Groupoid C*-Algebras with Hausdorff spectrum

Suppose that G is a second countable, locally compact Hausdorff groupoid with abelian stabiliser subgroups and a Haar system. We provide necessary and sufficient conditions for the groupoid C* -algebra to have Hausdorff spectrum. In particular, we show that the spectrum of C*(G) is Hausdorff if and only if the stabilisers vary continuously with respect to the Fell topology, the orbit space {G}⁽⁰⁾ G is Hausdorff, and, given convergent sequences χ_i → χ and γ_i ̇ χ_i →ω in the dual stabiliser groupoid S where the γ_i G act via conjugation, if χ and ω are elements of the same fibre then χ = ω.