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 étale 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 $C_r^{\ast }(\mathcal{G};\mathcal{E})$ is injective if and only if its restriction to $C_r^{\ast }({\mathcal{I}}^{\mathcal{G}};{\mathcal{I}}^{\mathcal{E}})$ 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.

26 pages. This version matches the version in the Journal of Functional Analysis. The author would like to thank the anonymous referee for their careful reading and helpful suggestions

References in corpus (4)

• A non-amenable groupoid whose maximal and reduced $C^{\ast }$-algebras are the same
• An elementary approach to C*-algebras associated to topological graphs
• Reconstruction of twisted Steinberg algebras
• Simplicity of twisted C*-algebras of Deaconu--Renault groupoids

Cited by in corpus (6)

• Alexandrov groupoids and the nuclear dimension of twisted groupoid ${\mathrm{C}}^{\ast }$-algebras
• The local bisection hypothesis for twisted groupoid C*-algebras
• Simplicity of twisted C*-algebras of Deaconu--Renault groupoids
• A twist over a minimal étale groupoid that is topologically nontrivial over the interior of the isotropy
• Relative topological principality and the ideal intersection property for groupoid C*-algebras
• Detecting ideals in reduced crossed product C*-algebras of topological dynamical systems