Rademacher Averages on Noncommutative Symmetric Spaces

Let E be a separable (or the dual of a separable) symmetric function space, let M be a semifinite von Neumann algebra and let E(M) be the associated noncommutative function space. Let (εk)k≥1 be a Rademacher sequence, on some probability space Ω. For finite sequences (xk)k≥1 of E(M), we consider the Rademacher averages ∑ k εk ⊗ xk as elements of the noncommutative function space E(L(Ω)⊗M) and study estimates for their norms ‖ ∑ k εk ⊗ xk‖E calculated in that space. We establish general Khintchine type inequalities in this context. Then we show that if E is 2-concave, ‖ ∑ k εk⊗xk‖E is equivalent to the infimum of ‖( ∑ y kyk) 1 2 ‖+‖( ∑ zkz ∗ k) 1 2 ‖ over all yk, zk in E(M) such that xk = yk+zk for any k ≥ 1. Dual estimates are given when E is 2-convex and has a non trivial upper Boyd index. In this case, ‖ ∑ k εk ⊗ xk‖E is equivalent to ‖( ∑ x∗kxk) 1 2 ‖+ ‖( ∑ xkx ∗ k) 1 2 ‖. We also study Rademacher averages ∑ i,j εi ⊗ εj ⊗ xij for doubly indexed families (xij)i,j of E(M). Mathematics Subject Classification : Primary 46L52; Secondary 46M35, 47L05