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 $G_e$. 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.