ ) and η ∈ L p ( Y , ν ). We will denote this operator by a ⊗ b .Moreover, || a ⊗ b || = || a |||| b || . 5 The tensor product of operators is associative, bilinear, and satisfies( a ⊗ b )( a ⊗ b ) = a a ⊗ b b .If A ⊆ B ( L p ( X , µ )) and B ⊆ B ( L p ( Y , ν )) are norm-closed subalgebras, we thendefine A ⊗ B ⊆ B ( L p ( X × Y , µ × ν )) to be the closed linear span of all a ⊗ b with a ∈ A and b ∈ B .Regarding M n ( C ) as B ( ℓ pn ), we may view M n ( A ) as M n ( C ) ⊗ A when A is an L p operator algebra and the tensor product is as described above (see Remark1.14 and Example 1.15 in [32]). Writing M p ∞ for S n ∈ N M n ( C ) B ( ℓ p ) , we see that M p ∞ is a closed subalgebra of B ( ℓ p ). Let P n be the projection onto the first n coordinates with respect to the standard basis in ℓ p . When p ∈ (1, ∞ ), wehave lim n →∞ || a − P n aP n || = a ∈ K ( ℓ p ). It followsthat M p ∞ = K ( ℓ p ) for p ∈ (1, ∞ ). However, when p =
1, we can only say thatlim n →∞ || a − P n a || = a ∈ K ( ℓ ). In fact, there is a rank one operator on ℓ that is not in M ∞ . We refer the reader to Proposition 1.8 and Example 1.10in [32] for details.The standard proof in the p = A is an L p operator algebra for some p ∈ [1, ∞ ), then K ∗ ( M p ∞ ⊗ A ) ∼= K ∗ ( A ).In particular, when p ∈ (1, ∞ ), we have K ∗ ( K ( ℓ p ) ⊗ A ) ∼= K ∗ ( A ).We refer to [32, Lemma 6.6] for details.The following lemma is well-known when p = Lemma 6.
Let Γ be a discrete group acting freely, properly, and cocompactlyby 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 fundamentaldomain for the Γ -action on Z. Then for p ∈ (1, ∞ ) there is an isomorphism ψ D : B p ( X ) Γ → B pr ( Γ ) ⊗ K ( ℓ p ( D , ℓ p )). Moreover, the induced isomorphism on K -theory is independent of the choiceof D.Proof.
Let K f ( ℓ p ( D , ℓ p )) be the dense subalgebra of K ( ℓ p ( D , ℓ p )) consistingof those operators ( K x , 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 ( ℓ p ( D , ℓ p )) by the matrixformula T ( g ) x , y : = T x , gy for all x , y ∈ D .6efine a homomorphism ψ D : C p [ X ] Γ → C Γ ⊙ K f ( ℓ p ( D , ℓ p ))by the formula T X g ∈ Γ λ g ⊙ T ( g ) .Note that only finitely many T ( g ) are nonzero since T has finite propagation.Moreover, ψ D is an isomorphism.In fact, ψ D is implemented by conjugating T by the isometric isomor-phism U : ℓ p ( Z , ℓ p ) → ℓ p ( Γ ) ⊗ ℓ p ( D , ℓ p ), ξ X g ∈ Γ δ g ⊗ χ D U g ξ ,i.e., ψ D ( T ) = U TU − , and so ψ D extends to an isometric isomorphism be-tween the completions.If D ′ ⊆ Z is another precompact fundamental domain for the Γ -action,then ψ D ( T ) and ψ D ′ ( T ) differ by conjugation by an invertible multiplier of B pr ( Γ ) ⊗ K ( ℓ p ( D ′ , ℓ p )), which induces the identity map on K -theory (cf. [22,Lemma 4.6.1]).The domain of the coarse p -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 coarse p -Baum-Connes conjec-ture. Definition 7.
