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

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.