Exact Crossed-Products : Counter-example Revisited

Transcription

1 Exact Crossed-Products : Counter-example Revisited Ben Gurion University of the Negev Sde Boker, Israel Paul Baum (Penn State) 19 March, 2013

2 EXACT CROSSED-PRODUCTS : COUNTER-EXAMPLE REVISITED An expander or expander family is a sequence of finite graphs X 1, X 2, X 3,... which is efficiently connected. A discrete group G which contains an expander in its Cayley graph is a counter-example to the Baum-Connes (BC) conjecture with coefficients. Some care must be taken with the definition of contains. M. Gromov outlined a method for constructing such a group. G. Arjantseva and T. Delzant completed the construction. The group so obtained is known as the Gromov group and is the only known example of a non-exact group.

3 The left side of BC with coefficients sees any group as if the group were exact. This talk will indicate how to make a change in the right side of BC with coefficients so that the right side also sees any group as if the group were exact. This corrected form of BC with coefficients uses the unique minimal exact intermediate crossed-product. For exact groups (i.e. all groups except the Gromov group) there is no change in BC with coefficients.

4 In the corrected form of BC with coefficients the Gromov group acting on the coefficient algebra obtained from an expander is not a counter-example. Thus at the present time (March, 2013) there is no known counter-example to the corrected form of BC with coefficients. The above is joint work with E. Kirchberg and R. Willett. This work is based on and inspired by a result of R. Willett and G. Yu.

5 A discrete group Γ which contains an expander in its Cayley graph is a counter-example to the usual (i.e. uncorrected) BC conjecture with coefficients. Some care must be taken with the definition of contains.

6 An expander or expander family is a sequence of finite graphs which is efficiently connected. X 1, X 2, X 3,...

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10 The isoperimetric constant h(x) For a finite graph X with vertex set V and V vertices h(x) =: min{ F F F V and F V 2 } F = number of vertices in F. F V. F = number of edges in X having one vertex in F and one vetex in V F.

11 Definition of expander An expander is a sequence of finite graphs such that Each X j is conected. X 1, X 2, X 3,... a positive integer d with each X j d-regular. X n as n. a positive real number ɛ > 0 with h(x j ) ɛ > 0 j = 1, 2, 3,....

12 G topological group G is assumed to be : locally compact, Hausdorff, and second countable. (second countable = The topology of G has a countable base.) Examples Lie groups SL(n, R) p-adic groups SL(n, Q p ) adelic groups SL(n, A) discrete groups SL(n, Z)

13 Ordinary BC and BC with coefficients are for such topological groups G. EG denotes the universal example for proper actions of G. EXAMPLE. If Γ is a (countable) discrete group, then EΓ can be taken to be the convex hull of Γ within l 2 (Γ ).

14 Example Give Γ the measure in which each γ Γ has mass one. Consider the Hilbert space l 2 (Γ ). Γ acts on l 2 (Γ ) via the (left) regular representation of Γ. Γ embeds into l 2 (Γ ) Γ l 2 (Γ ) γ Γ γ [γ] where [γ] is the Dirac function at γ. Within l 2 (Γ ) let Convex-Hull(Γ) be the smallest convex set which contains Γ. The points of Convex-Hull(Γ) are all the finite sums t 0 [γ 0 ] + t 1 [γ 1 ] + + t n [γ n ] with t j [0, 1] j = 0, 1,..., n and t 0 + t t n = 1 The action of Γ on l 2 (Γ ) preserves Convex-Hull(Γ). Γ Convex-Hull(Γ) Convex-Hull(Γ) EΓ can be taken to be Convex-Hull(Γ) with this action of Γ.

15 Kj G (EG) denotes the Kasparov equivariant K-homology with G-compact supports of EG. Definition A closed subset of EG is G-compact if: 1. The action of G on EG preserves. and 2. The quotient space /G (with the quotient space topology) is compact.

16 Definition K G j (EG) = lim KK j G (C 0( ), C). EG G-compact The direct limit is taken over all G-compact subsets of EG. Kj G (EG) is the Kasparov equivariant K-homology of EG with G-compact supports.

17 Ordinary BC Conjecture For any G which is locally compact, Hausdorff and second countable K G j (EG) K j (C r G) j = 0, 1 is an isomorphism

18 Let A be a G C algebra i.e. a C algebra with a given continuous action of G by automorphisms. BC with coefficients Conjecture G A A For any G which is locally compact, Hausdorff, and second countable and any G C algebra A is an isomorphism. K G j (EG, A) K j (C r (G, A)) j = 0, 1

19 Definition K G j (EG, A) = lim KK j G (C 0( ), A). EG G-compact The direct limit is taken over all G-compact subsets of EG. Kj G (EG, A) is the Kasparov equivariant K-homology of EG with G-compact supports and with coefficient algebra A.

20 Basic property of C algebra K-theory SIX TERM EXACT SEQUENCE Let 0 I A B 0 be a short exact sequence of C algebras. Then there is a six term exact sequence of abelian groups K 0 I K 0 A K 0 B K 1 B K 1 A K 1 I

21 DEFINITION. G is exact if whenever 0 I A B 0 is an exact sequence of G C algebras, then 0 Cr (G, I) Cr (G, A) Cr (G, B) 0 is an exact sequence of C algebras.