Let X be a proper metric space, and let C p [ X ] be its algebraicRoe algebra. Let C pL [ X ] be the algebra of bounded, uniformly continuous func-tions f : [0, ∞ ) → C p [ X ] such that prop( f ( t )) → as t → ∞ . Equip C pL [ X ] withthe norm || f || : = sup t ∈ [0, ∞ ) || f ( t ) || B p ( X ) . The completion of C pL [ X ] under this norm, denoted by B pL [ X ] , is the ℓ p local-ization algebra of X .Moreover, if Γ is a countable discrete group acting freely and properly onX by isometries, we define the equivariant ℓ p localization algebra B pL ( X ) Γ inthe same way by considering functions f : [0, ∞ ) → C p [ X ] Γ instead, and com-pleting in the norm || f || : = sup t ∈ [0, ∞ ) || f ( t ) || B p ( X ) Γ .7 efinition 8. Let ( X , d ) be a bounded geometry metric space, and let R > .The Rips complex of X at scale R, denoted P R ( X ) , is the simplicial complexwith vertex set X and such that a finite set { x , . . ., x n } ⊆ X spans a simplex ifand only if d ( x i , x j ) ≤ R for all i , j =
1, . . ., n.Equip P R ( X ) with the spherical metric defined by identifying each n-simplexwith the part of the n-sphere in the positive orthant, and equipping P R ( X ) withthe associated length metric. For any R >
0, there is a homomorphism i R : K ∗ ( B p ( P R ( X ))) → K ∗ ( B p ( X )),(cf. [42, Lemma 2.8] for the p = p -Baum-Connes as-sembly map µ : lim R →∞ K ∗ ( B pL ( P R ( X ))) → K ∗ ( B p ( X ))is defined to be the limit of the composition K ∗ ( B pL ( P R ( X ))) e ∗ → K ∗ ( B p ( P R ( X ))) i R → K ∗ ( B p ( X )),where e : B pL ( P R ( X )) → B p ( P R ( X )) is the evaluation-at-zero map.The coarse p -Baum-Connes conjecture for a bounded geometry metricspace X is the statement that µ is an isomorphism.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 L p equivariant assembly maplim R →∞ K ∗ ( B pL ( P R ( X )) Γ ) → K ∗ ( B p ( X ) Γ ) ∼= K ∗ ( B pr ( Γ )),which is (a model for) the Baum-Connes assembly map for Γ when p = p -operator norm localization, andsurjectivity of the assembly map Definition 9.
Let X be a metric space, and let π : ˜ X → X be a surjective map.Let ε > . Then ( ˜ X , π ) is called an ε -metric cover of X if for all x ∈ ˜ X , therestriction of π to the open ball B ( x , ε ) of radius ε around x in ˜ X is an isometryonto the open ball B ( π ( x ), ε ) of radius ε around π ( x ) in X . .1 The map φ L We first consider a lifting/induction map based on the one considered by Wil-lett and Yu in [41, page 1403].Let Y be a finite CW complex, and let π : ˜ Y → Y be a normal coveringspace with deck transformation group Γ . Let ε > π : e Y → Y is an ε -metric cover. Let T be a locally compact operator on ℓ p ( Z , ℓ p ) thathas propagation less than ε /2, where Z is a countable dense subset of Y con-taining a finite net { z i } Ni = in Y with bounded geometry. Consider a cover U = { U i } Ni = of Y by uniformly bounded, pairwise disjoint Borel sets of diame-ter less than ε /2 such that z i ∈ U i for each i ∈ I , and let χ i be the characteristicfunction of U i . Define T i , j = χ i T χ j : χ j ℓ p ( Z , ℓ p ) = ℓ p ( U j ∩ Z , ℓ p ) → ℓ p ( U i ∩ Z , ℓ p ) = χ i ℓ p ( Z , ℓ p ).The cover U can be lifted to a cover f U of the covering space e Y by setting e U i = π − ( U i ). Then e U i = F g ∈ G U i , g ∼= U i × Γ , where π restricted to each U i , g isan isometry onto