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We introduce the notion of Zimmer amenability for actions of discrete quantumgroups on von Neumann algebras. We prove generalizations of several fundamentalresults of the theory in the noncommutative case. In particular, we give acharacterization of Zimmer amenability of an action $\alpha:\BbbG\curvearrowright N$ in terms of $\hat{\Bbb{G}}$-injectivity of the von Neumannalgebra crossed product $N\ltimes_\alpha\Bbb G$. As an application we show thatthe actions of any discrete quantum group on its Poisson boundaries are alwaysamenable.

Amenable actions of discrete quantum groups on von Neumann algebras