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On Hausdorff covers for non-Hausdorff groupoids

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arxiv 2503.23203 v1 pith:NPGCBCQE submitted 2025-03-29 math.OA math.DSmath.RA

Kevin Aguyar Brix , Julian Gonzales , Jeremy B. Hume , Xin Li This is my paper · ORCID
classification math.OAmath.DSmath.RA
keywords groupoidsidealsnon-hausdorffsingularhausdorffwhenalgebraicalgebras
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abstract

We develop a new approach to non-Hausdorff \'etale groupoids and their algebras based on Timmermann's construction of Hausdorff covers. As an application, we completely characterise when singular ideals vanish in Steinberg algebras over arbitrary rings. We also completely characterise when $C^*$-algebraic singular ideals have trivial intersection with the non-Hausdorff analogue of subalgebras of continuous, compactly supported functions. This leads to a characterisation when $C^*$-algebraic singular ideals vanish for groupoids satisfying a finiteness condition. Moreover, our approach leads to further sufficient vanishing criteria for singular ideals and reduces questions about simplicity, the ideal intersection property, amenability and nuclearity for non-Hausdorff \'etale groupoids to the Hausdorff case.

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Cited by 4 Pith papers

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