Direct integrals and Hilbert W*-Modules

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.