Amenability and singular functions on non-Hausdorff groupoids

Gonzales, Julian J. (2026) Amenability and singular functions on non-Hausdorff groupoids. PhD thesis, University of Glasgow.

Abstract

This thesis investigates structural properties of non-Hausdorff groupoid C∗-algebras. The two main points of study are groupoid amenability, and the nature of functions belonging to the singular ideal. Conclusive results are obtained relating amenability of a non-Hausdorff étale groupoid to nuclearity of its C∗-algebras, generalising well-known results from the Hausdorff setting. In relation to the singular ideal, algebraic techniques are established for the building of explicit elementary singular functions within it. For a large class of amenable groupoids, these elementary functions are shown to span a dense subspace of the singular ideal. In the context of Steinberg algebras, an analogous family of elementary singular functions are shown to linearly span the algebraic singular ideal of any ample groupoid.

A key tool throughout is the Hausdorff cover groupoid. It is used to obtain the aforementioned results regarding amenability, and to study ideals in non-Hausdorff groupoid C∗-algebras. In most instances, the Hausdorff cover is used to reduce a given problem to the Hausdorff setting. Under certain conditions on the Hausdorff cover, an explicit formula is provided for computing the singular ideal.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics
• Supervisor's Name: Li, Professor Xin
• Date of Award: 2026
• Depositing User: Theses Team
• Unique ID: glathesis:2026-86177
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 13 Aug 2026 15:42
• Last Modified: 13 Aug 2026 15:45
• Thesis DOI: 10.5525/gla.thesis.86177
• URI: https://theses.gla.ac.uk/id/eprint/86177