A uniqueness theorem for twisted groupoid C*-algebras

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 $C_r^{\ast }({\mathcal{I}}^{\mathcal{G}};{\mathcal{I}}^{\mathcal{E}})$ embeds in $C_r^{\ast }(\mathcal{G};\mathcal{E})$. 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}$; $\mathcal{E}$)$isinjectiveifandonlyifitsrestrictionto$C_r^($\mathcal{I}$^$\mathcal{G}$; $\mathcal{I}$^$\mathcal{E}$)$ is injective. We also show that if $\mathcal{G}$ is effective, then $C_r^{\ast }(\mathcal{G};\mathcal{E})$ is simple if and only if $\mathcal{G}$ is minimal.