Categorising the operator algebras of groupoids and higher-rank graphs

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, to have locally closed orbits, to be integrable, to be Cartan and to be proper.