Group amenability properties for von Neumann algebras

TL;DR

This paper extends fixed-point theorems for amenable groups to the setting of G-amenable von Neumann algebras, broadening the understanding of amenability in operator algebras.

Contribution

It proves a fixed-point theorem for G-amenable von Neumann algebras, generalizing classical fixed-point properties to a broader algebraic context.

Findings

Established a fixed-point theorem for G-amenable von Neumann algebras

Extended F{\

Abstract

In his study of amenable unitary representations, M. E. B. Bekka asked if there is an analogue for such representations of the remarkable fixed-point property for amenable groups. In this paper, we prove such a fixed-point theorem in the more general context of a $G$-amenable von Neumann algebra $M$, where $G$ is a locally compact group acting on $M$. The F{\o}lner conditions of Connes and Bekka are extended to the case where $M$ is semifinite and admits a faithful, semifinite, normal trace which is invariant under the action of $G$.