aa r X i v : . [ m a t h . OA ] J u l NON- K -EXACT UNIFORM ROE C*-ALGEBRAS J ´AN ˇSPAKULAA
BSTRACT . We prove that uniform Roe C*-algebras C ∗ u X associated to some expandergraphs X coming from discrete groups with property ( t ) are not K -exact. In particular,we show that this is the case for the expander obtained as Cayley graphs of a sequence ofalternating groups (with appropriately chosen generating sets). Keywords: uniform Roe C*-algebras, K-exactness, expanders
MSC 2000:
Primary 46L80
1. I
NTRODUCTION
Uniform Roe C*-algebras (also called uniform translation C*-algebras) provide, amongother things, a link between coarse geometry and C*-algebra theory via the followingtheorem, which connects a coarse–geometric property of a discrete group G with a purelyanalytic property of its reduced C*-algebra C ∗ r G : Theorem 1 (Guentner–Kaminker [5] and Ozawa [12]) . Let G be a finitely generated dis-crete group. Then the following are equivalent: • G has property A (see [22] ), • the reduced group C*-algebra C ∗ r G is exact, • the uniform Roe C*-algebra C ∗ u | G | is nuclear. The only known examples of groups which do not have property A are Gromov’s ran-dom groups [4]. The theorem also characterizes nuclearity of uniform Roe C*-algebras ofdiscrete groups.More generally, one may ask if an analogue of the above theorem for general boundedgeometry metric spaces X is true, i.e. if property A of X is equivalent to nuclearity of C ∗ u X . One proof was obtained by representing C ∗ u X as a groupoid C*-algebra (see [15])and referring to the general groupoid C*-algebra theory (see [1]). For a more elementaryargument for one direction see [14].In further attempt to generalize, we may ask about a K -theoretic analogue of C*-algebraic property of nuclearity. Namely, we may seek some (coarse) geometric conditionsof X that would imply K -nuclearity of uniform Roe C*-algebras. The first step beyond therealm of property A is coarse embeddability into a Hilbert space. Using groupoid lan-guage, Ulgen proved that if X admits a coarse embedding into a Hilbert space, then C ∗ u X is K -EXACT UNIFORM ROE C*-ALGEBRAS 2 K -nuclear (see [18, proof of Theorem 3.0.12]). The argument uses the fact that groupoidswith the Haagerup property are K -amenable, and that crossed products of K -nuclear alge-bras with K -amenable groupoids are again K -nuclear (see [16] and [17]).On the other side, we may look for examples of spaces, whose uniform Roe C*-algebrasare not nuclear. An unpublished result of Higson asserts that uniform Roe C*-algebras ofexpander graphs constructed from groups with property (T) are not exact, and thereforenot nuclear.Wandering into the K -theoretic territory, we can ask for examples of spaces, whoseuniform Roe C*-algebras are not K -nuclear. Ulgen defined K -exactness in [18] as a gen-eralization of exactness in the context of K -theory. She also proved that if a separableC*-algebra is K -nuclear, then it is also K -exact. Unfortunately, this cannot be applied touniform Roe C*-algebras, which are usually not separable. In this paper, we show that forsome spaces X , C ∗ u X is not even K -exact. Our examples are expander graphs, constructedout of groups with property ( t ) with respect to a family of subgroups L (see [9]). Certainassumption on the family L is required at this point: Theorem 2.
Let G be a finitely generated discrete group with property t ( L ) . Assume that ( ⋆ ) G has t ( L ′ ) , where L ′ = { N ∩ N | N , N ∈ L } .Then C ∗ u X is not K-exact, where X = ⊔ N ∈ L G / N. Using results of Kassabov [7], we obtain the following corollary:
Corollary 3.
There is a sequence ( n i ) i ∈ N , such that the uniform Roe C*-algebra of theexpander obtained as a coarse disjoint union of Cayley graphs of the alternating groups Alt ( n i ) (with appropriately chosen generating sets) is not K-exact. The question of K -exactness for uniform Roe C*-algebras of expander graphs is closelyrelated to the same question for C*-algebras of the type (cid:213) q M q ( C ) . This has been set-tled negatively in various contexts by Ozawa [13] and by Manuilov–Thomsen [11]. Bothconstructions extend the work of Wassermann [21].Projections in (uniform) Roe C*-algebras similar to the one that is used in the construc-tion in this paper are called ghosts and were studied in the context of the ideal structure ofRoe C*-algebras [2, 20].The paper is organized as follows: In the next section, we recall the definitions of prop-erties and objects involved. Section 3 is devoted to the proof of Theorem 2 and in the lastsection we show how to deduce Corollary 3. Acknowledgment:
The author would like to thank Guoliang Yu for helpful and en-lightening conversations and never-ending encouragement and support, the reviewer ofthe previous version of this paper for enormous simplification and generalization of pre-vious argument, and finally Martin Kassabov for helpful comments about the alternatinggroups.
