Non-commutative Khintchine type inequalities associated with free groups

Abstract

Let F n denote the free group with n generators g 1 , g 2 ,..., g n . Let A stand for the left regular representation of F n and let T be the standard trace associated to A. Given any positive integer d, we study the operator space structure of the subspace W p (n, d) of L p (τ) generated by the family of operators λ(g i ,g 2 ... g id ) with 1 < i k ≤ 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 L p (τ) generated by the image under A 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 K p formed of all block matrices a = (a ij ) with entries in the Schatten class S p , such that a is in S p relative to l 2 ⊗ l 2 and, moreover, such that (Σ ij a* ij a ij ) 1/2 and (Σ ij a ij a* ij ) 1/2 both belong to S p . We equip K p with the maximum of the three corresponding norms. Then, for 2 ≤ p ≤ oo, we have K p ≃ (K 2 , K∞)θ with 1/p = (1 - θ)/2.