U i .This allows us to lift the operator T i , j to the operator g T i , j : χ j , h ( ℓ p ( Γ ) ⊗ ℓ p ( Z ) ⊗ ℓ p ) → χ i , g ( ℓ p ( Γ ) ⊗ ℓ p ( Z ) ⊗ ℓ p ),where χ i , g is the characteristic function of U i , g , i.e., g T i , j may be identifiedwith T i , j via the following diagram: χ j , h ( ℓ p ( Γ ) ⊗ ℓ p ( Z ) ⊗ ℓ p ) g T i , j −−−−→ χ i , g ( ℓ p ( Γ ) ⊗ ℓ p ( Z ) ⊗ ℓ p ) °°° °°° ℓ p ( { h } ) ⊗ ℓ p ( U j ∩ Z , ℓ p ) T i , j −−−−→ ℓ p ( { g } ) ⊗ ℓ p ( U i ∩ Z , ℓ p )Let now L δ [ Y ] denote the set of locally compact operators on ℓ p ( Z , ℓ p )of propagation less than δ ; let also L δ [ e Y ] Γ denote the set of locally compact, Γ -equivariant operators on ℓ p ( Γ ) ⊗ ℓ p ( Z ) ⊗ ℓ p of propagation less than δ .For an operator T ∈ L ε /2 [ Y ], define its lift φ L ( T ) by the formula φ L ( T ) ( i , g ),( j , h ) = ½ g T i , j d ( U i , g , U j , h ) < ε φ L ( T ) ∈ L ε /2 [ e Y ] Γ .Note that if S , T ∈ L ε /2 [ Y ] are such that ST ∈ L ε /2 [ Y ], then φ L ( ST ) = φ L ( S ) φ L ( T ). Also, by the definition of φ L , if f : [0, ∞ ) → L ε /2 [ Y ] is a bounded,9niformly continuous function, then so is φ L ◦ f : [0, ∞ ) → L ε /2 [ e Y ] Γ , and thus φ L ◦ f ∈ B pL ( ˜ Y ) Γ if f ∈ B pL ( Y ). Indeed, this follows from the fact that since e Y contains a net with bounded geometry { z i , g } ≤ i ≤ N , g ∈ Γ such that z i , g ∈ U i , g , if T ∈ L δ [ e Y ], then there exists c δ > || φ L ( T ) || ≤ c δ sup ( i , g ),( j , h ) || φ L ( T ) ( i , g ),( j , h ) || = c δ sup i , j || T i , j || (see [28, Lemma 2.6] for the p = p ).Every class in K ∗ ( B pL ( Y )) can be represented by a bounded, uniformlycontinuous function f : [0, ∞ ) → L ε /4 [ Y ] such that prop( f ( s )) → s → ∞ .Moreover, if [ f ] = [ f ] in K ( B pL ( Y )), then (up to stabilization) there is apiecewise linear homotopy of quasi-idempotents between f and f . In par-ticular, there is a homotopy ( f t ) t ∈ [0,1] such that f t is a function taking val-ues in L ε /4 [ Y ] for each t ∈ [0, 1]. Then ( φ L ◦ f t ) t ∈ [0,1] is a homotopy of quasi-idempotents between φ L ◦ f and φ L ◦ f . Hence φ L induces a homomorphism K ( B pL ( Y )) → K ( B pL ( ˜ Y ) Γ ), which we shall still denote by φ L . p -operator norm localization and the map φ Definition 10.
Let ( X , ν ) be a metric space equipped with a positive locallyfinite Borel measure ν , let p ∈ [1, ∞ ) , and let E be an infinite-dimensionalBanach space. Let f : N → N be a non-decreasing function. We say that X hasthe p-operator norm localization property relative to f (and E) with constant < c ≤ if for all r ∈ N and every T ∈ B ( L p ( X , ν ; E )) with prop( T ) ≤ r, thereexists a nonzero ξ ∈ L p ( X , ν ; E ) satisfying1. diam(supp( ξ )) ≤ f ( r ) ,2. || T ξ || ≥ c || T |||| ξ || . Definition 11.
A metric space X is said to have the p-operator norm localiza-tion property if there exists a constant < c ≤ and a non-decreasing functionf : 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 12. (cf. [3, Proposition 2.4]) If a metric space has the p -operatornorm localization property, then it has the property with constant c for all0 < c < Proposition 13.
The metric sparsification property implies the p-operatornorm localization property for every p ∈ [1, ∞ ) . The following corollary is obtained by combining the proposition with theresult in [37].
Corollary 14.
