Katsura algebra simplicity is completely characterized by explicit matrix conditions (Theorems 5.7, 6.14) and shown to be decidable in polynomial space.
Hume, and Xin Li,On Hausdorff covers for non-Hausdorff groupoids (2025), 33 pp., available at2503.23203
3 Pith papers cite this work. Polarity classification is still indexing.
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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.
representative citing papers
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On the simplicity of Katsura algebras
Katsura algebra simplicity is completely characterized by explicit matrix conditions (Theorems 5.7, 6.14) and shown to be decidable in polynomial space.
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Algebraic singular functions are not always dense in the ideal of $C^*$-singular functions
Provides the first counterexamples showing algebraic singular functions are not always dense in the ideal of C*-singular functions for certain étale non-Hausdorff groupoids, including a bundle of groups and one from a self-similar action.
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A Baum-Connes assembly map for essential semigroup crossed products
Constructs equivariant E-theory and a natural Baum-Connes assembly map for Fell bundles of inverse semigroups, covering maximal, reduced, and essential cases with applications to groupoids and Cartan pairs.