Amenable actions of compact and discrete quantum groups on von Neumann algebras

TL;DR

This paper establishes the equivalence of amenability notions for actions of compact and discrete quantum groups on von Neumann algebras, extending duality results and providing explicit examples of amenable quantum groups with non-amenable actions.

Contribution

It proves the equivalence of strong and standard amenability for quantum group actions on von Neumann algebras and offers the first explicit examples of amenable quantum groups with non-amenable actions.

Findings

Strong equivariant amenability is equivalent to equivariant amenability for quantum group actions.

A discrete quantum group is inner amenable if and only if it is strongly inner amenable.

Explicit examples of amenable quantum groups acting non-amenably on von Neumann algebras.

Abstract

Let $\mathbb{G}$ be a compact quantum group and $A\subseteq B$ an inclusion of $\sigma$-finite $\mathbb{G}$-dynamical von Neumann algebras. We prove that the $\mathbb{G}$-inclusion $A\subseteq B$ is strongly equivariantly amenable if and only if it is equivariantly amenable, using techniques from the theory of non-commutative $L^p$-spaces. In particular, if $(A, \alpha)$ is a $\mathbb{G}$-dynamical von Neumann algebra with $A$ $\sigma$-finite, the action $\alpha: A \curvearrowleft \mathbb{G}$ is strongly (inner) amenable if and only if the action $\alpha: A \curvearrowleft \mathbb{G}$ is (inner) amenable. By duality, we also obtain the same result for $\mathbb{G}$ a discrete quantum group, so that, in particular, a discrete quantum group is inner amenable if and only it is strongly inner amenable. This result can be seen as a dynamical generalization of Tomatsu's result on the amenability/co-amenability duality. We also provide the first explicit examples of amenable discrete quantum groups that act non-amenably on a von Neumann algebra.