The pure extension property for discrete crossed products

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