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

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