Survey on Weak Amenability

Transcription

1 Survey on Weak Amenability OZAWA, Narutaka RIMS at Kyoto University Conference on von Neumann Algebras and Related Topics January 2012 Research partially supported by JSPS N. OZAWA (RIMS at Kyoto University) Weak Amenability 1 / 19

2 Fejér s theorem T = {z C : z = 1}. Let f C(T) and f k= Does the partial sum s m := Not necessarily! But, Theorem (Fejér) The Cesáro mean 1 n c k z k be the Fourier expansion. m k= m c k z k converges to f? n s m converges to f uniformly. m=1 Note: For ϕ n (k) = (1 k n ) 0, one has 1 n n s m = m=1 k= ϕ n (k)c k z k. N. OZAWA (RIMS at Kyoto University) Weak Amenability 2 / 19

3 (Reduced) Group C -algebra Γ a countable discrete group λ: Γ l 2 Γ the left regular representation (λ s ξ)(x) = ξ(s 1 x), λ(f )ξ = f ξ for f = f (s)δ s CΓ C λ Γ the C -algebra generated by λ(cγ) B(l 2 Γ) For Γ = Z, Fourier transform l 2 Z = L 2 T implements C λ Z = C(T), λ(f ) k Z f (k)zk. Fejér s theorem means that multipliers m ϕn : λ(f ) λ(ϕ n f ) converge to the identity on the group C -algebra Cλ Z, where ϕ n (k) = (1 k n ) 0 has finite support and is positive definite. N. OZAWA (RIMS at Kyoto University) Weak Amenability 3 / 19

4 Amenability Definition An approximate identity on Γ is a sequence (ϕ n ) of finitely supported functions such that ϕ n 1. A group Γ is amenable if there is an approximate identity consisting of positive definite functions. ϕ positive definite m ϕ is completely positive on C λ Γ m ϕ is completely bounded and m ϕ cb = ϕ(1). A group Γ is weakly amenable (or has the Cowling Haagerup property) if there is an approximate identity (ϕ n ) such that sup m ϕn cb <. Definition for locally compact groups is similar. Fejér s theorem implies that Z is amenable. In fact, all abelian groups and finite groups are amenable. The class of amenable groups is closed under subgroups, quotients, extensions, and limits. N. OZAWA (RIMS at Kyoto University) Weak Amenability 4 / 19

5 Cowling Haagerup constant Recall that a group Γ is weakly amenable if there is an approximate identity (ϕ n ) such that C := sup m ϕn cb <. The optimal constant C 1 is called the Cowling Haagerup constant and denoted by Λ cb (Γ). Λ cb (Γ) is equal to the CBAP constant for C λ Γ and the W CBAP constant for the group von Neumann algebra LΓ: If Λ Γ, then Λ cb (Λ) Λ cb (Γ). If Γ is amenable, then Λ cb (Γ) = 1. Λ cb (Γ) = Λ cb (C λ Γ) = Λ cb(lγ). F 2 is not amenable, but is weakly amenable and Λ cb (F 2 ) = 1. Λ cb (Γ 1 Γ 2 ) = Λ cb (Γ 1 ) Λ cb (Γ 2 ) (Cowling Haagerup 1989). If Λ cb (Γ i ) = 1, then Λ cb (Γ 1 Γ 2 ) = 1 (Ricard Xu 2006). Λ cb is invariant under measure equivalence. OPEN: Is weak amenability preserved under free products? N. OZAWA (RIMS at Kyoto University) Weak Amenability 5 / 19

6 Herz Schur multipliers Theorem (Grothendieck, Haagerup, Bożejko Fendler) For a function ϕ on Γ and C 0, the following are equivalent. The multiplier m ϕ : λ(f ) λ(ϕf ) is completely bounded on C λ Γ and m ϕ cb C. The Schur multiplier M ϕ : [A x,y ] x,y Γ [ϕ(y 1 x)a x,y ] x,y Γ is bounded on B(l 2 Γ) and M ϕ C. There are a Hilbert space H and ξ, η l (Γ, H) such that and ξ η C. ϕ(y 1 x) = ξ(x), η(y) A function ϕ satisfying the above conditions is called a Herz Schur multiplier, and the optimal constant C 0 is denoted by ϕ cb. N. OZAWA (RIMS at Kyoto University) Weak Amenability 6 / 19

7 An application to convolution operators on l p Γ If ϕ is a Herz Schur multiplier, then the Schur multiplier M ϕ : [A x,y ] x,y Γ [ϕ(y 1 x)a x,y ] x,y Γ is bounded on B(l p Γ) for all p [1, ]. Indeed, suppose ϕ(y 1 x) = ξ(x), η(y) for ξ, η l (Γ, H). Then, using H L p and H L q, we define V ξ : l p Γ l p Γ L p by V ξ δ x = δ x ξ(x 1 ), and likewise V η. One has V η ( A 1 ) Vξ = M ϕ (A). Now we wonder B(l p Γ) ρ(γ) = {λ(f ) : λ(f ) is bounded on l p Γ} =? SOT-cl{λ(f ) : f is finitely supported}. No counterexample is known. Theorem (von Neumann, Cowling) It is true if p = 2 or Γ is weakly amenable (or has the weaker property AP). Indeed, if Γ is weakly amenable, then m ϕn (λ(f )) λ(f ) in SOT. N. OZAWA (RIMS at Kyoto University) Weak Amenability 7 / 19

