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

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 provide an example of a co-amenable (non-Kac) compact quantum group that acts non-amenably on a von Neumann algebra. By duality, this gives an explicit example of an amenable discrete quantum group that acts non-amenably on a von Neumann algebra.