ON- K -EXACT UNIFORM ROE C*-ALGEBRAS 3
2. D
EFINITIONS K -exactness. Recall that a C*-algebra A is exact, if · ⊗ min A is an exact functor, i.e.if we min–tensor every term in a short exact sequence with A , the sequence stays exact.If it does, then we obtain a 6-term exact sequence in K -theory (as below). It may happenthat the tensored short exact sequence is not exact for a non–exact C*-algebra, but there isalways an exact 6-term sequence in K -theory. Let us be more precise: Proposition 4 ([18, 2.3.2]) . For a C*-algebra A, the following are equivalent: • for any exact sequence of C*-algebras → I → B → B / I → , there is a cyclic6-term exact sequence in K-theory:K ( I ⊗ min A ) / / K ( B ⊗ min A ) / / K ( B / I ⊗ min A ) (cid:15) (cid:15) K ( B / I ⊗ min A ) O O K ( B ⊗ min A ) o o K ( I ⊗ min A ) . o o (Note that in general, there might be no such 6-term K-theory sequence at all.), • for any exact sequence of C*-algebras → I → B → B / I → , the sequencesK i ( I ⊗ min A ) → K i ( B ⊗ min A ) → K i ( B / I ⊗ min A ) , are exact in the middle for both i = , . Definition 5.
We say that a C*-algebra A is K-exact , if it satisfies the conditions in theprevious proposition.For separable C*-algebras, a sufficient condition for K -exactness is K -nuclearity. Theargument [18, proposition 3.4.2] can be summarized as follows: the max–tensor productalways preserves exact sequences, and if a C*-algebra is K -nuclear, then min–tensor prod-ucts and max–tensor products with it are KK –equivalent. However, this relies on the keyproperties of KK -theory (existence and associativity of Kasparov product), which havebeen proved only for separable C*-algebras in general.2.2. Uniform Roe C*-algebras.
A metric space X has bounded geometry, if for each r > r is uniformly bounded. We say that X isuniformly discrete, if there exists c >
0, such that any two distinct points of X are at least c apart. Definition 6.
Let X be a uniformly discrete metric space with bounded geometry. Wesay that an X -by- X matrix ( t yx ) x , y ∈ X with complex entries has finite propagation , if thereexists R ≥
0, such that t yx = d ( x , y ) ≥ R . We say that such a matrix is uniformlybounded , if there exists T ≥
0, such that | t yx | ≤ T for all x , y ∈ X . ON- K -EXACT UNIFORM ROE C*-ALGEBRAS 4 Let A ( X ) be the algebra of all finite propagation matrices which are uniformly bounded.It is easy to see that each element of A ( X ) represents a bounded operator on ℓ ( X ) (see[14, lemma 4.27]). This yields a representation l : A ( X ) → B ( ℓ ( X )) . Definition 7.
The uniform Roe C*-algebra C ∗ u X of X is defined to be the norm closure of l ( A ( X )) ⊂ B ( ℓ ( X )) .2.3. Expanders and property ( t ) .Definition 8. An expander is a sequence X n of finite graphs with the properties: • The maximum number of edges emanating from any vertex is uniformly bounded. • The number of vertices of X n tends to infinity as n increases. • The first nonzero eigenvalue of the Laplacian, l ( X n ) , is uniformly bounded awayfrom zero, say by l > X n as of a discrete metric space, where the points are vertices of the graph,and the metric is given by the path distance in the graph. We understand the sequence asone metric space ⊔ n X n via the coarse disjoint union construction.Let us recall one possibility of how to construct a coarse disjoint union of finite spaces:Given a sequence ( X q ) q ∈ N of finite metric spaces, we define their coarse disjoint union ⊔ q X q to be the set ∪ q X q endowed with the metric inherited from individual X q ’s togetherwith the condition d ( X q , X q ′ ) = max ( q , q ′ ) for q = q ′ .The first explicit examples of expanders were constructed by Margulis as ⊔ q G / G q ,where G is a finitely generated group with property (T) (with a fixed generating set),and G q ≤ G is a decreasing sequence of normal subgroups with finite index, such that T q G q = { } . This construction eventually led to Lubotzky’s property ( t ) [9]. Definition 9.
