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A Peter-Weyl theorem for compact group bundles and the geometric representation of relatively ergodic compact extensions

Nikolai Edeko, Asgar Jamneshan, Henrik Kreidler

TL;DR

The paper develops a comprehensive geometric representation theory for relatively ergodic extensions with relative discrete spectrum by transferring ergodic questions to topological dynamics and introducing a Peter–Weyl framework for bundles of compact groups. It proves a Kenney–Mackey–Zimmer type classification: such extensions are precisely skew-products built from open compact group bundles acting fiberwise on homogeneous spaces, with a robust topological and operator-algebraic foundation. A key technical advance is a Peter–Weyl theorem for open compact group bundles and homogeneous-space bundles, enabling a fiberwise harmonic analysis that underpins the representation results. The work unifies uncountable-group and non-separable settings, provides topological models, and yields a clear ergodic-theoretic realization via relative skew-products, with potential applications to Furstenberg–Zimmer–Host–Kra style structure theory in broad contexts.

Abstract

We show that a relatively ergodic extension of measure-preserving dynamical systems has relative discrete spectrum if and only if it can be represented as a skew-product by a bundle of compact homogeneous spaces. Our result holds without restrictions on the acting group or the underlying probability spaces. This generalizes previous work by Mackey, Zimmer, Ellis, Austin, and the second author and Tao, and is inspired by the Furstenberg-Zimmer and Host-Kra structure theories for actions of uncountable groups. Our approach translates the ergodic-theoretic question into topological dynamics, where we establish a corresponding classification: an extension in topological dynamics has relative discrete spectrum precisely when it admits a skew-product representation by bundles of compact homogeneous spaces. A key step in our argument is establishing a Peter-Weyl-type theorem for bundles of compact groups which might be of independent interest.

A Peter-Weyl theorem for compact group bundles and the geometric representation of relatively ergodic compact extensions

TL;DR

The paper develops a comprehensive geometric representation theory for relatively ergodic extensions with relative discrete spectrum by transferring ergodic questions to topological dynamics and introducing a Peter–Weyl framework for bundles of compact groups. It proves a Kenney–Mackey–Zimmer type classification: such extensions are precisely skew-products built from open compact group bundles acting fiberwise on homogeneous spaces, with a robust topological and operator-algebraic foundation. A key technical advance is a Peter–Weyl theorem for open compact group bundles and homogeneous-space bundles, enabling a fiberwise harmonic analysis that underpins the representation results. The work unifies uncountable-group and non-separable settings, provides topological models, and yields a clear ergodic-theoretic realization via relative skew-products, with potential applications to Furstenberg–Zimmer–Host–Kra style structure theory in broad contexts.

Abstract

We show that a relatively ergodic extension of measure-preserving dynamical systems has relative discrete spectrum if and only if it can be represented as a skew-product by a bundle of compact homogeneous spaces. Our result holds without restrictions on the acting group or the underlying probability spaces. This generalizes previous work by Mackey, Zimmer, Ellis, Austin, and the second author and Tao, and is inspired by the Furstenberg-Zimmer and Host-Kra structure theories for actions of uncountable groups. Our approach translates the ergodic-theoretic question into topological dynamics, where we establish a corresponding classification: an extension in topological dynamics has relative discrete spectrum precisely when it admits a skew-product representation by bundles of compact homogeneous spaces. A key step in our argument is establishing a Peter-Weyl-type theorem for bundles of compact groups which might be of independent interest.
Paper Structure (30 sections, 64 theorems, 166 equations)

This paper contains 30 sections, 64 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Related literature
  3. Innovations and main results
  4. Convention and notation
  5. Topological bundles
  6. Basic concepts
  7. Continuous fiber maps
  8. Fiber measures and relative measures
  9. Banach bundles
  10. Representations of compact group bundles
  11. Compact group bundles
  12. Group bundle C*-algebras
  13. Representations and the dual space
  14. Positive definite functions
  15. Topology of the dual
  16. ...and 15 more sections

Key Result

Theorem 1.1

For an open relatively topologically ergodic extension $q \colon (K;\varphi) \rightarrow (L;\psi)$ with $L$ being Stonean consider the following assertions. Then (a) implies (b). Conversely, if (b) holds and the subgroup bundle $p|_U\colon U \rightarrow L_{\operatorname{fix}}$ in (ii) can also be chosen to be open, then (a) holds.

Theorems & Definitions (140)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Example 2.5
  • Corollary 2.6
  • Definition 2.7
  • ...and 130 more