Actions of quantum groups on dual operator spaces and their crossed products

We study the category of dual operator spaces equipped with an action of a locally compact quantum group $\mathbb{G}$. The Fubini crossed product functor $-\rtimes^\mathcal{F} \mathbb{G}$ and the weak$^*$-crossed product functor $-\bar{\rtimes}\mathbb{G}$ are shown to be equal if and only if $\mathbb{G}$ has the approximation property of Haagerup and Kraus. Using the natural isomorphism $-\rtimes^\mathcal{F}\mathbb{G}\cong {}_{L^1(\mathbb{G})}\mathcal{CB}(B(L^2(\mathbb{G}))_*, -)$, this leads to a characterization of the approximation property of $\mathbb{G}$ via an $L^1(\mathbb{G})$-module approximation property for $B(L^2(\mathbb{G}))_*$. Finally, exactness of the Fubini crossed product functor is investigated and related to amenability properties of $\mathbb{G}$.