Let G be a finitely generated group and L a countable family of finite indexnormal subgroups of G . We also assume that L is infinite, and that [ G : N ] → ¥ as N → ¥ † , N ∈ L . We say that G has property ( t ) with respect to the family L (written also t ( L ) )if the trivial representation is isolated in the set of all unitary representations of G , whichfactor through G / N , N ∈ L . We say that G has property ( t ) if it has this property withrespect to the family of all finite index normal subgroups.Property t ( L ) is equivalent to ⊔ N ∈ L G / N being an expander [9, Theorem 4.3.2].3. P ROOF OF T HEOREM G , a finitely generated discrete group anda countable family L of normal subgroups of G with finite index. We also fix a finitesymmetric generating set S of G . † By N → ¥ we mean “outside the finite sets”. ON- K -EXACT UNIFORM ROE C*-ALGEBRAS 5 For each N ∈ L , we denote G N = G / N and by q N : G → G N the quotient map. We let X to be a coarse disjoint union of the Cayley graphs of G N ’s with respect to the generatingsets { q N ( g ) | g ∈ S } .Let l N : G N → B ( ℓ G N ) be the left regular representation of G N . Denote also ˜ l N = l N ◦ q N : G → B ( ℓ G N ) and L = ⊕ N ∈ L ˜ l N : G → B ( ℓ X ) . Claim 1.
For each N ∈ L , we can choose an irreducible representation p N : G N → B ( H N ) ,so that( ⋆⋆ ) dim ( H N ) → ¥ as N → ¥ . Proof.
We shall use the fact that if G has t ( L ) , then for each fixed d >
0, there areonly finitely many non-equivalent irreducible d ′ -dimensional ( d ′ ≤ d ) representations of G factoring through some G N , N ∈ L . The same conclusion is known for groups withproperty (T) [19, 3], the argument for groups with ( t ) is outlined also in [10, Theorem3.11].Now assume that the claim doesn’t hold, that is, there is a sequence N n ∈ L , such thatall irreducible representations of G N n ’s are at most d -dimensional. By the above fact, allof them come from a finite set of irreducible representations { r , . . . , r m } of G . Conse-quently, each G N n embeds as a subgroup of K = image ( r ⊕ · · · ⊕ r m ) . Since every r i factors through some G N k , its image is a finite group. Hence K is finite, so we obtain acontradiction with the assumption that | G N n | → ¥ as n → ¥ . (cid:3) Denote ˜ p N = p N ◦ q N , H = ⊕ N ∈ L H N and p = ⊕ N ∈ L p N ◦ q N : G → B ( H ) . Let ussummarize the notation in the following diagram: G q N / / ˜ l N GGGGGGGGGG G N l N (cid:15) (cid:15) B ( ℓ G N ) G q N / / ˜ p N " " EEEEEEEEE G N p N (cid:15) (cid:15) B ( H N ) Finally, we let B = (cid:213) N ∈ L B ( H N ) and J = ⊕ N ∈ L B ( H N ) . We obtain an exact sequenceof C*-algebras 0 → J → B → B / J → . We shall use this sequence to show that C ∗ u X is not K -exact. We construct a projection e ∈ C ∗ u X ⊗ B , whose K -class will violate the exactness of the K -theory sequence K ( C ∗ u X ⊗ J ) → K ( C ∗ u X ⊗ B ) → K ( C ∗ u X ⊗ ( B / J )) . To construct such e , we let T = | S | (cid:229) g ∈ S ( L ⊗ p )( g ) ∈ C ∗ u X ⊗ B ⊂ B ( ℓ X ⊗ H ) . A few remarks are in order:
