Direct integrals and Hilbert W*-Modules

TL;DR

This paper explores the decomposition of von Neumann algebras and Hilbert W*-modules, establishing a connection between their direct integral theories and proving a Weyl--Berg--Murphy type theorem for certain operator decompositions.

Contribution

It introduces an equivalence principle linking direct disintegration of von Neumann algebras with representations on self-dual Hilbert modules, and proves a new decomposition theorem for bounded operators.

Findings

Established a connection between two theories of von Neumann algebras and Hilbert W*-modules.

Proved a decomposition theorem for normal bounded A-linear operators on self-dual Hilbert A-modules.

Provided examples illustrating bounds and possibilities for generalizations.

Abstract

Investigating the direct integral decomposition of von Neumann algebras of bounded module operators on self-dual Hilbert W*-moduli an equivalence principle is obtained which connects the theory of direct disintegration of von Neumann algebras on separable Hilbert spaces and the theory of von Neumann representations on self-dual Hilbert {\bf A}-moduli with countably generated {\bf A}-pre-dual Hilbert {\bf A}-module over commutative separable W*-algebras {\bf A}. Examples show posibilities and bounds to find more general relations between these two theories, (cf. R. Schaflitzel's results). As an application we prove a Weyl--Berg--Murphy type theorem: For each given commutative W*-algebra {\bf A} with a special approximation property (*) every normal bounded {\bf A}-linear operator on a self-dual Hilbert {\bf A}-module with countably generated {\bf A}-pre-dual Hilbert {\bf A}-module is decomposable into the sum of a diagonalizable normal and of a ''compact'' bounded {\bf A}-linear operator on that module.