Conjugacy co-amenability

TL;DR

This paper introduces and explores a property related to co-amenability in groups, focusing on invariant states for conjugation actions within von Neumann algebras, with examples and connections to proper proximality.

Contribution

It defines a new analytic property of group inclusions akin to co-amenability and provides examples and insights into its relation to proper proximality.

Findings

Identifies a property of invariant states for conjugation actions

Provides examples illustrating this property in various group settings

Discusses the relation to proper proximality

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.