Year

2022

Degree Name

Doctor of Philosophy

Department

School of Mathematics and Applied Statistics

Abstract

A graph of groups consists of an undirected graph with each edge and vertex assigned a group. There is a one-to-one correspondence between graphs of groups and group actions on trees. In this way, the study of graphs of groups is equivalent to the study of group actions on trees.

In 2017, Brownlowe, Mundey, Pask, Spielberg and Thomas developed a generators and relations construction of a C∗-algebra associated to a graph of groups, which they linked to a specific crossed-product C∗-algebra to show that the correspondence between graphs of groups and group actions on trees extends to their respective C∗- algebras. A Cuntz-Pimsner model for this graph of groups C∗-algebra was given by Mundey and Rennie in 2021. We compare a generators and relations approach to Toeplitz algebras to the Toeplitz-Pimsner construction.

KMSβ states on a C∗-dynamical system with a real action are invariant states that characterise equilibria. In this thesis we provide a characterisation of KMSβ states on the Toeplitz and Cuntz-Pimsner algebras associated to a graph of groups. While a complete classification of KMSβ states on these C∗-algebras is an extremely large and difficult problem, we are able to classify a certain class of KMSβ states, namely those which arise from the Haar state on group 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.