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 $\mathcal{ A} \rtimes_r G$.
Categories: math.OA