INCLUSIONS OF C*-ALGEBRAS OF GRADED GROUPOIDS

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 Ge. 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.