Magdalena Musat University of Memphis

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1 Noncommutative Khintchine-type inequalities and applications (Joint work with Uffe Haagerup) Magdalena Musat University of Memphis GPOTS 2008 University of Cincinnati June 8, 2008

2 Khintchine inequalities: Let r n (t) := sgn(sin(2 n tπ)), n be the Rademacher functions on [0, ]. Then for every 0 <p<, there exist A p,b p > 0 such that for arbitrary n N and a,..., a n R, ( n /2 A p ak) /p 2 n p ( n ) /2 a k r k dt B p a 2 k k= k= k= 0 Suppose that A p,b p already denote the best constants above. By elementary methods, B p =, 0 <p 2 and A p =, 2 p<. Furthermore, Szarek (974): A =/ 2. Young (976): computed B p for p 3. Haagerup (982): computed A p and B p in the remaining cases. Kahane proved that for any Banach space X, A p,x ak r k L 2 ([0,];X) ak r k L p ([0,];X) B p,x ak r L2([0,];X) k In particular, if X = L p (Ω,µ), p<, then A p ) /2 ( ak 2 L p (Ω) L a k r k p ([0,];L p (Ω)) ) /2 B p ( ak 2 L p (Ω) 2

3 Next step: Generalize this to the case of noncommutative L p -spaces, e.g., the Schatten classes S p. Recall that for n, ( /p x S n p := Tr((x x) )) p/2, x Mn (C), were Tr denotes the non-normalized trace on M n (C). These results were obtained by Lust-Piquard (986) for < p <, and by Lust-Piquard and Pisier (99) for p =. Theorem (Lust-Piquard-Pisier 99): Given d, n N and x,..., x d M n (C), then + 2 {x j} d j= x j e i2n t {x j } d j= () L ([0,];S n) where, by definition, j= {x i } d i= := inf i= y i y i S n + z i zi ; x i = y i + z i. S n i= Remark: By classical results of Maurey and Pisier, inequalities () will then also hold when replacing the sequence {e i2nt } n by a sequence of Rademacher functions, or independent standard complex Gaussian random variables, or Steinhauss random variables, but with possibly different constants. 3

4 In joint work with U. Haagerup (JFA 250 (2007)) we obtain a direct proof of inequalities () both in the Gaussian and the Rademacher case. The Gaussian case: Theorem (Haagerup-M. 2007) Let d, n and consider x,..., x d M n (C). Let {γ i } in be independent, standard complex-valued Gaussian random variables on a probability space (Ω, P). Then 2 {x i } d i= i= x i γ i L (Ω;S n ) {x i } d i=. Remark: The conclusion of Theorem remains valid when replacing {γ n } n by {e i2nt } n, or by a sequence of Steinhauss r. v. Moreover, we show that in all these cases, both lower and upper bound are sharp. Theorem 2 (Haagerup-M. 2007) Denote by c, c 2 the best constants in the inequalities c {x i } d i= x i γ i c 2 {x i } i= L d i=. (Ω;S n) Then c =/ 2 and c 2 =. 4

5 The Rademacher case: Theorem 3 (Haagerup-M. 2007) Let d, n and consider x,..., x d M n (C). Let {r i } in be Rademacher functions on [0, ]. Then 3 {x i } d i= i= x i r i L ([0,];S n ) {x i } d i=. Remark: Let c,c 2 denote the best constants in the inequalities c {x i } d i= x i r i c 2 {x i } i= L d i=. ([0,];S n) Then the following estimates hold c, c 2 =

6 Embedding of OH via Khintchine inequalities for subspaces of R C It is a classical result that L 2 (Ω) embeds isometrically into L (Ω). Question: What about the noncommutative analogue of this fact? The noncommutative analogue of L (Ω) is the predual M of a von Neumann algebra M, while the noncommutative analogue of L 2 (Ω) is Pisier s operator Hilbert space OH. Theorem (Pisier): There exists an operator space OH B(K) (where K is a separable Hilbert space) such that () OH is isometric to l 2 (N) (as a Banach space) (2) The canonical identification between OH and OH (corresponding to the canonical identification between l 2 (N) and l 2 (N) is a complete isometry. Moreover, OH is the unique operator space (up to complete isometry) satisfying () and (2). 6

7 Some results: ) OH does not embed (cb-isomorphically) into the predual of any semifinite von Neumann algebra (Pisier 2004). 2) OH admits a cb-embedding into the predual of a type III von Neumann algebra (Junge 2005). Later, Junge (2006) showed that OH cb-embeds into the predual of the hyperfinite type III -factor, with cbisomorphism constant 200. (3) OH admits a cb-embedding into the predual of the hyperfinite type III -factor, with cb-isomorphism constant 2(Haagerup- M. 2007). The proof is based on Khintchine inequalities for subspaces of R C, proved in the same JFA (2007) paper. The bound of 2 comes from the estimate of the constant in this Khintchine-type inequality. (Unfortunately), we showed that this estimate is, again, sharp. The fascinating question whether the analogue of the classical isometric embedding result of L 2 into L holds (or not) in the noncommutative setting remains open. 7

8 Idea of the proof of the OH embedding: Let H be a closed subspace of R C, and associate to it A B(H), 0 A I H so that the operator space structure on H is given by r x i ξ i = i= Mn(H) 2 2 r r = max (I H A)ξ i,ξ j H x i x j, Aξ i,ξ j H x i x j i,j= for all n,r N, x i M n (C), ξ i H. i,j= Let A be the CAR-algebra built on H, and let ω A be the gaugeinvariant quasi-free state on A corresponding to A, that is, for all n,m N, ω A (a(f n )... a(f ) a(g )... a(g m )) = δ nm det( Ag i,f j H, i, j), for all f,..., f n,g,..., g m H. Use Riesz representation theorem to define a map E A : A H by for all b A. f, E A (b) H = ω A (a(f)b + b a(f)), f H, Let π A be the unital -representation from the GNS construction for (A,ω A ). Then E A extends to a bounded linear operator on M := π A (A) sot. 8

9 By a result of Powers-Størmer (970), M is a hyperfinite factor. Theorem (Haagerup-M. 2007) The map E A : A H yields a complete isomorphism H = A/Ker(E A ) with cb-isomorphism constant 2. Furthermore, the dual map E A is a complete isomorphism of H onto a subspace of M. Now, let P denote the hyperfinite type III -factor. Then M P = P, and hence M cb-embeds into P. Therefore by the above Theorem, H cb-embeds into P, with cb-isomorphism constant 2. We deduce that any quotient (and further, any subspace of a quotient) of (R C) cb-embeds into P, with cb-isomorphism constant 2. Pisier (2004) showed that OH is a subspace of a quotient of R C. Since OH is self-dual, OH is also a sub-quotient of (R C). Hence OH has this property. 9