KMS states on generalised Bunce-Deddens algebras and their Toeplitz extensions

TL;DR

This paper characterizes the structure and KMS states of generalized Bunce-Deddens algebras and their Toeplitz extensions, providing universal properties, uniqueness theorems, and conditions for simplicity.

Contribution

It introduces novel universal properties for these algebras, proves new uniqueness theorems without aperiodicity assumptions, and characterizes simplicity via KMS states and coprimality conditions.

Findings

KMS states are explicitly calculated for finite graphs.

Simplicity of the algebra is characterized by the uniqueness of KMS states.

Uniqueness theorems are established without the need for aperiodicity when the supernatural number is infinite.

Abstract

We study the generalised Bunce-Deddens algebras and their Toeplitz extensions constructed by Kribs and Solel from a directed graph and a sequence $\omega$ of positive integers. We describe both of these $C^*$-algebras in terms of novel universal properties, and prove uniqueness theorems for them; if $\omega$ determines an infinite supernatural number, then no aperiodicity hypothesis is needed in our uniqueness theorem for the generalised Bunce-Deddens algebra. We calculate the KMS states for the gauge action in the Toeplitz algebra when the underlying graph is finite. We deduce that the generalised Bunce-Deddens algebra is simple if and only if it supports exactly one KMS state, and this is equivalent to the terms in the sequence $\omega$ all being coprime with the period of the underlying graph.