p-operator Spaces Zhong-Jin Ruan University of Illinois at Urbana-Champaign GPOTS 2008 June 18-22, 2008

Transcription

1 p-operator Spaces Zhong-Jin Ruan University of Illinois at Urbana-Champaign GPOTS 2008 June 18-22,

2 Operator Spaces Operator spaces are spaces of operators on Hilbert spaces. A concrete operator space is a norm closed subspace of B(H) together with a mantrix norm n on each M n (V ) given by M n (V ) M n (B(H)) = B(l n 2 (H)). Theorem [R 1988]: Let V be a Banach space with a norm n on each matrix space M n (V ). Then V is completely isometrically isomorphic to a concrete operator space if and only it satisfies M1. [ x 0 0 y ] n+m = max{ x n, y m } M2. αxβ n α x n β for all x M n (V ),y M m (V ) and α, β M n (C) =B(l n 2 ). 2

3 Completely Bounded Maps Let ϕ : V W be a bounded linear map. For each n N, we can define a linear map ϕ n :[x ij ] M n (V ) [ϕ(x ij )] M n (W ). The map ϕ is called completely bounded if ϕ cb = sup{ ϕ n : n N} <. We let CB(V, W ) denote the space of all completely bounded maps from V into W, which is again an operator space with matrix norm given by M n (CB(V, W )) = CB(V, M n (W )). In particular, the dual space V = CB(V,C) space matrix norm given by has a natural operator M n (V )=CB(V, M n (C)). 3

4 In the category of operator spaces, it is important to consider Matrix Norms and Completely Bounded Maps 4

5 Examples of Operator Spaces: C*-algebras or von Neumann algebras are operator spaces. In particular, if G is a locally compact/discrete group, C λ (G), C (G) and VN(G) Duals A of C*-algebras and preduals M of von Neumann algebras A(G) =VN(G), B λ (G) =C λ (G), and B(G) =C (G) Moreover, Herz-Schur multiplier algebra M cb A(G) CB(A(G),A(G)): M cb A(G) ={ϕ : G C : m ϕ : ψ A(G) ϕψ A(G) with m ϕ cb < }. 5

6 Approximation Properties A C*-algebra A is said to be nuclear if there exists comletely contractive maps ϕ α : A M n(α) and ψ α : M n(α) A such that in the point-norm topology. ψ α ϕ α id A A C*-algebra A is said to have the CBAP (resp. CCAP) if there exists finite rank maps ϕ α : A A such that ϕ α cb k (resp. ϕ α cb 1) and ϕ α id A in the point-norm topology. A C*-algebra is said to have the OAP if for every x =[x ij ] K(l 2 )ˇ A and ε> 0, there exists a finite rank map T : A A such that [T (x ij )] [x ij ] K(l2 )ˇ A < ε. A C*-algebra is said to be exact if we have the short exact sequence 0 K(l 2 )ˇ A B(l 2 )ˇ A Q(l 2 )ˇ A 0, where Q(H) =B(l 2 )/K(l 2 ). 6

7 Let G be a discrete group. We have the following implications for group C*-algebra C λ (G) Nuclearity CCAP/CBAP OAP Exactness. If we consider the Fourier algebra A(G), then we have Cλ (G) is nuclear G is amenable A(G) has a BAI/CAI. Cλ (G) has the CBAP/CCAP A(G) has the CBAP/CCAP Cλ (G) has the OAP A(G) has the OAP Cλ (G) is exact A(G) has???. 7

8 Finite Representatility in {M n } and in {T n } Theorem [Pisier 1995]: A C*-algebra A is exact if and only if it is finitely representable in {M n }, i.e. for every f.d. subspace E A and ε> 0, there exist n(ε) N and F M n(ε) such that E 1+ε = cb F, i.e., there exists a linear isomorphism T : E F such that T cb T 1 cb < 1+ε. Question: It is natural to ask whether a C*-algebra A is exact if and only if A is finitely representable in {T n }; whether Cλ (G) is exact if and only if A(G) is finitely represnetable in {T n }. 8

9 Actually, it is shown Theorem [Effros- Junge-Ruan 2000]: A (resp., M ) is finitely represnetable in {T n } if and only if A (resp., M) has QWEP. [Theorem Kirchberg 1993]: QWEP is equivalent to Connes embedding conjecure. A. Connes conjecture 1976: Every finite von Neumann algebra with separable predual is -isomorphic to a von Neumann subalgebra of the ultrapower of the hyperfinite II 1 factor. 8

10 Operator Algebras on L p Spaces Let G be a locally compact group with a left invariant Haar measure. For 1 p, we may consider L p (G) ={f : G C : measurable f p < }. For p = 1, we get the convolution algebra L 1 (G) and for p = we get the commutative von Neumann algebra L (G). For general 1 < p <, we can consider a (regular) representation λ p : G B(L p (G)) defined by λ p s(ξ)(t) =ξ(s 1 t) for ξ L p (G). For f L 1 (G), we have λ p (f) = G f(s)λp sds. 9

