Existence and uniqueness of inductive limit Cartan subalgebras in inductive limit C*-algebras

Raad, Ali (2021) Existence and uniqueness of inductive limit Cartan subalgebras in inductive limit C*-algebras. PhD thesis, University of Glasgow.

Abstract

The main focus of this thesis is to answer the question of existence and uniqueness of inductive limit Cartan subalgebras in certain inductive limit C*-algebras. The classes of inductive limit C*-algebras considered are the unital AF, AI, and AT- algebras. The results obtained are then generalized to certain AX-algebras.

This thesis shows that all the aforementioned classes (and AX-algebras for planar finite connected graphs X ⊂ ℂ which arise from unital and injective connecting maps contain inductive limit Cartan subalgebras. It also shows that for all these classes except for the AF-algebras, uniqueness of the inductive limit Cartan subalgebras fails. We construct two non-isomorphic AI-Cartan subalgebras inside both a non-simple and simple AI-algebra. We provide a class of simple and unital AIalgebras for which uniqueness of AI-Cartan subalgebras fails. For the AF-algebras, we give a K-theoretic proof of the uniqueness of the AF-Cartan subalgebras.

Additionally, this thesis generalizes a theorem by Renault which characterises Cartan pairs in separable C*-algebras by twisted étale second countable groupoids. The generalization captures all Cartan pairs, not just the separable ones.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Keywords: C*-algebra, Cartan, Cartan subalgebra, groupoid, inductive limit.
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Supervisor's Name: Li, Professor Xin
• Date of Award: 2021
• Depositing User: Theses Team
• Unique ID: glathesis:2021-82456
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 24 Sep 2021 11:18
• Last Modified: 29 Sep 2021 07:49
• Thesis DOI: 10.5525/gla.thesis.82456
• URI: https://theses.gla.ac.uk/id/eprint/82456