Rademacher averages on noncommutative symmetric spaces

AbstractLet 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 ∑kεk⊗xk as elements of the noncommutative function space E(L∞(Ω)⊗¯M) and study estimates for their norms ‖∑kεk⊗xk‖E calculated in that space. We establish general Khintchine type inequalities in this context. Then we show that if E is 2-concave, ‖∑kεk⊗xk‖E is equivalent to the infimum of ‖(∑yk∗yk)12‖+‖(∑zkzk∗)12‖ 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, ‖∑kεk⊗xk‖E is equivalent to ‖(∑xk∗xk)12‖+‖(∑xkxk∗)12‖. We also study Rademacher averages ∑i,jεi⊗εj⊗xij for doubly indexed families (xij)i,j of E(M)