11 We let PF p (G) ={λ p (f) :f L 1 (G)} B(L p (G)) and let PM p (G) =span{λ p s : s G} w.o.t B(L p (G)). We note that PM p (G) is a dual space with a predual, the Figà-Talamanca- Herz algebra, A p (G) ={f : G C : f(s) = n <η n,λ p s(ξ n ) > with η n Lp ξ n Lp < }. If we let then we can identify Λ p : η ξ L p (G) π L p (G) < η, λ p s(ξ) >. A p (G) =L p (G) π L p (G)/kerΛ p with the quotient of L p (G) π L p (G). It is known (by Herz 1971) that A p (G) is a commutative Banach algebra and G is amenable if and only if A p (G) has a BAI. 10

12 Classical Case p=2 Case 1 < p < Case C 0 (G) L (G) C λ (G) VN(G) PF p(g) PM p (G) L 1 (G) M(G) A(G) B λ (G) B(G) A p (G) B λ p(g) B p (G) M cb A(G) MA(G) M cb A p (G) MA p (G) where M(G) is the measure algebra on G, MA p (G) is the multiplier algebra of A p (G), and M cb A p (G) is the completely bounded multiplier algebra on A p (G). For p = 2, M cb A 2 (G) =M cb A(G) is the completely bounded Herz-Schur multiplier algebra on G. What can we say about the corresponding properties on p-cases? 11

13 Uniformly Embeddability Let (Ω,d) be a metric space. Then (Ω,d) is said to be uniformly embbedable into a Banach space V if there exist a map f :Ω V and two monotone increasing functions ρ 1 ρ 1 on [0, ) [0, ) such that 1) ρ 1 (d(x, y)) f(x) f(y) ρ 2 (d(x, y)) 2) lim t ρ 1 (t) =. It is clear that Uniformaly embbedable into Hilbert spaces Uniformly embbedable into L p spaces Uniformly embbedable into uniformly convex Banach spaces. Theorem [Nowak 2005] Let (X, d) be a metric space. Then (X, d) is uniformly embbedable into L p space for some 1 p<2 if and only if (X, d) is uniformly embbedable into H. 12

14 Remark : For p>2, this is false for general metric space: Johnson and Randrianarivony 2006 showed that l p does not uniformly embbedable into H. However, it is still an open question for finitely generated groups (G, d l ). 13

15 Let G be a finitely generated group with a finite generator set S = S 1. Then we can obtain a length function l(x) =min{n : x = g 1 g n,g i S} and a metric d l (x, y) =l(x 1 y) on G. Then we can consider unformly embeeding problem for (G, d l ). 14

16 p-operator Spaces What are p-operator spaces? One possible definition is the spaces of operators on L p -spaces i.e. subspaces of B(L p (µ)). These are spaces we are interested in this talk! 15

17 General Definition of p-operator Spaces Let SQ p denote the set of all subspaces of quotients of L p -spaces, which is equal to the set QS p of all quotients of subspaces of L p -spaces. Let E SQ p and n N, we define p-column norm on E n = l n p(e), i.e. we define We define M n (B(E)) = B(E n ). [x i ] p =( x i p ) 1 p. A concrete p-operator space V is a norm closed subspace of B(E) for some E SQ p together with a mantrix norm n on each M n (V ) given by M n (V ) M n (B(E)) = B(l n p(e)). 16

18 Theorem [Le Merdy 1996]: Let V be a Banach space with a norm n on each matrix space M n (V ). Then V is completely isometrically isometric to a p-concrete operator space if and only it satisfies M1. [ x 0 0 y ] n+m = max{ x n, y m } Mp2. αxβ n α x n β for all x M n (V ),y M m (V ) and α, β M n (C) =B(l n p). 17

19 p-completely Bounded Maps Let V and W be p-operator spaces and let ϕ : V linear map. Then ϕ is called p-completely bounded if W be a bounded ϕ pcb = sup{ ϕ n : n N} <. We let CB p (V, W ) denote the space of all p-completely bounded maps from V into W, which is again a p-operator space with matrix norm given by M n (CB p (V, W )) = CB p (V, M n (W )). In particular, the dual space V = CB p (V,C) space matrix norm given by has a natural p-operator M n (V )=CB p (V, B(l n p)). Now we can consider the p-analogues of operator spaces. 18

20 Second Duals Let V be a p-operator space. We can consider the natural dual matricial norm induced on V, i.e. we have M n (V )=CB(V,B(l n p)). It is clear that the canonical inclusion κ : x V ˆx V is an isometry and is a complete contraction. Theorem [Daws 2007] For a p-operator space V, the canonical inclusion x V ˆx V is a p-completely isometric injection i.e. have for every [x ij ] M n (V ), we [x ij ] Mn (V ) = sup{ [Φ(x ij)] : all c.c. Φ = [Φ kl ] M m (V ) 1,m N} if and only if V B(L p ) for some L p -space. 19

