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

Abstract

The Haagerup approximation property for a von Neumann algebra equipped with a faithful normal state $φ$ is shown to imply existence of unital, $φ$-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.

26 pages; v3 adds Corollary 5.8, corrects a mistake in Section 3 and updates references; all the main results remain unchanged. The article will appear in the Communications in Mathematical Physics