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

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.

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