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Let $G$ be a discrete group acting on a unital $C^*$-algebra $\mathcal{A}$ by $*$-automorphisms. We characterize (in terms of the dynamics) when the inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes_r G$ has a unique conditional expectation, and when it has a unique pseudo-expectation (in the sense of Pitts). Likewise for the inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes G$. As an application, we (slightly) strengthen results of Kishimoto and Archbold-Spielberg concerning $C^*$-simplicity of $\mathcal{A} \rtimes_r G$.

We initiate a study of operator approximate biprojectivity for quantum group algebra $L^{1}({\Bbb{G}})$ , where $\mathbb{G}$ is a locally compact quantum group. We show that if $L^{1}({\Bbb{G}})$ is operator approximately biprojective, then $\mathbb{G}$ is compact. We prove that if $\mathbb{G}$ is a compact quantum group and $\mathbb{H}$ is a non-Kac-type compact quantum group such that both $L^{1}({\Bbb{G}})$ and $L^{1}({\Bbb{H}})$ are operator approximately biprojective, then $L^{1}({\Bbb{G}})\widehat{\otimes}L^{1}({\Bbb{H}})$ is operator approximately biprojective, but not operator biprojective.