Simplicity of algebras associated to non-Hausdorff groupoids

Clark, Lisa; Exel, Ruy; Pardo, Enrique; Sims, Aidan; Starling, Charles

doi:10.1090/tran/7840

Simplicity of algebras associated to non-Hausdorff groupoids
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by Lisa Orloff Clark, Ruy Exel, Enrique Pardo, Aidan Sims and Charles Starling PDF

Trans. Amer. Math. Soc. 372 (2019), 3669-3712 Request permission

Abstract:

We prove a uniqueness theorem and give a characterization of simplicity for Steinberg algebras associated to non-Hausdorff ample groupoids. We also prove a uniqueness theorem and give a characterization of simplicity for the $C^{*}$-algebra associated to non-Hausdorff étale groupoids. Then we show how our results apply in the setting of tight representations of inverse semigroups, groups acting on graphs, and self-similar actions. In particular, we show that the $C^{*}$-algebra and the complex Steinberg algebra of the self-similar action of the Grigorchuk group are simple but the Steinberg algebra with coefficients in $\mathbb {Z}_2$ is not simple.

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