# Algebraic tensor products and internal homs of noncommutative Lp-spaces

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## Abstract

We prove that the multiplication map La(M)⊗MLb(M)→La+b(M) is an isometric isomorphism of (quasi)Banach M-M-bimodules. Here La(M)=L1/a(M) 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 La(M)→HomM(Lb(M),La+b(M)) 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 Lp(M)-modules of Junge and Sherman for all p≥0, as well as identifying subspaces of the space of bilinear forms on Lp-spaces. This paper is also available at arXiv:1309.7856v2.

Original language English 229-244 16 Journal of Mathematical Analysis and Applications 456 1 https://doi.org/10.1016/j.jmaa.2016.11.060 Published - Dec 1 2017

## Keywords

• Internal homs
• Noncommutative L-spaces
• Quasi-Banach spaces
• Tensor products
• Tomita–Takesaki theory
• Von Neumann algebras