The pure extension property for discrete crossed products

Abstract

Let $G$ be a discrete group acting on a unital $C^*$-algebra $\mathcal{A}$ by $*$-automorphisms. In this note, we show that the inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes_r G$ has the pure extension property (so that every pure state on $\mathcal{A}$ extends uniquely to a pure state on $\mathcal{A} \rtimes_r G$) if and only if $G$ acts freely on $\mathcal{\widehat{A}}$, the spectrum of $\mathcal{A}$. The same characterization holds for the inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes G$. This generalizes what was already known for $\mathcal{A}$ abelian.