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 and study estimates for their norms ‖k_∑εk⊗xkE_‖ calculated in that space. We establish general Khintchine type inequalities in this context. Then we show that if E is 2-concave, ‖k_∑εk⊗xkE_‖ is equivalent to the infimum of 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⊗xkE_‖ is equivalent to . We also study Rademacher averages _∑i,jεi⊗εj⊗xij for doubly indexed families (xij)_i,j of E(M).