Given an amenable second countable Hausdorff locally compact étale groupoid $\mathcal G$, it is known that every primitive ideal of the corresponding C$^*$-algebra is induced from an isotropy group. For certain classes of groupoids, we refine this result by describing the Jacobson topology on such induced ideals, thus obtaining a description of the ideal structure of $C^*(\mathcal G)$. (Joint work with Johannes Christensen.)
