Non-commutative Khintchine Type Inequalities Associated with Free Groups

Let Fn denote the free group with n generators g1, g2, . . . , gn. Let λ stand for the left regular representation of Fn and let τ be the standard trace associated to λ. Given any positive integer d, we study the operator space structure of the subspace Wp(n, d) of Lp(τ) generated by the family of operators λ(gi1gi2 · · · gid ) with 1 ≤ ik ≤ n. Moreover, our description of this operator space holds up to a constant which does not depend on n or p, so that our result remains valid for infinitely many generators. We also consider the subspace of Lp(τ) generated by the image under λ of the set of reduced words of length d. Our result extends to any exponent 1 ≤ p ≤ ∞ a previous result of Buchholz for the space W∞(n, d). The main application is a certain interpolation theorem, valid for any degree d (extending a result of the second author restricted to d = 1). In the simplest case d = 2, our theorem can be stated as follows: consider the space Kp formed of all block matrices a = (aij ) with entries in the Schatten class Sp, such that a is in Sp relative to l2 ⊗ l2 and moreover such that ( ∑ ij a ∗ ijaij) 1/2 and ( ∑ ij aija ∗ ij) 1/2 both belong to Sp. We equip Kp with the maximum of the three corresponding norms. Then, for 2 ≤ p ≤ ∞ we have Kp ≃ (K2,K∞)θ with 1/p = (1− θ)/2. Introduction Let Rp(n) be the subspace of Lp[0, 1] generated by the classical Rademacher functions r1, r2, . . . , rn. As is well-known, for any exponent 1 ≤ p < ∞, the classical Khintchine inequalities provide a linear isomorphism between Rp(n) and l2(n) with constants independent of n. However, the operator space structure of Rp(n) is not so simple. It is described by the so-called non-commutative Khintchine inequalities, introduced by F. Lust-Piquard in [6] and extended in [7] to the case p = 1, see also [2] for an analysis of the optimal constants. Let us write eij to denote the natural basis of the Schatten class Sp. To describe these inequalities, we define R n p to be the operator space generated by {e1j | 1 ≤ j ≤ n} in S p . Similarly, C p denotes the space generated by {ei1 | 1 ≤ i ≤ n}. Then, it turns out that Rp(n) is completely isomorphic to R p +C n p whenever 1 ≤ p ≤ 2 and Rp(n) is completely isomorphic to R p ∩Cn p for 2 ≤ p < ∞. Again, the constants do not depend on n. More explicitly, we have the following equivalences of norms • For 1 ≤ p ≤ 2, ∥∥ n ∑ k=1 xkrk ∥∥ Lp([0,1];S p ) ≃ inf xk=yk+zk {∥∥∥ ( n ∑ k=1 y kyk 1/2∥∥ Sn p + ∥∥ ( n ∑ k=1 zkz ∗ k 1/2∥∥ Sn p } . ∗ Partially supported by the Project BFM 2001/0189, Spain. † Partially supported by the NSF and by the Texas Advanced Research Program 010366-163.