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 $\phi$ is shown to imply existence of unital, $\phi$-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.

arXiv comment26 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

References in corpus (4)

• The Haagerup property for locally compact quantum groups
• Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory
• Haagerup approximation property and positive cones associated with a von Neumann algebra
• Haagerup approximation property for arbitrary von Neumann algebras

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• Convolution semigroups on locally compact quantum groups and noncommutative Dirichlet forms
• Maximal subgroups and von Neumann subalgebras with the Haagerup property
• Derivations and KMS-Symmetric Quantum Markov Semigroups
• Quantum dynamics on Orlicz spaces
• Pointwise convergence of noncommutative Fourier series
• Haagerup approximation property via bimodules
• $L_2$-cohomology, derivations and quantum Markov semi-groups on $q$-Gaussian algebras
• The emergence of Noncommutative Potential Theory
• Riesz transforms on compact quantum groups and strong solidity
• BMO spaces of $\sigma$-finite von Neumann algebras and Fourier-Schur multipliers on $S{U}_q(2)$