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

Abstract

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.

Convolution semigroups on locally compact quantum groups and noncommutative Dirichlet forms

“…and for α > 0, consider the positive contraction R α,n := (1 + α∆ n ) −1 ∈ M(A). By [CaS,Lemma 6.4], C i is invariant under R α,n for every i ∈ I. Since θ n ≤ 1, we have…”

“…The approach of our proof is different from that of [AkW, DFSW], and instead relies on Proposition 4.10: roughly speaking, instead of seeking a generator with the desirable properties, we construct the semigroup directly, following the technique of [Sau], applied later for example in [CaS]. We note that Theorem 4.12 compares with the characterisation of the Haagerup Property of von Neumann algebras obtained recently in [CaS,Theorem 6.7].…”

“…Proof. We follow closely the proof of [CaS,Proposition 6.5], explaining which changes should be made and where. Write 1 := 1 M(A) .…”

“…In the next corollary, which extends [CaS,Proposition 6.6], we use the following notation. For k ∈ N, write L 2 (M k ) := L 2 (M k , tr k ), where tr k is the canonical (non-normalised) trace on M k .…”

Harmonic analysis and BMO-spaces of free Araki--Woods factors

“…[CiSa03], [CFK14]. Semi-groups naturally appear in approximation properties of von Neumann algebras [JoMa04], [CaSk15]. Also the approach by Ozawa-Popa [OzPo10] and Peterson [Pet09] yields new deformation-rigidity properties of von Neumann algebras through the theory of semi-groups and derivations (see also [Avs11]).…”