ON- K -EXACT UNIFORM ROE C*-ALGEBRAS 6 • If we denote s = | S | (cid:229) g ∈ S g ∈ C G , then T = ( L ⊗ p )( s ) . • T is “diagonal” with respect to the decomposition ℓ X ⊗ H = ⊕ M , N ∈ L ℓ G N ⊗ H M .This is clear from the fact that each ℓ G N ⊗ H M is a L ⊗ p –invariant subspace. Wedenote its “entries” by T NM ∈ B ( ℓ G N ⊗ H M ) . • With our choice of metric on X , L ( g ) ∈ B ( ℓ X ) has propagation 1 for each g ∈ S .The construction is finished by proving three claims, which we state and give some re-marks about them. The proofs are spelled out afterward. Claim 2. ∈ spec ( T ) is an isolated point.For proving this, we need to use some form of property t . The necessary conditionfor Claim 2 is that G has t ( L ) . However, this by itself is not sufficient, since the Claimrequires uniform bound on the spectral gap for all l N ⊗ p M , not just l N ’s. The condition ( ⋆ ) ensures this.Claim 2 allows us to define the projection e ∈ C ∗ u X ⊗ B to be the spectral projectionof T corresponding to 1 ∈ spec ( T ) . Note that e is also “diagonal” as T , hence we candecompose it into projections e NM ∈ B ( ℓ G N ⊗ H M ) . It is clear from the definition of T that each e NM is in fact the projection onto the subspace of G -invariant vectors in ℓ G N ⊗ H M . In fact, if G has property (T), then e is the image of the Kazhdan projection p ∈ C ∗ max G under L ⊗ p . Claim 3. e maps to 0 ∈ C ∗ u X ⊗ ( B / J ) .The key observation here is that if for a fixed N , e NM are eventually 0, then Claim 3holds. This is where the condition ( ⋆⋆ ) is used. Claim 4. [ e ] ∈ K ( C ∗ u X ⊗ B ) does not come from any class in K ( C ∗ u X ⊗ J ) .This claim is proved by “detecting the diagonal” of e . More precisely, observe that e NN = N ∈ L , since p N is conjugate to a subrepresentation of l N and hence l N ⊗ p N has nonzero invariant vectors. However, any element coming from C ∗ u X ⊗ J willhave the NN -entries eventually 0. The construction of a *-homomorphism that detects thisis essentially due to Higson [6]. Proof of Claim 2.
Taking N ∈ L , the G N -action on ℓ G N ⊗ H N is via l N ⊗ p N . This rep-resentation contains the trivial representation (since l N contains the conjugate of p N , as itdoes any irreducible representation of G N ), so there are nonzero G N -invariant vectors in ℓ G N ⊗ H N . Therefore, 1 ∈ spec ( T NN ) .To show that 1 is actually isolated in each spec ( T NM ) with the uniform bound on the sizeof the gap, we shall use the condition ( ⋆ ). Property t ( L ′ ) says that we have such a uniformbound on the size of the spectral gap of the image of r ( T ) for all the representations r of G which factor through some of G L , L ∈ L ′ [9, Theorem 4.3.2]. Using that ˜ p M is contained ON- K -EXACT UNIFORM ROE C*-ALGEBRAS 7 in ˜ l M , we obtainker ( ˜ l N ⊗ ˜ p M ) = ker ( ˜ l N ) ∩ ker ( ˜ p M ) ⊇ ker ( ˜ l N ) ∩ ker ( ˜ l M ) = N ∩ M . This shows that ˜ l N ⊗ ˜ p M factors through G / ( N ∩ M ) and the proof is finished. (cid:3) Proof of Claim 3.
Denote A = (cid:213) N ∈ L B ( ℓ G N ) , a product of matrix algebras. It is clearthat T ∈ ( C ∗ u X ⊗ B ) ∩ ( A ⊗ B ) , and so also e ∈ A ⊗ B ⊂ B ( ℓ X ⊗ H ) .For N ∈ L , let us examine the B ( ℓ G N ) ⊗ B ( H ) –component of e . Denote by P N ∈ B ( ℓ X ) the projection onto ℓ G N . It suffices to show that e N = ( P N ⊗ B ) e ( P N ⊗ B ) ∈ B ( ℓ G N ⊗ H ) actually belongs to B ( ℓ G N ) ⊗ J , since that shows that e ∈ C ∗ u X ⊗ J , andtherefore maps to 0 ∈ C ∗ u X ⊗ ( B / J ) .Further decompose e N into e NM ∈ B ( ℓ G N ⊗ H M ) . Recall that e NM = ℓ G N ⊗ H M has nonzero invariant vectors. Since the representation ˜ p M is irreducible, thisis further equivalent to ˜ p M being conjugate to a subrepresentation of ˜ l N . But by ( ⋆⋆ ), thiscan only happen for finitely many M ’s, since ˜ l N is fixed and dim ( H M ) → ¥ . (cid:3) Proof of Claim 4.
