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

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