The Haagerup property for groups and for tracial von Neumann algebras in terms of invariant and mixing states

TL;DR

This paper characterizes the Haagerup property for groups and von Neumann algebras through approximations of invariant states by mixing states, offering new insights into their structure and properties.

Contribution

It introduces novel characterizations of the Haagerup property using invariant and mixing states for groups and von Neumann algebras.

Findings

Provides characterizations of the Haagerup property via state approximations.

Connects invariant states with mixing states in the context of operator algebras.

Enhances understanding of the Haagerup property in non-ergodic and tracial settings.

Abstract

The aim of the article is to provide characterizations of the Haage-rup property for locally compact, second countable groups in terms of approximations of some non-ergodic invariant states by mixing ones for actions on unital $C^*$-algebras one the one hand, and for pairs of tracial von Neumann algebras by mixing binormal states on the other hand.