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

2021/04/10 by Edeko, Nikolai, Haase, Markus, Kreidler, Henrik · 1 citation
#06E15 #43A65 #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Primary 37A15 #Secondary 46H25

paper · doi:10.48550/arxiv.2104.04865

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.

Cited by

• On the Jacobs-de Leeuw-Glicksberg decomposition

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