Hilbert C*-modules and amenable actions

TL;DR

This paper investigates the properties of group actions on Hilbert C*-modules derived from topological actions, demonstrating non-amenability under certain conditions related to the group's amenability and measure invariance.

Contribution

It introduces new results linking group non-amenability and non-a-T-menable groups to their actions on Hilbert C*-modules with quasi-invariant measures.

Findings

Non-amenability of actions of non-amenable groups.

Non-a-T-menable groups also exhibit non-amenable actions.

Existence of quasi-invariant measures close to invariant measures influences amenability.

Abstract

We study actions of discrete groups on Hilbert $C^*$-modules induced from topological actions on compact Hausdorff spaces. We show non-amenability of actions of non-amenable and non-a-T-menable groups, provided there exists a quasi-invariant probability measure which is sufficiently close to being invariant.