Amenable actions of compact and discrete quantum groups on von Neumann algebras
K. De Commer, J. De Ro
TL;DR
The paper shows that for actions of compact (and by duality, discrete) quantum groups on σ-finite von Neumann algebras, equivariant amenability coincides with strong equivariant amenability. It develops a cotensor-product framework for equivariant correspondences, linking L^2-containment to algebraic representations via the corner $ heta_{ aisebox{0pt}{$oxed{ }$}}$ on cotensor algebras and non-commutative $L^p$-space techniques. The results yield that strong and ordinary amenability (and inner amenability) are equivalent in this dynamical setting and provide new explicit examples of amenable discrete quantum groups acting non-amenably on von Neumann algebras, including Podleś-sphere phenomena. These findings extend Tomatsu's amenability/co-amenability duality into a dynamical context and hint at deeper connections with tensor categories and quasi-subgroups of quantum groups.
Abstract
Let $\mathbb{G}$ be a compact quantum group and $A\subseteq B$ an inclusion of $σ$-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 $(A, α)$ is a $\mathbb{G}$-dynamical von Neumann algebra with $A$ $σ$-finite, the action $α: A \curvearrowleft \mathbb{G}$ is strongly (inner) amenable if and only if the action $α: A \curvearrowleft \mathbb{G}$ 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.