Relative tensor products for modules over von Neumann algebras

Details Relative tensor products for modules over von Neumann algebras Relative tensor products for modules over von Neumann algebras

2003-01-08

arxiv.org/abs/math/0301061v1

Mathematics

Operator Algebras

17 pages; to appear in Contemporary Mathematics

Scientific paper

We give an overview of relative tensor products (RTPs) for von Neumann algebra modules. For background, we start with the categorical definition and go on to examine its algebraic formulation, which is applied to Morita equivalence and index. Then we consider the analytic construction, with particular emphasis on explaining why the RTP is not generally defined for every pair of vectors. We also look at recent work justifying a representation of RTPs as composition of unbounded operators, noting that these ideas work equally well for L^p modules. Finally, we prove some new results characterizing preclosedness of the map (\xi, \eta) \mapsto \xi \otimes_\phi \eta.

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