A remark about the Connes fusion tensor product

Abstract

We analyze the algebraic structure of the Connes fusion tensor product (CFTP) in the case of bi-finite Hilbert modules over a von Neumann algebra M. It turns out that all complications in its definition disappear if one uses the closely related bi-modules of bounded vectors. We construct an equivalence of monoidal categories with duality between a category of Hilbert bi-modules over M with CFTP and some natural category of bi-modules over M with the usual relative algebraic tensor product. The results are surely well-known to the experts. We hope that the exposition and presentation is useful for those who want to learn about the CFTP .