Categorising the operator algebras of groupoids and higher-rank graphs

TL;DR

This dissertation classifies groupoid and higher-rank graph C*-algebras by establishing equivalences of convergence notions, characterising their properties, and providing conditions for various algebraic and topological features.

Contribution

It introduces new characterisations and conditions for the properties of groupoid and higher-rank graph C*-algebras, advancing their classification and understanding.

Findings

Equivalence of convergence notions in groupoid orbit spaces.

Conditions for C*-algebras to be liminal, postliminal, or have bounded trace.

Characterisations of orbit properties and algebraic features of higher-rank graph C*-algebras.

Abstract

This dissertation concerns the classification of groupoid and higher-rank graph C*-algebras and has two main components. Firstly, for a groupoid it is shown that the notions of strength of convergence in the orbit space and measure-theoretic accumulation along the orbits are equivalent ways of realising multiplicity numbers associated to a sequence of induced representations of the groupoid C*-algebra. Examples of directed graphs are given, showing how to determine the multiplicity numbers associated to various sequences of induced representations of the directed graph C*-algebras. The second component of this dissertation uses path groupoids to develop various characterisations of the C*-algebras of higher-rank graphs. Necessary and sufficient conditions are developed for the Cuntz-Krieger C*-algebras of row-finite higher-rank graphs to be liminal and to be postliminal. When Kumjian and Pask's path groupoid is principal, it is shown precisely when these C*-algebras have bounded trace, are Fell, and have continuous trace. Necessary and sufficient conditions are provided for the path groupoids of row-finite higher-rank graphs without sources to have closed orbits and to have locally closed orbits. When these path groupoids are principal, necessary and sufficient conditions are provided for them to be integrable, to be Cartan and to be proper.