22 LEMMA. Let 0 I A B 0 be an exact sequence of G C algebras. Assume that G is exact. Then there is a six term exact sequence of abelian groups K 0 C r (G, I) K 0 C r (G, A) K 0 C r (G, B) K 1 Cr (G, B) K 1 Cr (G, A) K 1 Cr (G, I)

23 The left side of BC with coefficients sees any G as if G were exact.

24 LEMMA. For any locally compact Hausdorff second countable topological group G and any exact sequence 0 I A B 0 of G C algebras, there is a six term exact sequence of abelian groups K0 G(EG, I) K0 G(EG, A) K0 G (EG, B) K G 1 (EG, B) K1 G (EG, A) K1 G (EG, I)

25 QUESTION. Do non-exact groups exist? ANSWER. If a discrete group Γ contains an expander in its Cayley graph, then Γ is not exact. contains = There exists an expander X and a map f : X Cayley graph(γ) such that f comes close to being a uniform embedding (in the sense of coarse geometry of metric spaces).

26 Definition of expander An expander is a sequence of finite graphs such that Each X j is conected. X 1, X 2, X 3,... a positive integer d with each X j d-regular. X n as n. a positive real number ɛ > 0 with h(x j ) ɛ > 0 j = 1, 2, 3,....

27 Precise meaning of contains Let X 1, X 2, X 3,... be an expander. maps (of sets) ϕ 1, ϕ 2, ϕ 3,... with ϕ j : vertices(x j ) Γ j = 1, 2, 3,... There is a constant K such that d(ϕ j (x), ϕ j (x )) Kd(x, x ) j and x, x X j. limit n (max{ ϕ 1 n (γ) / vertices(x n ) γ Γ) = 0.

28 M.Gromov indicated how a discrete group Γ which contains an expander in its Cayley graph might be constructed. Several mathematicians (Silberman, Arjantseva, Delzant etc etc) then worked on the problem of constructing such a Γ. For a complete proof that such a Γ exists, see the paper of G. Arjantseva and T. Delzant.

29 If Γ, contains an expander in its Cayley graph, then there exists an exact sequence of Γ C algebras, such that 0 I A B 0 K 0 C r (Γ, I) K 0 C r (Γ, A) K 0 C r (Γ, B) is not exact Since K Γ 0 (EΓ, I) K Γ 0 (EΓ, A) K Γ 0 (EΓ, B) is exact, such a Γ is a counter-example to BC with coefficients.

30 For the construction (given such a Γ) of the relevant exact sequence 0 I A B 0 of Γ C algebras, see the paper of N.Higson and V. Lafforgue and G. Skandalis. A is (the closure of) the sub-algebra of L (Γ) consisting of functions which are supported in R-neighborhoods of the expander.

31 Theorem (N. Higson and G. Kasparov) If Γ is a discrete group which is amenable (or a-t-menable), then BC with coeficients is true for Γ. Theorem ( V. Lafforgue) If Γ is a discrete group which is hyperbolic (in Gromov s sense), then BC with coefficients is true for Γ.

32 Possible Happy Ending A possible happy ending is : If G is exact, then BC with coefficients is true for G. PROBLEM. Is BC (i.e.ordinary BC = BC without coefficients) true for SL(3, Z)?

33 STOP!!!! HOLD EVERYTHING!!!! Consider the recent result of Rufus Willett and Guoliang Yu: Theorem Let Γ be the Gromov group and let A be the Γ - C algebra obtained by mapping an expander to the Cayley graph of Γ. Then is an isomorphism. K Γ j (EΓ, A) K j (C max(γ, A)) j = 0, 1 This theorem indicates that for non-exact groups the right side of BC with coefficients has to be reformulated. For exact groups (i.e. all groups except the Gromov group) no change should be made in the statement of BC with coefficients. SL(3, Z)????

34 With G fixed, {G C algebras} denotes the category whose objects are all the G C algebras. Morphisms in {G C algebras} are -homomorphisms which are G-equivariant. {C algebras} denotes the category of C algebras. Morphisms in {C algebras} are -homomorphisms.

35 A crossed-product is a functor, denoted A C τ (G, A) from {G C algebras} to {C algebras} C τ : {G C algebras} {C algebras}

36 intermediate = between the max and the reduced crossed-product For an intermediate crossed-product C τ there are surjections: C max(g, A) C τ (G, A) C r (G, A) Denote by τ(g, A) the kernel of the surjection C max(g, A) C τ (G, A) is exact. 0 τ(g, A) C max(g, A) C τ (G, A) 0

37 Denote by ɛ(g, A) the kernel of C max(g, A) C r (G, A). is exact. 0 ɛ(g, A) C max(g, A) C r (G, A) 0 An intermediate crossed-product C τ is then a function τ which assigns to each G C algebra A a norm closed ideal τ(g, A) in C max(g, A) such that :

