On the reduction theory of $W^{*}$-algebras by Hilbert modules

TL;DR

This paper explores the reduction theory of $W^*$-algebras using Hilbert modules constructed from spatial representations, providing a detailed framework for reducing a $W^*$-algebra along its central subalgebras.

Contribution

It introduces a comprehensive approach to the reduction of $W^*$-algebras via Hilbert modules and applies this to the standard form reduction along central subalgebras.

Findings

Develops a detailed theory of Hilbert modules for $W^*$-algebras.

Shows how to reduce the standard form of a $W^*$-algebra along central subalgebras.

Prepares groundwork for analyzing $W^*$-tensor products over common centers.

Abstract

We deal with the reduction theory of a $W^*$-algebra $M$ along a $W^*$-subalgebra $Z$ of the centre of $M$. This is done by using Hilbert modules naturally constructed by suitable spatial representations of the abelian $W^*$-algebra $Z$. We start with an exhaustive investigation of such kind of Hilbert modules, which is also of self-contained interest. After explaining the notion of the reduction in this framework, we exhibit the reduction of the standard form of a $W^*$-algebra $M$ along any $W^*$-subalgebra of its centre, containing the unit of $M$. In a forthcoming paper, this result is applied to study the structure of the standard representation of the $W^*$-tensor product $M_1\ots_Z M_2$ of two $W^*$-algebras $M_1$ and $M_2$ over a common $W^*$-subalgebra $Z$ of the centres.