A uniqueness theorem for twisted groupoid C*-algebras

Abstract

We present a uniqueness theorem for the reduced C*-algebra of a twist $\mathcal{E}$ over a Hausdorff \'etale groupoid $\mathcal{G}$. We show that the interior $\mathcal{I}^\mathcal{E}$ of the isotropy of $\mathcal{E}$ is a twist over the interior $\mathcal{I}^\mathcal{G}$ of the isotropy of $\mathcal{G}$, and that the reduced twisted groupoid C*-algebra $C_r^*(\mathcal{I}^\mathcal{G}; \mathcal{I}^\mathcal{E})$ embeds in $C_r^*(\mathcal{G}; \mathcal{E})$. We also investigate the full and reduced twisted C*-algebras of the isotropy groups of $\mathcal{G}$, and we provide a sufficient condition under which states of (not necessarily unital) C*-algebras have unique state extensions. We use these results to prove our uniqueness theorem, which states that a C*-homomorphism of $C_r^*(\mathcal{G}; \mathcal{E})$ is injective if and only if its restriction to $C_r^*(\mathcal{I}^\mathcal{G}; \mathcal{I}^\mathcal{E})$ is injective. We also show that if $\mathcal{G}$ is effective, then $C_r^*(\mathcal{G}; \mathcal{E})$ is simple if and only if $\mathcal{G}$ is minimal.