8 Examples of Herz Schur multipliers Every coefficient of a uniformly bounded representation (π, H) of Γ is a Herz Schur multiplier: For every ξ, η H, the function ϕ(s) = π(s)ξ, η on Γ has ϕ cb sup π(x) 2 ξ η. Indeed, one has ϕ(y 1 x) = π(x)ξ, π(y 1 ) η. Conversely, every Herz Schur multiplier on an amenable group Γ is a coefficient of some unitary representation. Indeed, if Γ is amenable, then the unit character τ 0 is continuous on C λ Γ and ω ϕ : C λ Γ λ(f ) ϕ(s)f (s) C = τ 0 m ϕ m ϕ. For the GNS rep n π : Cλ Γ B(H), there are ξ, η H such that π(λ(s))ξ, η = ω ϕ (λ(s)) = ϕ(s). N. OZAWA (RIMS at Kyoto University) Weak Amenability 8 / 19

9 Relation to Dixmier s and Kadison s problems Thus, for any group Γ one has where B(Γ) UB(Γ) B 2 (Γ) Theorem B(Γ) UB(Γ) B 2 (Γ), the space of coefficients of unitary representations the space of coefficients of uniformly bounded representations the space of Herz Schur multipliers (Bożejko 1985) A group Γ is amenable iff B(Γ) = B 2 (Γ). B(Γ) UB(Γ) if F 2 Γ. Is this true for any non-amenable Γ? (Haagerup 1985) A Herz Schur multiplier need not be a coefficient of a uniformly bounded representation, i.e., UB(F 2 ) B 2 (F 2 ). A uniformly bounded representation π of Γ extends on the full group C -algebra C Γ if all of its coefficients belong to B(Γ). N. OZAWA (RIMS at Kyoto University) Weak Amenability 9 / 19

10 Examples of weakly amenable subgroups 1 Theorem (De Cannière Haagerup, Cowling, Co. Ha., Ha. 80s) For a simple connected Lie group G, one has 1 if G = SO(1, n) or SU(1, n), } Haagerup Λ cb (G) = 2n 1 if G = Sp(1, n), property (T) + if rk R (G) 2, e.g., SL(3, R). For a lattice Γ G, one has Λ cb (Γ) = Λ cb (G). The idea of the proof: If G = PK with P amenable and K compact, then for any bi-k-invariant function ϕ on G, one has ϕ cb = ϕ P cb = C (P) λ(f ) f ϕ dµ C C (P). Further results: If rk R (G) 2, then G and its lattices even fail the AP. (Lafforgue de la Salle 2010 and Haagerup de Laat 2012). N. OZAWA (RIMS at Kyoto University) Weak Amenability 10 / 19

11 Examples of weakly amenable subgroups 2 Theorem (Oz. 2007, Oz. Popa 2007, Oz. 2010) Hyperbolic groups are weakly amenable. If G is weakly amenable and N G is an amenable closed normal subgroup, then there is an Ad(G)-invariant N-invariant states on L (N), or equivalently G is co-amenable in G N. SL(2, R) R 2 and SL(2, Z) Z 2 are not w.a. (Haagerup 1988), nor any wreath product Γ with 1 and Γ non-amenable. Proof of Corollary (non weak amenability of SL(2, Z) Z 2 ). Consider Γ = SL(2, Z) Z 2. Then, the stabilizer of every non-neutral element is amenable. (The stabilizer of [ m 0 ] Z 2, m 0, is {[ ]}.) If P is amenable, then Γ l 2 (Γ/P) is weakly contained in Γ l 2 (Γ). Hence, any Γ-invariant mean on Z 2 has to be concentrated on [ 0 0 ]. No mean on Z 2 is at the same time Γ-invariant and Z 2 -invariant. N. OZAWA (RIMS at Kyoto University) Weak Amenability 11 / 19

12 Applications to von Neumann algebras For a vn subalgebra M N which is a range of a conditional expectation, Λ cb (M) Λ cb (N). In particular, any non-weakly amenable von Neumann algebra, e.g. L(SL(2, Z) Z 2 ), does not embed into an weakly amenable II 1 -factor. Theorem (Oz. Popa 2007, Oz. 2010) Let M be an weakly amenable finite von Neumann algebra and P M be an amenable von Neumann subalgebra. Then P is weakly compact in M, or equivalently, there is a state ω on B(L 2 (P)) such that ω Ad u = ω for every u U(P) σ(n M (P)), where N M (P) = {u U(M) : upu = P} is the normalizer of P, acting on L 2 (P) by conjugation. Applications to strong solidity and uniqueness of Cartan subalgebras by Oz. Popa (2007), Chifan Sinclair (2011) and Popa Vaes ( ). N. OZAWA (RIMS at Kyoto University) Weak Amenability 12 / 19

