A Decomposition Theorem for Unitary Group Representations on Kaplansky-Hilbert Modules and the Furstenberg-Zimmer Structure Theorem

TL;DR

This paper extends the decomposition of unitary group representations from Hilbert spaces to Kaplansky-Hilbert modules and applies it to generalize the Furstenberg-Zimmer structure theorem for measure-preserving group actions.

Contribution

It introduces a new decomposition theorem for unitary representations on Kaplansky-Hilbert modules and generalizes the Furstenberg-Zimmer structure theorem to broader group actions.

Findings

Decomposition theorem for covariant unitary representations on Kaplansky-Hilbert modules.

Generalization of Furstenberg-Zimmer structure theorem to arbitrary groups and probability spaces.

Operator theory techniques applied to spectral analysis on Kaplansky-Hilbert modules.

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, de Leeuw 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.