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Journal of Operator Theory

Volume 76, Issue 2, Fall 2016  pp. 271-283.

Weak* tensor products for von Neumann algebras

Authors:  Matthew Wiersma
Author institution:Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada, N2L 1W5

Summary: The category of $C^*$-algebras is blessed with many different tensor products. In contrast, virtually the only tensor product ever used in the category of von Neumann algebras is the normal spatial tensor product. In this paper, we propose a definition of what a generic tensor product in this category should be. We call these {\it weak* tensor products}. A complete characterization for an analogue of nuclearity for weak* tensor products is given and we construct $2^{\mathfrak c}$ nonequivalent weak* tensor product completions of $L^\infty(\mathbb R)\odot L^\infty(\mathbb R)$.

Keywords: von Neumann algebras, tensor products

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