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

Let G be a compact quantum group and A⊆ B an inclusion of σ-finite G-dynamical von Neumann algebras. We prove that the G-inclusion A⊆ B is strongly equivariantly amenable if and only if it is equivariantly amenable, using techniques from the theory of non-commutative Lᵖ-spaces. In particular, if (A, α) is a G-dynamical von Neumann algebra with A σ-finite, the action α: A G is strongly (inner) amenable if and only if the action α: A G is (inner) amenable. By duality, we also obtain the same result for 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.