Inclusions of C*-algebras of graded groupoids

Becky Armstrong, Lisa Orloff Clark, Astrid an Huef

Published 2021-07-08Version 1

We consider a locally compact Hausdorff groupoid $G$ which is graded over a discrete group. Then the fibre over the identity is an open and closed subgroupoid $G_e$. We show that the full C*-algebra of this subgroupoid embeds isometrically into the full C*-algebra of the groupoid; this extends a theorem of Kaliszewski--Quigg--Raeburn from the \'etale to the non-\'etale setting. We use the same ideas to investigate a possible embedding of the reduced C*-algebra of the subgroupoid in the reduced C*-algebra of the groupoid, and find that there is an obstruction in the kernel of the quotient map from the full to the reduced C*-algebras of $G_e$. As an application we show that the full and reduced C*-algebras of $G$ are topologically graded in the sense of Exel, and we discuss the full and reduced C*-algebras of the associated bundles.