On Primitive Ideal Spaces of C*-Algebras over Certain Locally Compact Groupoids

Abstract

Let F be a locally compact Hausdorff second countable groupoid with a left Haar system {v ^x} _x∈X in the sense of [9] (X = the unit space of Γ). By analogy with Fell’s algebraic bundles over groups, we define the notion of C*-algebras over F and, given a C*-algebra A over Γ, we can form a C*-algebra C*(Γ, A) as the completion of the cross sectional algebra of A. In this note, under some stringent assumptions on Γ, we present a concrete realization of the primitive ideal space of C*(Γ, A). This is a C*-version of [12].