Inclusions of C*-algebras of graded groupoids

TL;DR

This paper extends the embedding results of C*-algebras from subgroupoids to the entire graded groupoid in a non-étale setting, showing that these algebras are topologically graded and exploring their associated bundles.

Contribution

It generalizes a theorem on C*-algebra embeddings from étale to non-étale graded groupoids and demonstrates their topological grading.

Findings

Full and reduced C*-algebras of subgroupoids embed isometrically into those of the groupoid

The C*-algebras are topologically graded in the sense of Exel

Discussion of C*-algebras of associated bundles

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 \'etale to the non-\'etale 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.