Simplicity of algebras associated to non-Hausdorff groupoids

By L. O. Clark, R. Exel, Enrique Pardo, Aidan Sims, Charles Starling. Transactions of the American Mathematical Society, 2018

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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 \'etale 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 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

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