Inclusions of C*-algebras of graded groupoids

Abstract

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 both the full and reduced C*-algebras of this subgroupoid embed isometrically into the full and reduced C*-algebras of $G$; this extends a theorem of Kaliszewski--Quigg--Raeburn from the étale to the non-étale setting. 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.

12 pages. This version matches the version in the Journal of Operator Theory

References in corpus (3)

• On ${\mathrm{C}}^{\ast }$-algebras associated to product systems
• Groupoid algebras as covariance algebras
• Strongly graded groupoid and directed graph algebras