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

Nikolai Edeko, Markus Haase, Henrik Kreidler

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.

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, 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.

Paper Structure

This paper contains 37 sections, 56 theorems, 220 equations.

Table of Contents

  1. Stone Algebras and Kaplansky--Hilbert Modules
  2. Lattice-Normed Spaces and Stone Algebras
  3. Hilbert Modules
  4. Lattice-Normed Spaces
  5. Stone Algebras and Stonean Spaces
  6. Kaplansky--Hilbert Modules
  7. Support of Elements and Modules
  8. Bounded Module Homomorphisms
  9. (Sub)Orthonormal Systems
  10. Operator Theory on KH-Modules
  11. Modules and Homomorphisms of Finite Rank
  12. Hilbert--Schmidt Homomorphisms
  13. Order-Completion and Tensor Products
  14. Hilbert Bundle Representation
  15. Spectral Theory of Hilbert--Schmidt Homomorphisms on KH-Modules
  16. ...and 22 more sections

Key Result

lemma 1

Let $E,F$ be lattice-normed modules over a unital, commutative C$^*$-algebra $\A$. Then the following assertions hold:

Theorems & Definitions (126)

  • definition 1
  • definition 2
  • definition 3
  • lemma 1
  • proof
  • definition 4
  • proposition 1
  • proof
  • lemma 2
  • proof
  • ...and 116 more