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 $\varphi$ 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. 
Martijn Caspers, Adam Skalski
math.OA