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

KMS States on Generalised Bunce–Deddens Algebras and their Toeplitz Extensions