If X is a metric space with bounded geometry, then the 2-operator norm localization property implies the p-operator norm localizationproperty for all p ∈ [1, ∞ ) . We consider another lifting map φ defined in [41, Lemma 3.8] (and alsoin [3, Section 7]). Let G be a finitely generated, residually finite group with asequence of normal subgroups of finite index N ⊇ N ⊇ · · · such that T i N i = { e } . Let ä G = F i G / N i be the box space, i.e., the disjoint union of the finitequotients G / N i , endowed with a metric d such that its restriction to each G / N i is the quotient metric, while d ( G / N i , G / N j ) ≥ i + j if i j .Let T ∈ C p [ ä G ] have propagation S , and let M be such that for all i , j ≥ M ,we have d ( G / N i , G / N j ) ≥ S and π i : G → G / N i is a 2 S -metric cover. We maythen write T = T (0) ⊕ Q i ≥ M T ( i ) , where T (0) ∈ B ( ℓ p ( G / N ⊔ · · · ⊔ G / N M − , ℓ p )),and each T ( i ) ∈ B ( ℓ p ( G / N i , ℓ p )). For each i ≥ M , define an operator g T ( i ) ∈ C p [ G ] N i by g T ( i ) x , y = ( T ( i ) π i ( x ), π i ( y ) if d ( x , y ) ≤ S φ ( T ) to be the image of Q i ≥ M g T ( i ) in Q i C p [ G ] Ni L i C p [ G ] Ni . This defines ahomomorphism φ : C p [ ä G ] → Q i C p [ G ] N i L i C p [ G ] N i . Lemma 15.
If G has the p-operator norm localization property, then φ ex-tends to a bounded homomorphism φ : B p ( ä G ) → Q ∞ i = B p ( | G | ) N i L ∞ i = B p ( | G | ) N i .11 roof. Suppose G has the p -operator norm localization property relative to f with constant c . Let T ∈ C p [ ä G ], and suppose T has propagation r . For eachsufficiently large i , there exists a nonzero ξ ∈ ℓ p ( G , ℓ p ) with diam(supp( ξ )) ≤ f ( r ) and || T ( i ) |||| ξ || ≥ || g T ( i ) ξ || ≥ c || g T ( i ) |||| ξ || . Hence || T || ≥ || T ( i ) || ≥ c || g T ( i ) || forall such i , so || φ ( T ) || ≤ lim sup i || g T ( i ) || ≤ c || T || . Definition 16.
Let X be a proper metric space, and let Z be a countable densesubset of X used to define C p [ X ] . An operator T ∈ B p ( X ) is said to be a ghostif for all R , ε > , there exists a bounded set B ⊆ X such that if ξ ∈ ℓ p ( Z , ℓ p ) isof norm one and supported in the open ball of radius R about some x ∉ B, then || T ξ || < ε . The proof of the following lemma is exactly the same as that of [41, Lemma5.5], and we include it for the convenience of the reader.
Lemma 17.
Suppose that G has the p-operator norm localization property.Let φ : B p ( ä G ) → Q ∞ i = B p ( | G | ) N i L ∞ i = B p ( | G | ) N i be the homomorphism in Lemma 15. Then φ ( T G ) = for any ghost operatorT G .Proof. Fix ε >
0. Let T G be a ghost operator, and let T ∈ C p [ ä G ] have prop-agation R and be such that || T G − T || < ε . Let g T ( i ) be as in the definition of φ ( T ), and note that each g T ( i ) has propagation at most R . Suppose G has the p -operator norm localization property relative to f with constant c . Then foreach i , there exists a nonzero e ξ i ∈ ℓ p ( G , ℓ p ) of norm one with support diame-ter at most f ( R ) such that || g T ( i ) e ξ i || ≥ c || g T ( i ) || .On the other hand, for all sufficiently large i , there exists ξ i ∈ ℓ p ( G / N i , ℓ p ) ofnorm one such that || g T ( i ) e ξ i || = || T ( i ) ξ i || . For such i , since T G is a ghost, wehave c || g T ( i ) || ≤ || T ( i ) ξ i || ≤ || T G − T || + || T G ξ i || < ε .Hence || φ ( T G ) || < ε + || φ ( T ) || ≤ ε + lim sup i || g T ( i ) || < ε + ε c .Since ε is arbitrary, and c is independent of ε , this completes the proof.12 .3 Kazhdan projections in B p ( X ) At this point an interlude is necessary in order to introduce and discuss Kazd-han projections in the setting of ℓ p -spaces. Our description follows that of [7].A representation π of G on a Banach space E induces a representation of C Γ on E by the formula π ( f ) = X g ∈ G f ( g ) π g ,for every f ∈ C Γ . On C Γ consider the following norm k f k max, p = sup © k π ( f ) k ℓ p : π isometric representation of G on ℓ p ª .The completion of C Γ in this norm will be denoted B p max ( G ) and called the ℓ p -maximal group Banach algebra.Recall that given an isometric representation π of a locally compact group Γ on a Banach space E the dual space E ∗ is naturally equipped with therepresentation π g = ( π − g ) ∗ . We have a canonical decomposition of π into thetrivial representation and its complement, E = E π ⊕ π E π ,where E π is the subspace of invariant vectors of π and E π = Ann(( E ∗ ) π ). Definition 18.
