Amenability and subexponential spectral growth rate of Dirichlet forms on von Neumann algebras

Abstract

Abstract. In this work we apply Noncommutative Potential Theory to characterize (relative) amenability and the (relative) Haagerup Property (H) of von Neumann algebras in terms of the spectral growth of Dirichlet forms. Examples deal with (inclusions of) countable discrete groups and free orthogonal compact quantum groups.

Gradient forms and strong solidity of free quantum groups

“…We collect some final corollaries. Firstly, we recall the following result from [22]. We give their proof in terms of Stinespring dilations.…”

“…their generators) to obtain strong solidity for all free orthogonal and unitary quantum groups. Dirichlet forms have been studied extensively [18,[20][21][22]29,35,58,60]. In particular in [21] it was shown that in the tracial case a Dirichlet form always leads to a derivation as a square root.…”

“…Theorem 3.15 of[22]) Let M be a von Neumann algebra and suppose that there exists a conservative completely Dirichlet form Q associated with M such that Q has a complete set of eigenvectors with eigenvalues 0 ≤ λ 1 ≤ λ 2 ≤ λ 3 ≤ . .…”

The emergence of Noncommutative Potential Theory

“…( CS5]). To illustrate a first connection between approximation properties of von Neumann algebras and spectral properties of Dirichlet form, we first recall a definition.…”

“…Definition 7.3. (Spectral growth rate of Dirichlet forms [CS5]). Let (N, ω) be an infinite dimensional, σ-finite, von Neumann algebra with a fixed faithful, normal state on it.…”