# Journal of Operator Theory

Volume 89, Issue 1, Winter 2023  pp. 287-301.

Inclusions of $C^*$-algebras of graded groupoids

Authors:  Becky Armstrong (1), Lisa Orloff Clark (2), Astrid an Huef (3)
Author institution:(1) Mathematical Institute, WWU Munster, Einsteinstr. 62, 48149 Munster, Germany
(2) School of Mathematics and Statistics, Victoria Univ. of Wellington, PO Box 600, Wellington 6140, Aotearoa New Zealand
(3) School of Mathematics and Statistics, Victoria University of Wellington, PO Box 600, Wellington 6140, Aotearoa New Zealand

Summary: 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 nonetale 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.

DOI: http://dx.doi.org/10.7900/jot.2021aug26.2353
Keywords: groupoid, $C^*$-algebra, topological grading, isometric inclusion