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Ultracoproducts and weak containment for flows of topological groups

Andy Zucker

Abstract

We develop the theory of ultracoproducts and weak containment for flows of arbitrary topological groups. This provides a nice complement to corresponding theories for p.m.p. actions and unitary representations of locally compact groups. For the class of locally Roelcke precompact groups, the theory is especially rich, allowing us to define for certain families of $G$-flows a suitable compact space of weak types. When $G$ is locally compact, all $G$-flows belong to one such family, yielding a single compact space describing all weak types of $G$-flows.

Ultracoproducts and weak containment for flows of topological groups

Abstract

We develop the theory of ultracoproducts and weak containment for flows of arbitrary topological groups. This provides a nice complement to corresponding theories for p.m.p. actions and unitary representations of locally compact groups. For the class of locally Roelcke precompact groups, the theory is especially rich, allowing us to define for certain families of -flows a suitable compact space of weak types. When is locally compact, all -flows belong to one such family, yielding a single compact space describing all weak types of -flows.
Paper Structure (16 sections, 40 theorems, 27 equations)

This paper contains 16 sections, 40 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Ultracoproducts of compact spaces
  4. $G$-continuity and $G$-compactification
  5. Ultracoproducts of $G$-flows
  6. The Vietoris topology and weak containment
  7. Fubini sums and Tietze extensions
  8. Gleason complete flows and their relatives
  9. The Gleason completion
  10. Lower semi-continuous metrics via seminorms
  11. Weak Rigidity
  12. LRPC groups
  13. Weak types for flows of LRPC groups
  14. Discrete groups
  15. LRPC groups
  16. ...and 1 more sections

Key Result

Proposition 4.2

If $G$ is a non-discrete topological group, then there are an infinite set $I$ and $\mathcal{U}\in \beta I$ with $\mathrm{C}(\Sigma_\mathcal{U}^G \mathrm{Sa}(G))\subsetneq \mathrm{C}_G(\Sigma_\mathcal{U} \mathrm{Sa}(G))$

Theorems & Definitions (119)

  • Claim
  • proof
  • Claim
  • proof
  • Example 2.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.5
  • Example 3.6
  • Definition 3.7
  • ...and 109 more