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.