The pure extension property for discrete crossed products

Abstract

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

A gap in the proof of the implication (iii => i) in Theorem 2.4 has been eliminated

Cited by in corpus (1)

• Aperiodicity: the almost extension property and uniqueness of pseudo-expectations