A decomposition theorem for unitary group representations on Kaplansky–Hilbert modules and the Furstenberg–Zimmer structure theorem

Abstract

In this paper, a decomposition theorem for (covariant) unitary group representations on Kaplansky–Hilbert modules over Stone algebras is established, which generalizes the well-known Hilbert space case (where it coincides with the decomposition of Jacobs, deLeeuw and Glicksberg).

The proof rests heavily on the operator theory on Kaplansky–Hilbert modules, in particular the spectral theorem for Hilbert–Schmidt homomorphisms on such modules.

As an application, a generalization of the celebrated Furstenberg–Zimmer structure theorem to the case of measure-preserving actions of arbitrary groups on arbitrary probability spaces is established.