The pure extension property for discrete crossed products

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...

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Bibliographic Details
Published in:arXiv.org 2017-08
Main Author: Zarikian, Vrej
Format: Article
Language:English
Subjects:
Automorphisms
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Summary: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.
ISSN:2331-8422