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

We study tensor products of the Schatten classes SpSp, and their duals. It is shown that if p1−1+⋯+pk−1+q−1=1, then the space of multilinear forms B(Sp1,…,Spk;C)B(Sp1,…,Spk;C) contains a complemented subspace isometric to SqSq. We construct explicit embeddings of Srn into Spn⊗ˆSqn for r−1=p−1+q−1r−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−1r−1+1=p−1+q−1, with p,q≥1p,q≥1, then, given an n-by-n matrix ϕ  , the nuclear norm of the multiplication operator h∈Spn↦h⋅ϕ∈Sqn is n times the norm of ϕ   in Srˆn, where rˆ=max⁡(1,r). A number of results on general noncommutative LpLp spaces are also included.