The Haagerup approximation property for von Neumann algebras via quantum Markov semigroups and Dirichlet forms

TL;DR

This paper characterizes the Haagerup approximation property for von Neumann algebras using quantum Markov semigroups and Dirichlet forms, extending previous results to the non-tracial setting.

Contribution

It introduces a new characterization of the Haagerup property via quantum Markov semigroups and Dirichlet forms for von Neumann algebras with a faithful normal state.

Findings

Existence of unital, $$-preserving, KMS-symmetric approximating maps

Extension of tracial case results to non-tracial setting

Characterization of the Haagerup property via quantum Dirichlet forms

Abstract

The Haagerup approximation property for a von Neumann algebra equipped with a faithful normal state $\varphi$ is shown to imply existence of unital, $\varphi$-preserving and KMS-symmetric approximating maps. This is used to obtain a characterisation of the Haagerup approximation property via quantum Markov semigroups (extending the tracial case result due to Jolissaint and Martin) and further via quantum Dirichlet forms.