## Abstract

We prove that the multiplication map L^{a}(M)⊗_{M}L^{b}(M)→L^{a+b}(M) is an isometric isomorphism of (quasi)Banach M-M-bimodules. Here L^{a}(M)=L^{1/a}(M) is the noncommutative L^{p}-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 L^{a}(M)→Hom_{M}(L^{b}(M),L^{a+b}(M)) is an isometric isomorphism of (quasi)Banach M-M-bimodules, where Hom_{M} 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 L^{p}(M)-modules of Junge and Sherman for all p≥0, as well as identifying subspaces of the space of bilinear forms on L^{p}-spaces. This paper is also available at arXiv:1309.7856v2.

Original language | English |
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Pages (from-to) | 229-244 |

Number of pages | 16 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 456 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 2017 |

## Keywords

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