38 For each G C algebra A, τ(g, A) ɛ(g, A). For each morphism A B in {G C algebras} the resulting -homomorphism C max(g, A) C max(g, B) maps τ(g, A) to τ(g, B). C τ (G, A) = C max(g, A) / τ(g, A)

39 An intermediate crossed-product C τ is exact if whenever 0 A B C 0 is an exact sequence in {G C algebras} the resulting sequence in {C algebras} is exact. 0 C τ (G, A) C τ (G, B) C τ (G, C) 0

40 Equivalently : An intermediate crossed-product C τ is exact if whenever 0 A B C 0 is an exact sequence in {G C algebras} the resulting sequence in {C algebras} is exact. 0 τ(g, A) τ(g, B) τ(g, C) 0

41 QUESTION. Given G, does there exist a unique minimal intermediate crossed-product which is exact? PROPOSITION. (E. Kirchberg) For any locally compact Hausdorff second countable topological group G there exists a unique minimal intermediate crossed-product which is exact. Denote the unique minimal intermediate exact crossed-product by C exact.

42 Reformulation of BC with coefficients. CONJECTURE. Let G be a locally compact Hausdorff second countable topological group, and let A be a G C algebra, then K G j (EG, A) K j (C exact(g, A)) j = 0, 1 is an isomorphism.

43 Kirchberg s proof is based on two lemmas. Definition Let 0 A 1 A 2 A 3 0 be an exact sequence in {C algebras}. Let J 1, J 2, J 3 be ideals (possibly not norm closed) in A 1, A 2, A 3 J i A i i = 1, 2, 3. Then J 1, J 2, J 3 is adapted to the exact sequence 0 A 1 A 2 A 3 0 if 1 The -homomorphism A 1 A 2 maps J 1 to J 2. 2 The -homomorphism A 2 A 3 maps J 2 to J 3. 3 The sequence 0 J 1 J 2 J 3 0 is exact.

44 Kirchberg s proof is based on the following two lemmas. Lemma 1. Assume that J 1, J 2, J 3 is adapted to 0 A 1 A 2 A 3 0. Denote the norm closure of J i in A i by J i. Then : J 1, J 2, J 3 is adapted to 0 A 1 A 2 A 3 0. Lemma 2. Assume that J 1, J 2, J 3 is adapted to 0 A 1 A 2 A 3 0. Assume that L 1, L 2, L 3 is adapted to 0 A 1 A 2 A 3 0. Assume that J i and L i are norm closed in A i. Then : J 1 + L 1, J 2 + L 2, J 3 + L 3 is adapted 0 A 1 A 2 A 3 0.

45 With G fixed, an intermediate crossed-product C τ is exact iff whenever 0 A B C 0 is an exact sequence in {G C algebras}, then τ(g, A), τ(g, B), τ(g, C) is adapted to the exact sequence of C algebras 0 C max(g, A) C max(g, B) C max(g, C) 0

46 With G fixed, C exact is defined by C exact(g, A) := C max(g, A) / τ(g, A) where the union (i.e. set-theoretic union within C max(g, A)) is taken over all exact intermediate crossed-products C τ. Closure is norm closure in C max(g, A).

47 Lemma 1. Assume that J 1, J 2, J 3 is adapted to 0 A 1 A 2 A 3 0. Denote the norm closure of J i in A i by J i. Then : J 1, J 2, J 3 is adapted to 0 A 1 A 2 A 3 0. Proof of Lemma 1 uses a standard fact about ideals in C algebras: Standard Fact. Let J and I be ideals in a C algebra A. Assume that I is norm closed. Then : J I = J I Proof of Lemma 1. A 1 is a norm closed ideal in A 2. The problem is to prove J 2 A 1 = J 1 Since J 1 = J 2 A 1 this is implied by QED J 2 A 1 = J 2 A 1

48 Lemma 2. Assume that J 1, J 2, J 3 is adapted to 0 A 1 A 2 A 3 0. Assume that L 1, L 2, L 3 is adapted to 0 A 1 A 2 A 3 0. Assume that J i and L i are norm closed in A i. Then : J 1 + L 1, J 2 + L 2, J 3 + L 3 is adapted 0 A 1 A 2 A 3 0. Proof of Lemma 2 uses two Standard Facts about norm closed ideals in C algebras. Let J, L be norm closed ideals in a C algebra A. Then : 1 The (purely algebraic) J + L is a norm closed ideal in A. 2 The (purely algebraic) JL is a norm closed ideal in A and is equal to J L.

49 Proof of Lemma 2 Since 0 J i L i J i L i J i + L i 0 is exact, it will suffice to prove exactness for 0 J 1 L 1 J 2 L 2 J 3 L 3 0 The problem is to prove surjectivity of J 2 L 2 J 3 L 3 This surjectivity is implied by J L = JL. QED

50 Theorem (PB and R. Willett) Let Γ be the Gromov group and let A be the Γ - C algebra obtained by mapping an expander to the Cayley graph of Γ. Then is an isomorphism. K Γ j (EΓ, A) K j (C exact(γ, A)) j = 0, 1