Complemented subspaces of spaces of multilinear forms and tensor products, II. Noncommutative Lp spaces

We study tensor products of the Schatten classes S-p and their duals. It is shown that if p(1)(-1) + .... + p(k)(-1) + q(-1) = 1, then the space of multilinear forms B(S-p1 ,...., S-pk ;C) contains a complemented subspace isometric to S-q. We construct explicit embeddings of S-r(n) into S-p(n) (circle times) over cap S-q(n) for r(-1) = p(-1) + q(-1) whose range is complemented by a "natural" norm-one projection. As a byproduct we compute the nuclear norm of some multiplication operators: if r(-1) + 1 = p(-1) + q(-1), with p, q >= 1, then, given an n-by-n matrix phi, the nuclear norm of the multiplication Operator h is an element of S-p(n) -> h.phi is an element of S-q(n) is n times the norm of phi in S-<(tau)over cap(n), where (r) over cap = max(1, r). A number of results on general noncommutative L-p spaces are also included. (C) 2014 Elsevier Inc. All rights reserved.