The Primitive Ideal Space of Groupoid C*-Algebras for Groupoids with Abelain Isotropy

We study the topology of the primitive ideal space of groupoid C*-algebras for groupoids with abelian isotropy. Our results include the known results for action groupoids with abelain stabilizers. Furthermore, we obtain complete results when the isotropy map is continuous except for jump discontinuities, and also when $G$ is a unit space fixing extension of a proper groupoid by an abelain group bundle. We hope that our methods will be a springboard to further results of this type.