A Kazhdan projection p ∈ B p max ( Γ ) is an idempotent such that π ( p ) ∈ B ( ℓ p ) is the projection onto ( ℓ p ) π along ( ℓ p ) π . The following is a special case of [7, Theorem 1.2].
Theorem 19 ([7]) . Let Γ be a finitely generated group with property ( T ) . Thenfor every < p < ∞ , there exists a Kazhdan idempotent in B p max ( Γ ) . Let G be a residually finite group and let { N i } i ∈ N be a family of finiteindex, normal subgroups. Consider the box space X = F i G / N i , as before. Wehave Theorem 20.
There exists a non-compact ghost idempotent Q = Q in B p ( X ) .Sketch of proof. Consider the projection q i = G : N i ] M i , where M i is a squarematrix indexed by the elements G / N i with all entries equal to 1. Then q = L q i belongs to B p ( X ). Indeed, by the construction in [7] one can choose afinitely supported probability measure µ on G such that k µ n − q i k → i . As µ n are finite propagation operators, it suffices to take Q to be the matrix defined by Q ( x , y ) = q ( x , y ) P , where P is some finite rankprojection on ℓ p . 13iven 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 ) = inf A ⊆ V ∂ A min { A , V \ A } .Recall that a sequence of finite graphs { Γ n } is a sequence of expanders ifinf n h ( Γ n ) > G / N i form a sequence ofexpanders. See e.g. [23] for an overview. For any compact metric space Y , the algebra B p ( Y ) is isomorphic to the al-gebra of compact operators K ( ℓ p ), so it admits a canonical unbounded trace T r : B p ( Y ) → C ∪ { ∞ } , which induces a map on K -theory T r ∗ : K ( B p ( Y )) → R .If ˜ Y is a normal covering space of Y with deck transformation group Γ , then B p ( ˜ Y ) Γ is isomorphic to B pr ( Γ ) ⊗ K ( ℓ p ) by Lemma 6, so it admits an unboundedtrace τ : B p ( ˜ Y ) Γ → C ∪ { ∞ } obtained by taking the tensor product of the traceson B pr ( Γ ) and K ( ℓ p ). This then induces a map on K -theory τ ∗ : K ( B p ( ˜ Y ) Γ ) → R . Lemma 21.
Let Y be a finite CW complex, and π : ˜ Y → Y a normal coveringspace with deck transformation group Γ . Then we have the following commu-tative diagram: K ( B pL ( ˜ Y ) Γ ) e ∗ −−−−→ K ( B p ( ˜ Y ) Γ ) τ ∗ −−−−→ R φ L x °°° K ( B pL ( Y )) e ∗ −−−−→ K ( B p ( Y )) Tr ∗ −−−−→ R Proof.