For N ∈ L , denote C N = B ( ℓ G N ⊗ H N ) . We construct a *-homomor-phism f : C ∗ u X ⊗ B → (cid:213) N C N (cid:14) ⊕ N C N , such that f ∗ ([ e ]) = ∈ K ( (cid:213) N C N (cid:14) ⊕ N C N ) , but f ∗ ([ x ]) = [ x ] ∈ K ( C ∗ u X ⊗ J ) .We first embed C ∗ u X into a direct limit of C*-algebras A k , defined below. We enumerate L = { N k | k ∈ N } and put A q = B ( ℓ G N q ) , A k = B ( ℓ ( ⊔ q ≤ k G N q )) and finally A k = A k ⊕ (cid:213) q > k A q , k ≥
1. There are obvious inclusion maps A k ֒ → A l for k < l , so we canform a direct limit A = lim k A k . It follows from the condition on distances d ( G N q , G N p ) that each finite propagation operator on ℓ X is a member of some A k ⊂ B ( ℓ X ) , hence C ∗ u X ֒ → A .For k ≥
1, denote B k = B ( H N k ) (so that B = (cid:213) k ∈ N B k ) and define f k as the followingcomposition: A k ⊗ B = (cid:16) A k ⊕ (cid:213) q > k A q (cid:17) ⊗ B ֒ → (cid:16) A k ⊗ B (cid:17) ⊕ (cid:213) q > k [ A q ⊗ B q ] ⊕ (cid:20) A q ⊗ (cid:16) (cid:213) p = q B p (cid:17)(cid:21)! proj −→ proj −→ (cid:213) q > k A q ⊗ B q = (cid:213) q > k C N k quot −→ (cid:213) q C N q . (cid:229) q C N q . It is easy to see that f k ’s commute with inclusions A k ⊗ B ֒ → A l ⊗ B , k < l . Consequently,we obtain a *-homomorphism f : C ∗ u X ⊗ ( B ⋊ G ) → (cid:213) N C N (cid:14) ⊕ N C N .It is known that K ( (cid:213) N C N (cid:14) ⊕ N C N ) embeds into (cid:213) N Z (cid:14) ⊕ N Z . Examining the construc-tion of f , we see that f ∗ ([ e ]) is the class of the sequence k rank ( e N k N k ) in (cid:213) N k Z (cid:14) ⊕ N k Z .As already noted, every term of this sequence is nonzero. ON- K -EXACT UNIFORM ROE C*-ALGEBRAS 8 On the other hand, any projection p ∈ C ∗ u X ⊗ J has only finitely many nonzero C N -components, and so f ∗ ([ p ]) = (cid:213) N Z (cid:14) ⊕ N Z . This obviously extends to the whole of K ( C ∗ u X ⊗ J ) . (cid:3)
4. E
XPANDERS COMING FROM FINITE GROUPS
Start with a sequence G n = h S n i , n ∈ N , of finite groups with chosen generating sets S n of some fixed size. Let G be a subgroup of G = (cid:213) n ∈ N G n generated by a finite set S ⊂ G that projects onto S n in each factor. Denote by q n : G → G n the natural projection, N n = ker ( q − n ) ⊂ G and L = { N n | n ∈ N } .Now G has t ( L ) if and only if the Cayley graphs of G n ’s with respect to S n ’s constitutean expander. In order to apply Theorem 2, we need to verify the condition ( ⋆ ). It seems tobe open whether t ( L ) implies ( ⋆ ) in general. For instance, it would be sufficient to knowa positive answer to [10, Question 1.14], that is, whether t ( L ) implies t ( L ′′ ) , where L ′′ is the closure of L under finite intersections. See also [8, Question 6] for a discussionwhen there is a finitely generated dense G ⊂ (cid:213) n ∈ N G n which has property ( t ) (with respectto all finite index subgroups).To prove Corollary 3, we appeal to the result of Kassabov [7, p. 352], which says thatthere is a finitely generated dense subgroup G ⊂ (cid:213) s Alt (cid:0) ( s − ) (cid:1) which has property( t ). Hence taking a finite symmetric generating set of G projects to generating sets of theindividual factors, making their Cayley graphs into an expander. Moreover, the condition( ⋆ ) is obviously satisfied. Consequently, we have shown Corollary 3.R EFERENCES [1] C. A
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