The Haagerup Approximation Property for von Neumann Algebras via Quantum Markov Semigroups and Dirichlet Forms

Martijn Caspers,Adam Skalski

doi:10.1007/s00220-015-2302-3

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.

Highlights

The Haagerup approximation property for a finite von Neumann algebra M equipped with a faithful normal tracial state τ, motivated by the celebrated Haagerup property for discrete groups, was introduced in [Cho]
It asserts the existence of a family of completely positive, normal, τ -non-increasing maps on M whose L2implementations are compact and converge strongly to the identity. This property was later studied in depth by Jolissaint ([Jol]), who showed that it does not depend on the choice of the trace, and that the approximating maps can be chosen unital and trace preserving
If is a group with the Haagerup property it follows from the characterisation via a conditionally negative definite function ψ on that the corresponding approximating maps for the von Neumann algebra of —which is known to have the Haagerup approximation property—can be chosen so that they form a semigroup

Summary

Introduction

The Haagerup approximation property for a finite von Neumann algebra M equipped with a faithful normal tracial state τ , motivated by the celebrated Haagerup property for discrete groups (see [CCJJV]), was introduced in [Cho] It asserts the existence of a family of completely positive, normal, τ -non-increasing maps on M whose L2implementations are compact and converge strongly to the identity. In the current article we show that in the case of a von Neumann algebra M equipped with a faithful normal state φ, which has the Haagerup approximation property, one can always choose the approximating maps to be unital and φ-preserving (in other words Markov) This is perhaps surprisingly rather technical, and the most difficult part of the proof is showing that the maps can be chosen to be contractive (Lemma 4.10). As in [CS] we assume that all the von Neumann algebras considered in the paper have separable preduals

Preliminaries Regarding Haagerup L p-Spaces and Related Embeddings

Definition of the Symmetric Haagerup Property

Markov Property of the Approximating Maps and Weight Independence

Haagerup Property for Arbitrary Von Neumann Algebras Revisited

Haagerup Property via KMS Symmetric Markov Semigroups

Haagerup Property via Dirichlet Forms