Algebraic tensor products and internal homs of noncommutative Lp-spaces

Abstract

We prove that the multiplication map ▪ is an isometric isomorphism of (quasi)Banach M-M-bimodules. Here ▪ is the noncommutative Lp-space of an arbitrary von Neumann algebra M and ⊗M denotes the algebraic tensor product over M equipped with the (quasi)projective tensor norm, but without any kind of completion. Similarly, the left multiplication map ▪ is an isometric isomorphism of (quasi)Banach M-M-bimodules, where HomM denotes the algebraic internal hom. In particular, we establish an automatic continuity result for such maps. Applications of these results include establishing explicit algebraic equivalences between the categories of ▪-modules of Junge and Sherman for all p ≥ 0, as well as identifying subspaces of the space of bilinear forms on ▪-spaces. This paper is also available at arXiv:1309.7856v2.