It suffices to check that τ ( φ L ( S )) = T r ( S ) for S with small propagation.Indeed, if { U i } Ni = is the Borel cover of Y used in the definition of φ L , so that S = ( S i , j ) Ni = with S i , j = χ U i S χ U j , and φ L ( S ) = ( φ L ( S ) ( i , g ),( j , h ) ) ≤ i ≤ N , g ∈ Γ , then τ ( φ L ( S )) = T r ([ φ L ( S ) ( i , e ),( j , e ) ]) = N X i = T r ( g S i , i ) = N X i = T r ( S i , i ) = T r ( S ).14e will apply the lemma in the case where Y is the Rips complex of afinite quotient G / N of a residually finite group G , e Y is the Rips complex of G ,and Γ = N .Note that if n < R , then P R ( ä G ) = P R ( n − G i = G / N i ) ⊔ G i ≥ n P R ( G / N i ).Each N i acts properly on P R ( G ). Moreover, if B ( e , r ) ∩ N i = { e } , then the actionof N i on P R ( G ) is free, and π : P R ( G ) → P R ( G )/ N i is a covering map (cf. [28,Lemma 4.2]). Since N i has finite index in G , this covering is cocompact.For each i , let τ ( i ) ∗ : K ( B p ( | G | ) N i ) → R be the map induced by the trace τ ( i ) as discussed above. We may then define a homomorphism T = Q τ ( i ) ∗ L τ ( i ) ∗ : Q i K ( B p ( | G | ) N i ) L i K ( B p ( | G | ) N i ) → Q i R L i R We also have a homomorphism d : B p ( ä G ) → Q i K ( ℓ p ( G / N i , ℓ p )) L i K ( ℓ p ( G / N i , ℓ p )) ,which induces a homomorphism d ∗ : K ( B p ( ä G )) → K µ Q i K ( ℓ p ( G / N i , ℓ p )) L i K ( ℓ p ( G / N i , ℓ p )) ¶ ∼= Q i Z L i Z Lemma 22.
If p is an idempotent in B p ( ä G ) such that [ p ] ∈ K ( B p ( ä G )) is inthe image of the coarse p-Baum-Connes assembly map µ : lim R →∞ K ( B pL ( P R ( ä G ))) → K ( B p ( ä G )), then T ( φ ∗ ([ p ])) = d ∗ [ p ] ∈ Q i R L i R . Proof.
Fix R >
0. As discussed above, for n < R , we have P R ( ä G ) = P ⊔ G i ≥ n P R ( G / N i ),where P = P R ( F n − i = G / N i ). For large R , we may assume that n is largeenough so that N i acts freely and properly on P R ( G ), and P R ( G )/ N i = P R ( G / N i )for all i ≥ n . In particular, elements in K ( B pL ( P R ( ä G ))) may be regarded aselements in K ( B pL ( P )) ⊕ Q i ≥ n K ( B pL ( P R ( G / N i ))).15ow consider the following diagram: Q i ≥ n K ( B pL ( P R ( G )) N i ) Q e ∗ −−−−→ K ( Q i ≥ n B p ( | G | ) N i ) −−−−→ K ³ Q i B p ( | G | ) Ni L i B p ( | G | ) Ni ´ ⊕ Q i ≥ n φ L x φ ∗ x K ( B pL ( P R ( ä G ))) e ∗ −−−−→ K ( B p ( P R ( ä G ))) −−−−→ K ( B p ( ä G ))This diagram commutes by the way the maps φ L and φ are defined. We thenobtain the result by applying Lemma 21. Theorem 23.
Let p ∈ (1, ∞ ) . Let G be a residually finite group with prop-erty (T) and the p-operator norm localization property. Let N ⊇ N ⊇ · · · bea sequence of normal subgroups of finite index such that T i N i = { e } , and let ä G = F i G / N i be the box space. If q ∈ B p ( ä G ) is the Kazhdan projection asso-ciated to ä G, then [ q ] ∈ K ( B p ( ä G )) is not in the image of the coarse p-Baum-Connes assembly map.Proof. Since q = Q q ( i ) with q ( i ) ∈ B ( ℓ p ( G / N i , ℓ p )) given by q ( i ) x , y = | G / N i | Q ,where Q ∈ B ( ℓ p ) is a fixed rank one projection, we have d ∗ [ q ] = [1, 1, 1, . . .].On the other hand, since q is a ghost operator, φ ( q ) = 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 and theassociated ghost projection in B P ( X ). Indeed, the techniques used here andoriginally in [7] require uniform convexity of the underlying Banach space.Therefore the following question is natural in this context. Question 24.
