Amenable actions of discrete quantum groups on von Neumann algebras

TL;DR

This paper introduces Zimmer amenability for discrete quantum group actions on von Neumann algebras, generalizing key results and characterizing amenability via crossed product injectivity, with applications to actions on Poisson boundaries.

Contribution

It defines Zimmer amenability in the noncommutative setting and provides a characterization through crossed product injectivity, extending classical theory to quantum groups.

Findings

Zimmer amenability characterized by $ ext{hat} extbf{G}$-injectivity.

Actions on Poisson boundaries are always amenable.

Generalizes fundamental results of the theory to noncommutative case.

Abstract

We introduce the notion of Zimmer amenability for actions of discrete quantum groups on von Neumann algebras. We prove generalizations of several fundamental results of the theory in the noncommutative case. In particular, we give a characterization of Zimmer amenability of an action $\alpha:\Bbb G\curvearrowright N$ in terms of $\hat{\Bbb{G}}$-injectivity of the von Neumann algebra crossed product $N\ltimes_\alpha\Bbb G$. As an application we show that the actions of any discrete quantum group on its Poisson boundaries are always amenable.