Simplicity of algebras associated to non-Hausdorff groupoids.(English)Zbl 1491.16032
Summary: 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.
MSC:
| 16S99 | Associative rings and algebras arising under various constructions |
| 16S10 | Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) |
| 22A22 | Topological groupoids (including differentiable and Lie groupoids) |
| 46L05 | General theory of \(C^*\)-algebras |
| 46L55 | Noncommutative dynamical systems |
Cite
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