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Amenability and skew-amenability of actions of topological groups

Vadim Alekseev, Hiroshi Ando, Friedrich Martin Schneider, Andreas Thom

TL;DR

This work develops a robust framework for amenability of actions of topological groups on compact spaces, introducing AD-amenability and skew-amenability and linking them to exactness. It proves that for locally compact groups AD-amenability and skew-amenability coincide, and that for second-countable LC groups amenability of actions coincides with AD-amenability, tying these notions to measurewise amenability. The authors identify natural subgroup conditions—amenably embedded and well-placed—under which amenability of a group transfers to subgroups, and they show that metrizability of the universal minimal flow implies amenability of the action and hence exactness. They further establish that amenability behaves well under products and inverse limits, derive Kirschberg-type exactness results, and apply the theory to unitary groups and L^0-spaces, yielding broad consequences for universal minimal flows and exactness in non-locally compact settings.

Abstract

We define and study notions of amenability and skew-amenability of continuous actions of topological groups on compact topological spaces. Our main motivation is the question under what conditions amenability of a topological group passes to a closed subgroup. Other applications include the understanding of the universal minimal flow of various non-amenable groups.

Amenability and skew-amenability of actions of topological groups

TL;DR

This work develops a robust framework for amenability of actions of topological groups on compact spaces, introducing AD-amenability and skew-amenability and linking them to exactness. It proves that for locally compact groups AD-amenability and skew-amenability coincide, and that for second-countable LC groups amenability of actions coincides with AD-amenability, tying these notions to measurewise amenability. The authors identify natural subgroup conditions—amenably embedded and well-placed—under which amenability of a group transfers to subgroups, and they show that metrizability of the universal minimal flow implies amenability of the action and hence exactness. They further establish that amenability behaves well under products and inverse limits, derive Kirschberg-type exactness results, and apply the theory to unitary groups and L^0-spaces, yielding broad consequences for universal minimal flows and exactness in non-locally compact settings.

Abstract

We define and study notions of amenability and skew-amenability of continuous actions of topological groups on compact topological spaces. Our main motivation is the question under what conditions amenability of a topological group passes to a closed subgroup. Other applications include the understanding of the universal minimal flow of various non-amenable groups.

Paper Structure

This paper contains 14 sections, 39 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Uniform spaces and the UEB topology
  3. Amenability of actions and exactness of topological groups
  4. Comparison in the locally compact case
  5. AD-amenability vs. skew-amenability
  6. AD-amenability vs. amenability for second-countable locally compact groups
  7. Amenably embedded and well-placed subgroups
  8. Exactness of Polish groups with metrizable UMF
  9. Products and inverse limits
  10. Products of actions
  11. Inverse limits
  12. Further examples and applications
  13. Exactness of unitary groups
  14. Well-placedness in $L^{0}$ groups

Key Result

Lemma 3.4

Let $X$ be a compact space an let $E$ be a locally convex space. Let $D \subseteq E$ and $K \mathrel{\mathop:}= \overline{D}$. Then is dense in $C(X,K)$ with respect to the topology of uniform convergence.

Theorems & Definitions (89)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 3.1: Anantharaman-Delaroche
  • Definition 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 79 more