Connections between Hilbert W*-modules and direct integrals

Investigating the direct integral decomposition of von Neumann algebras of bounded module operators on self-dual Hilbert W*-modules 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 A-modules with countably generated Apre-dual Hilbert A-module over commutative separable W*-algebras 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 A with a special approximation property (*) every normal bounded A-linear operator on a self-dual Hilbert A-module with countably generated A-pre-dual Hilbert A-module is decomposable into the sum of a diagonalizable normal and of a ”compact” bounded A-linear operator on that module. The idea to investigate the subject treated in the present paper arose in discussions with K. Schmüdgen and J. Friedrich at the University of Leipzig. They suggested to the author that self-dual Hilbert W*-modules over commutative W*-algebras might be closely connected with direct integrals of measurable fields of Hilbert spaces or, respectively, with some topologically related objects. Moreover, von Neumann algebras of bounded module operators on these self-dual Hilbert W*-modules should be decomposable into direct integrals of measurable fields of von Neumann algebras in a very easy way. Following this line appropriate facts have been proved. One gets a new view on the nowadays well-known theory of direct integral decomposition of von Neumann algebras M on separable Hilbert spaces. This theory is shown to be equivalent to the theory of von Neumann representations of W*-algebras M on self-dual Hilbert W*modules H over W*-subalgebras B of the center of M, where B has to be separable and the Hilbert B-modules have to possess countably generated B-pre-dual Hilbert Bmodules. The most interesting point is that the basic structures, Hilbert W*-modules and direct integrals of Hilbert spaces, are quite different. However, this equivalence will not be preserved turning to direct integrals of von Neumann algebras on nonseparable Hilbert spaces, in general. It would be interesting to make further considerations in this direction taking in account recent results of R. Schaflitzel [25, 26], P. Richter [23] and other authors [15, 16, 37, 40]. Applicating this equivalence principle, a new result is found generalizing theorems of H. Weyl, I. D. Berg and G. J. Murphy. Last but not least one realizes that the forthcomming theory is closely related to the describtion of self-dual Hilbert AW*-modules over commutative AW*-algebras in terms of Boolean valued analysis and logic created by M. Ozawa and G. Takeuti [20, 21, 33, 34, 35] ([19]) during 1979-85. There are also relations to the work of H. Takemoto [29, 30, 31] who has described similar phenomena in terms of continuous fields of Hilbert spaces. In the present more special case the mathematical terminology describing the situation is taken from measure theory. The present paper is organized as follows: The first section is a short summary of facts from the theory of direct integrals of measurable fields of Hilbert spaces and of von Neumann algebras, at one side, and from the theory of Hilbert W*-modules over commutative Hilbert W*-algebras, at the other. We slightly modify the traditional denotations for our purposes and recall some necessary facts from the literature. The second section deals with the interrelation between self-dual Hilbert W*-modules over commutative W*-algebras A possessing a countably generated A-pre-dual Hilbert Amodule and special sets of mappings into measurable fields of Hilbert spaces, giving rise to isomorphisms. Considering von Neumann algebras of bounded module operators on those Hilbert W*-modules we obtain their direct integral decomposition. As an application for commutative W*-algebras A with a special property (*) we prove that on self-dual Hilbert A-modules which possess a countably generated A-pre-dual Hilbert A-module every normal bounded module operator T is decomposable into the sum of a normal, diagonalizable bounded module operator D and a ”compact” bounded module operator K. The author thanks J. Friedrich, A. Kasparek, P. Richter, R. Schaflitzel and K. Schmüdgen for helpful discussions and suggestions during the time of preparation of the paper.