What happens when p = ? (How to construct Kazhdan projec-tions?) 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 ℓ ( G ).It also seems that the argument used here would extend also to the case of ℓ p ( Z , E ), where p > E is a uniformly convex Banach space, or a Banach16pace of nontrivial type. In this case also ℓ p ( Z , E ) is uniformly convex, orhas nontrivial type, respectively. Lafforgue [25] and Liao [26] 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 25.
What happens if we consider operators on ℓ p ( Z , E ) for Banachspaces E other than ℓ p in the definition of the Roe algebra? As noted earlier, even E = L p [0, 1] results in an algebra that is not iso-morphic to the one we have used in this paper, although its K -theory may bethe same.Our formulation of the coarse p -Baum-Connes assembly map is based ona straightforward modification of a model of the original coarse Baum-Connesassembly map from [42]. One can check that for each p and for a metricspace X with bounded geometry, the functors X lim R →∞ K n ( B pL ( P R ( X )))form a coarse homology theory in the sense of [21, Definition 2.3]. What is ofinterest, and which goes back to one of the original motivations for studying L p analogs of Baum-Connes type assembly maps, is to identify the left-handside of these assembly maps. Question 26.
Is the K -theory of ℓ p localization algebras associated to boundedgeometry metric spaces independent of p? It is not clear to us how to approach this question even for finite dimen-sional simplicial complexes. For instance, one certainly has a Mayer-Vietorissequence for each p , but it is not clear how to connect these sequences fordifferent p .The next question is about the p -operator norm localization property. Asnoted in Corollary 14, 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 27.
For p ∈ [1, ∞ ) \ { } , 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? References [1] P. Baum, A. Connes, and N. Higson,
Classifying space for proper actions and K-theory ofgroup C ∗ -algebras , Contemp. Math. (1994), 241–291.[2] J. Brodzki, G.A. Niblo, J. Špakula, R. Willett, and N. Wright, Uniform local amenability ,J. Noncommut. Geom. (2013), 583–603.
3] X. Chen, R. Tessera, X. Wang, and G. Yu,
Metric sparsification and operator norm local-ization , Adv. Math. (2008), 1496–1511.[4] Y.C. Chung,
Dynamical complexity and K-theory of L p operator crossed products ,preprint (2018). arXiv:1611.09000.[5] Y.C. Chung and K. Li, Rigidity of ℓ p Roe-type algebras , Bull. Lond. Math. Soc. (to appear).DOI: https://doi.org/10.1112/blms.12201.[6] A. Defant and K. Floret,
Tensor norms and operator ideals , North-Holland MathematicsStudies, North-Holland Publishing Co., 1993.[7] C. Dru¸tu and P.W. Nowak,
Kazhdan projections, random walks and ergodic theorems , J.Reine Angew. Math. (to appear). DOI: https://doi.org/10.1515/crelle-2017-0002.[8] A Engel,
Banach strong Novikov conjecture for polynomially contractible groups , Adv.Math. (2018), 148–172.[9] T. Fukaya and S. Oguni,
The coarse Baum-Connes conjecture for relatively hyperbolicgroups , J. Topol. Anal. (2012), no. 1, 99–113.[10] T. Fukaya and S. Oguni, Coronae of product spaces and the coarse Baum-Connes conjec-ture , Adv. Math. (2015), 201–233.[11] T. Fukaya and S. Oguni,
The coarse Baum-Connes conjecture for Busemann nonpositivelycurved spaces , Kyoto J. Math. (2016), no. 1, 1–12.[12] T. Fukaya and S. Oguni, A coarse Cartan-Hadamard theorem with application to thecoarse Baum-Connes conjecture , preprint (2018). arXiv:1705.05588.[13] E. Gardella and M. Lupini,
