Year

2023

Degree Name

Doctor of Philosophy

Department

School of Mathematics and Applied Statistics

Abstract

In this thesis, we study self-similar actions of groupoids on row-finite directed graphs and their associated Cuntz-Krieger algebras. Roughly speaking, if for all scales of parts of the object reiterate the whole, then the object is self-similar. For algebraic objects such as groups or groupoids that have this self-similarity property, we simply call them selfsimilar groups or self-similar groupoids. As an illustration of this self-similarity property, we recall the addition algorithm of integers that we learned from primary school, there is a so-called carrying operation that takes place to deal with addition of larger integers. Analogous to this carrying operation there is so-called a restriction map that encodes self-similarity property.

In the 1980s, Grigorchuk, Gupta, and Sidki were among the pioneers who introduced the concept of self-similar groups to address the question of whether there exist groups with intermediate growth. These groups exhibit self-similarity in their actions on the path-spaces of graphs with a single vertex. In contrast, Laca, Raeburn, Rammage, and Whittaker (2018) extended this notion to encompass self-similar actions on more general directed graphs by introducing the concept of a self-similar groupoid. A self-similar groupoid is defined as a system of partial isomorphisms of the path-spaces of a finite directed graph. Laca et al. (2018) explored self-similar groupoids and their associated C∗-algebras to investigate the KMS states on their corresponding dynamical systems. In contrast to their approach, which employed Hilbert modules and Cuntz-Pimsner algebras, we focus solely on generators and relations, as well as the associated Cuntz-Krieger algebras. Moreover, we extend our analysis to a broad class of self-similar groupoids, with particular emphasis on the ideal structure of their associated C∗-algebras.

FoR codes (2020)

490408 Operator algebras and functional analysis

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Unless otherwise indicated, the views expressed in this thesis are those of the author and do not necessarily represent the views of the University of Wollongong.