A uniqueness theorem for twisted groupoid C*-algebras

Abstract

We present a uniqueness theorem for the reduced C*-algebra of a twist E over a Hausdorff étale groupoid G. We show that the interior IE of the isotropy of E is a twist over the interior IG of the isotropy of G, and that the reduced twisted groupoid C*-algebra Cr⁎(IG;IE) embeds in Cr⁎(G;E). We also investigate the full and reduced twisted C*-algebras of the isotropy groups of 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 Cr⁎(G;E) is injective if and only if its restriction to Cr⁎(IG;IE) is injective. We also show that if G is effective, then Cr⁎(G;E) is simple if and only if G is minimal.