Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Conjugacy co-amenability

Mehrdad Kalantar, Srivatsav Kunnawalkam Elayavalli

Abstract

In this note we study a natural analytic property of inclusions of groups akin to co-amenability: the property of existence of a non-compactly supported invariant state for the conjugation action of a group $G$ on the von Neumann algebra generated by the characteristic functions $\{\mathbf{1}_{gHg^{-1}}\}_{g\in G}$ viewed inside $\ell^\infty(G)$. Some interesting settings and examples of this phenomena are proved. We also comment on a consideration related to proper proximality, which motivated this property.

Conjugacy co-amenability

Abstract

In this note we study a natural analytic property of inclusions of groups akin to co-amenability: the property of existence of a non-compactly supported invariant state for the conjugation action of a group on the von Neumann algebra generated by the characteristic functions viewed inside . Some interesting settings and examples of this phenomena are proved. We also comment on a consideration related to proper proximality, which motivated this property.
Paper Structure (4 sections, 11 theorems, 5 equations)

This paper contains 4 sections, 11 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Conjugacy co-amenability
  3. Some (non)examples
  4. A comment on proper proximality

Key Result

Lemma 2.1

The family $\widetilde{{\mathcal{X}}}:=\{\widetilde{X} : X\in \mathcal{W}(H)\}$ is pairwise disjoint, and $X = \bigcup_{Y\in \mathcal{W}(H), Y\subseteq X}\widetilde{Y}$ for every $X\in \mathcal{W}(H)$.

Theorems & Definitions (28)

  • Definition 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Proposition 2.5
  • proof
  • ...and 18 more