Equivalence of tensor products over a category of W*-algebras

TL;DR

This paper proves the equivalence of two different tensor product constructions over a category of W*-algebras, clarifying their relationship and contrasting them with the standard tensor product based on weak topological completion.

Contribution

It establishes the categorical equivalence of two tensor products over W*-algebras, providing insight into their structural relationship and differences from the standard construction.

Findings

Proves the equivalence of Guichardet and Dauns tensor products.

Shows the difference from the Misonou--Takeda--Turumaru tensor product.

Clarifies the categorical properties of these tensor products.

Abstract

We prove the equivalence of two tensor products over a category of W*-algebras with normal (not necessarily unital) *-homomorphisms, defined by Guichardet and Dauns, respectively. This structure differs from the standard tensor product construction by Misonou--Takeda--Turumaru, which is based on weak topological completion, and does not have a categorical universality property.