Direct integrals and Hilbert W*-Modules

Details Direct integrals and Hilbert W*-Modules Direct integrals and Hilbert W*-Modules

1993-12-06

arxiv.org/abs/funct-an/9312002v1

Problems in algebra, geometry and discrete mathematics (in russian), Moscow State University, Fakulty of Mechanics and Mathema

Mathematics

Operator Algebras

20 pages, LATEX file, preprint 23/91, NTZ, Univ. Leipzig, Germany

Scientific paper

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.

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