  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.

Operator Algebras