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.