The pure extension property for discrete crossed products

TL;DR

This paper characterizes when the inclusion of a unital C*-algebra into its reduced crossed product by a discrete group has the pure extension property, linking it to the freeness of the group action on the spectrum of the algebra.

Contribution

It generalizes the known abelian case to non-abelian C*-algebras, providing a complete characterization of the pure extension property for both reduced and full crossed products.

Findings

Pure extension property holds iff the group acts freely on the spectrum.

Characterization applies to both reduced and full crossed products.

Generalizes previous results from abelian to non-abelian C*-algebras.

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.