Almost finiteness of groupoid actions and Z-stability of C*-algebras associated to tilings

Ito, Luke John (2019) Almost finiteness of groupoid actions and Z-stability of C*-algebras associated to tilings. PhD thesis, University of Glasgow.

Abstract

The property of almost finiteness was first introduced by Matui for locally compact Hausdorff totally disconnected étale groupoids with compact unit spaces, and has since been extended by Suzuki to drop the assumption of being totally disconnected. A property of the same name has been defined by Kerr for free actions of discrete groups on compact metric spaces. In this setting, Kerr shows that almost finiteness has direct relevance to the classification programme for simple, separable, unital, nuclear, infinite dimensional C*-algebras, as it implies that the associated crossed product is Z-stable.

The motivating example for this thesis comes from the theory of aperiodic tilings. A tiling is a covering of Euclidean space by a collection of sets (called tiles) which overlap only on their boundaries. A tiling is called aperiodic if it does not contain arbitrarily large periodic patterns. Such tilings find physical applications, acting as models for quasicrystals. One may associate a groupoid to certain aperiodic tilings, and the C*-algebras of such groupoids encode information about physical observables in quasicrystalline molecules.

In this thesis, we generalise Kerr's notion of almost finiteness of group actions to allow for actions of groupoids. We show that the canonical action of the groupoid associated to any aperiodic, repetitive tiling with finite local complexity on its unit space is almost finite, and we use this to show that the C*-algebra of the tiling is Z-stable. We develop a groupoid version of the Ornstein-Weiss quasitiling machinery, which we use to prove our Z-stability result. Finally, we give a direct proof that tiling C*-algebras are quasidiagonal, which eases the route to classification in the case that the algebra has unique trace.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Keywords: Tilings, C*-algebras, classification, Z-stability, groupoids, almost finiteness, quasidiagonality.
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Supervisor's Name: Zacharias, Dr. Joachim and Whittaker, Dr. Mike
• Date of Award: 2019
• Depositing User: Mr Luke Ito
• Unique ID: glathesis:2019-74404
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 18 Sep 2019 07:45
• Last Modified: 10 Dec 2019 10:56
• Thesis DOI: 10.5525/gla.thesis.74404
• URI: https://theses.gla.ac.uk/id/eprint/74404