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Groupoid C*-algebras with Hausdorff Spectrum

Suppose $G$ is a second countable, locally compact Hausdorff groupoid with abelian stabilizer 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 stabilizers vary continuously with respect to the Fell topology, the orbit space $G^{(0)}/G$ is Hausdorff, and, given convergent sequences $χ_i\to χ$ and $γ_i\cdotχ_i \to ω$ in the dual stabilizer groupoid $\hat{S}$ where the $γ_i\in G$ act via conjugation, if $χ$ and $ω$ are elements of the same fiber then $χ= ω$