Unique expectations for discrete crossed products

Abstract

Let $G$ be a discrete group acting on a unital $C^{\ast }$-algebra $\mathcal{A}$ by $\ast$-automorphisms. We characterize (in terms of the dynamics) when the inclusion $\mathcal{A}\subseteq \mathcal{A}{⋊}_{r}G$ has a unique conditional expectation, and when it has a unique pseudoexpectation in the sense of Pitts; we do likewise for the inclusion $\mathcal{A}\subseteq \mathcal{A}⋊G$. As an application, we re-prove (and potentially extend) some known $C^{\ast }$-simplicity results for $\mathcal{A}{⋊}_{r}G$.