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

Abstract

+ 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 n r into S n p ⊗ S n q 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 φ, the nuclear norm of the multiplication operator h ∈ S n p → h · φ ∈ S n q is n times the norm of φ in S n r , where r = max(1, r). A number of results on general noncommutative L p spaces are also included.