21 Daws showed that Every dual p-operator space V is a p-operator subspace of B(L p ). There exists a finite dimensional p-operator space V which is not a p- operator subspace of any B(L p ), i.e. the p-matrix norm on V is different from the induced dual matrix norm on V = V! Another problem is that we do not have the Arveson-Wittstock- Hahn-Banach extension theorem for general p-operator spaces., or for p-operator spaces V B(L p ). 20

22 Extension Theorem Hahn-Banach Extension Theorem: W V ϕ ϕ C with ϕ = ϕ. Arveson-Wittstock-Hahn-Banach Extension Theorem: W V ϕ ϕ B(H) with ϕ cb = ϕ cb. 21

23 Tensor Products for p-operator Spaces Le Merdy 1996: Haagerup tensor product h p is projective, but not necessarily injective. Daws 2007: Projective tensor product p is projective and we have the p-complete isometry CB p (V ˆ p W, Z) =CB p (V, CB p (W, Z)). In particular, (V ˆ p W ) = CB p (V, W ). 22

24 Injective Tensor Product We can define the injective tensor product V ˇ p W CB(V,W). If W B(L p ), then for [u ij ] M n (V W ), we can write [u ij ] p = sup{ [(Φ Ψ)(u ij )] :Φ M k (V ) 1, Ψ M l (W )}. Theorem [An and R 2008]: Let V and W be subspaces of B(L p ) spaces. Then V ˇ p W is again a subspace of some B(L p ). Remark: However, it is still not clear whether p is injective, due to the lack of p-analogue of the Hahn-Banach extension theorem. 23

25 Let E SQ p. We may define the column (respectively, row) p-operator space structure E c = B(C, E) (respectively, E r = B(E,C)). More precisely, we have M n (E c )=B(l n p,l n p(e)) (respectively,m n (E r )=B(l n p(e ),l n p)). Theorem [An and R 2008] Let V B(L p ). isometries We have the p-complete V h p L c p = V p L c p and L r p h p V = L r p p V L c p h p V = L c p p V and V h p L r p = V p L r p 24

26 Combining the above result, we get the following p-complete isometric isomorphisms T (L p ) p V = L r p p V p L c p = L r p h p V h p L c p K(L p ) p V = L c p p V p L r p = L c p h p V h p L r p. In particular, we get L c p h p L r p = L c p p L r p = K(L p ) and L r p h p L c p = L r p p L c p = T (L p ). K(L p ) = T (L p ) and T (L p ) = B(L p ). Remark: Though these results are quite similar to operator spaces, the proofs have already need some quite different techniques. For instance, to show K(L p ) = T (L p ) for separable L p space, we need the fact: L p is separable dual space with RNP and approximation property. Therefore, every integral map is nuclear, i.e. we have K(L p ) =(L p ɛ L p ) = I(L p,l p )=N(L p,l p )=T (L p ). 25

27 It is known (from defintion of ˆ p ) that for each n N (T n ˆ p V ) = CB p (V, B(l n p)) = M n (V )=M n ˇ p V. However, it is quite interesting to show Theorem [An and R 2008]: Let V be a subspace of B(L p ). For each n N, we have the isometry (M n ˇ p V ) = T n ˆ p V. Moreover, we have the complete isometry (K(L p )ˇ p V ) = T (L p )ˆ p V. 26

28 p-operator Approximation Property We can study the corresponding approximation properties for p-operator spaces. For instance, let V be a subspace of B(L p ), then V is said to have p- operator space approximation property (p-oap) if for every x =[x ij ] K(l p )ˇ p V and ε> 0, there exists a finite rank map T : V V such that [T (x ij )] [x ij ] K(lp )ˇ p V < ε. Theorem [An and R 2008]: Let V be a p-operator subspace of some B(L p ). Then TFAE: 1) V has the p-oap; 2) the canonical map Φ : V ˆ p V V ˇ p V is injective; 3) if u V ˆ p V is such that Φ(u) = 0, then trace(u) = 0. 27

29 Now let G a locally compact group. Then M cb A p (G) ={ϕ : G C,m ϕ : ψ A p (G) ϕψ A p (G) with m ϕ cb < }. Theorem [Daws 2007]: A continuous function ϕ : G C is a cb multiplier of A p (G) if and only if there exist bounded maps ξ : G E and η : G E (with E in SQ p and E in SQ p ) such that ϕ(st 1 )=<η(s),ξ(t) >. Moreover we have m ϕ pcb = inf{ η ξ }. Proposition [Miao and R 08]: Q p (G). M cb A p (G) is a dual space with a predual Therefore, we can define p-ap for G: 1 is contained in the σ(m cb A p (G),Q p (G)) closure of A p (G) in M cb A p (G). Now we are working on the p-oap for FP p (G) and A p (G). results by Haagerup-Kraus 1994 can be generalized to p-case. Most of 28

30 Thank you for your attention! 29