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

TL;DR

This paper explores how the spectral growth of Dirichlet forms relates to amenability and the Haagerup Property in von Neumann algebras, using noncommutative potential theory with applications to groups and quantum groups.

Contribution

It introduces a novel approach linking spectral growth of Dirichlet forms to amenability and property (H) in von Neumann algebras, expanding understanding in noncommutative analysis.

Findings

Spectral growth rates characterize amenability.

Established criteria for the Haagerup Property via Dirichlet forms.

Applied results to groups and quantum groups.

Abstract

In this work we apply Noncommutative Potential Theory to prove (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.