13 Some more examples Theorem The following groups have the Cowling Haagerup constant 1. Amenable groups, SO(n, 1), SU(n, 1) (Haagerup and his friends). Free products of groups with Λ cb = 1 (Ricard Xu). Finite-dim CAT(0) cube complex groups (Guentner Higson, Mizuta). SL(2, K) as a discrete group (Guentner Higson Weinberger); in particular, limit groups. Baumslag Solitar groups (Gal). Counterexample: SL(2, Z) Z 2 = (Z/4Z Z 2 ) Z 2 (Z/6Z Z 2 ) is not weakly amenable. OPEN: relatively hyperbolic groups, mapping class groups, Ã 2 -groups,... N. OZAWA (RIMS at Kyoto University) Weak Amenability 13 / 19

14 Proofs of (non-)weak amenability Proofs of (non-)weak amenability N. OZAWA (RIMS at Kyoto University) Weak Amenability 14 / 19

15 Cayley graph of F 2 The Cayley graph of F 2 = a, a 1, b, b 1 is a tree with the metric d. b ab a 1 e a a 2 ab 1 b 1 x o y d(x, y) = χ [o,x] χ [o,y] 2 2 = 6 Theorem (Haagerup 1978) d is a Hilbert space metric, and r d is positive definite for r [0, 1]. N. OZAWA (RIMS at Kyoto University) Weak Amenability 15 / 19

16 Weak amenability of hyperbolic groups Theorem (Pytlik Szwarc 1986 and Oz. 2007) For any hyperbolic graph K of bounded degree, there is C 1 satisfying: For every z D = {z C : z < 1}, the function θ z : K K (x, y) z d(x,y) C is a bounded Schur multiplier on B(l 2 K) with 1 z θ z cb C 1 z. Moreover z θ z is holomorphic. If Γ K properly, then ϕ r (g) = r d(go,o) is an approximate identity on Γ, and Γ is weakly amenable. OPEN: Is ϕ z a coefficient of a uniformly bounded representation? To prove Theorem, one needs to factorize z d as z d(x,y) = ξ z (x), η z (y), ξ z, η z l (K, H). N. OZAWA (RIMS at Kyoto University) Weak Amenability 16 / 19

17 Proof for trees Let K be a tree and fix an infinite geodesic ω in K. For every x K, let ω x be the unique geodesic that starts at x and eventually flows into ω. For z D and x, y K, define ξ z (x) = 1 z 2 z k δ ωx (k) l 2 K, Then, one has k=0 ξ z (x) 2 2 = η z (y) 2 2 = 1 z 2 k 0 ω ω x and η z (y) = ξ z (y). x = ω x (0) z 2k = 1 z2 1 z 1 z 2 1 z. N. OZAWA (RIMS at Kyoto University) Weak Amenability 17 / 19

18 Proof for trees, cont d ω ω x x = ω x (0) y ω x (k) = ω y (l) m x ξ z (x) = 1 z 2 z k δ ωx (k) l 2 K, and η z (y) = ξ z (y). For x, y K, one has k=0 ξ z (x), η z (y) = (1 z 2 ) z k+l δ ωx (k), δ ωy (l) k,l 0 = (1 z 2 ) m 0 z d(x,y)+2m = z d(x,y). N. OZAWA (RIMS at Kyoto University) Weak Amenability 18 / 19

19 Non weakly amenable groups Theorem (Oz. Popa 2007, Oz. 2010) If G is weakly amenable and N G is an amenable closed normal subgroup, then there is an Ad(G)-invariant N-invariant states on L (N). Proof (Assuming Γ = G is discrete, N is abelian and Λ cb (Γ) = 1.) Let ϕ n be fin. supp. functions on Γ s.t. ϕ n 1 and sup m ϕn = 1. Then, for every s Γ, one has lim m ϕn m ϕn Ad(s) = 0. Indeed, if ϕ n (y 1 x) = ξ n (x), η n (y) for ξ, η : Γ H of norm 1, then ξ n (x) η n (x) ξ n (xs) uniformly for x, and likewise for η. Let τ 0 : C λ N C be the unit character and ω n = τ 0 m ϕn : C λ N C. Recall C λ N = C( N) via the Fourier transform l 2 N = L 2 ( N). Then, ω n (C λ N) is nothing but ϕ n N L 1 ( N) and ϕ n N 1 = ω n. Thus, ( ϕ n N ) is an approximately Γ-invariant approximate unit for L 1 ( N). Consequently, for ζ n l 2 N that corresponds to ϕn N 1/2 L 2 ( N), ζ n 2 l 1 N is approximately Ad(Γ)- and N-invariant. N. OZAWA (RIMS at Kyoto University) Weak Amenability 19 / 19