Representations of étale groupoids on L p -spaces , Adv. Math. (2017), 233–278.[14] E. Gardella and H. Thiel, Group algebras acting on L p -spaces , J. Fourier Anal. Appl. (2015), no. 6, 1310–1343.[15] E. Gardella and H. Thiel, Representations of p-convolution algebras on L q -spaces , Trans.Amer. Math. Soc. (to appear). DOI: https://doi.org/10.1090/tran/7489.[16] G. Gong, Q. Wang, and G. Yu, Geometrization of the strong Novikov conjecture for residu-ally finite groups , J. Reine Angew. Math. (2008), 159–189.[17] E. Guentner, R. Tessera, and G. Yu,
Operator norm localization for groups and its appli-cations to K-theory , Adv. Math. (2010), no. 4, 3495–3510.[18] S. Hejazian and S. Pooya,
Simple reduced L p -operator crossed products with unique trace ,J. Operator Theory (2015), no. 1, 133–147.[19] N. Higson, Counterexamples to the coarse Baum-Connes conjecture , unpublished note(1999). Available on author’s webpage.[20] N. Higson, V. Lafforgue, and G. Skandalis,
Counterexamples to the coarse Baum-Connesconjecture , Geom. Funct. Anal. (2002), no. 2, 330–354.[21] N. Higson and J. Roe, On the coarse Baum-Connes conjecture , Novikov conjectures, indextheorems, and rigidity: Volume 2, 1995, pp. 227â ˘A ¸S254.[22] N. Higson and J. Roe,
Analytic K-homology , Oxford Mathematical Monographs, OxfordUniversity Press, 2001.[23] S. Hoory, N. Linial, and A. Wigderson,
Expander graphs and their applications , Bull.Amer. Math. Soc. (N.S.) (2006), no. 4, 439–561. MR2247919
24] V. Lafforgue,
K-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes , Invent. Math. (2002), no. 1, 1–95.[25] V. Lafforgue,
Un renforcement de la propriété (T) , Duke Math. J. (2008), no. 3, 559–602.[26] B. Liao,
Strong Banach property (T) for simple algebraic groups of higher rank , J. Topol.Anal. (2014), no. 1, 75–105.[27] B. Liao and G. Yu, K-theory of group Banach algebras and Banach Property RD , preprint(2017). arXiv:1708.01982.[28] H. Oyono-Oyono and G. Yu,
K-theory for the maximal Roe algebra of certain expanders ,J. Funct. Anal. (2009), 3239–3292.[29] W. Paravicini,
The Bost conjecture, open subgroups and groups acting on trees , J. K-Theory (2009), no. 3, 469–490.[30] W. Paravicini, The Bost conjecture and proper Banach algebras , J. Noncommut. Geom. (2013), no. 1, 191–202.[31] N.C. Phillips, Analogs of Cuntz algebras on L p spaces , preprint (2012). arXiv:1309.4196.[32] N.C. Phillips, Crossed products of L p operator algebras and the K-theory of Cuntz alge-bras on L p spaces , preprint (2013). arXiv:1309.6406.[33] N.C. Phillips and D.P. Blecher, L p operator algebras with approximate identities I ,preprint (2018). arXiv:1802.04424.[34] N.C. Phillips and M.G. Viola, Classification of L p AF algebras , preprint (2017).arXiv:1707.09257.[35] Y. Qiao and J. Roe,
On the localization algebra of Guoliang Yu , Forum Math. (2010),no. 4, 657–665.[36] J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds , Mem.Amer. Math. Soc. (1993), no. 497.[37] H. Sako,
Property A and the operator norm localization property for discrete metric spaces ,J. Reine Angew. Math. (2014), 207–216.[38] G. Skandalis,
Progrès récents sur la conjecture de Baum-Connes. Contribution de VincentLafforgue , Séminaire Bourbaki (1999), 105–135.[39] J. Špakula,
Uniform K-homology theory , J. Funct. Anal. (2009), no. 1, 88–121.[40] J. Špakula and R. Willett,
A metric approach to limit operators , Trans. Amer. Math. Soc. (2017), 263–308.[41] R. Willett and G. Yu,
Higher index theory for certain expanders and Gromov monstergroups I , Adv. Math. (2012), 1380–1416.[42] G. Yu,
Localization algebras and the coarse Baum-Connes conjecture , K -theory (1997),307–318.[43] G. Yu, The Novikov conjecture for groups with finite asymptotic dimension , Ann. of Math. (1998), no. 2, 325–355.[44] G. Yu,
The coarse Baum-Connes conjecture for spaces which admit a uniform embeddinginto Hilbert space , Invent. Math. (2000), 201–240.(2000), 201–240.