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

We study the generalised Bunce-Deddens algebras and their Toeplitz extensions constructed by Kribs and Solel from a directed graph and a sequence $ω$ of positive integers. We describe both of these $C^*$-algebras in terms of novel universal properties, and prove uniqueness theorems for them; if $ω$ 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 $ω$ all being coprime with the period of the underlying graph.