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 ∗ k yk)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 nontrivial upper Boyd index. In this case, k εk ⊗ xk E is equivalent to ( x ∗ k xk)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).