The Bost–Connes phase transition and unitary representations

Abstract

We construct a family of unitary representations of the $ax+b$ group $\mathbb{Q}⋊{\mathbb{Q}}_{+}^{\times }$. We show that this family of representations exhibits a “phase transition” analogous to that observed by Bost and Connes [2] and then explain how these representations are related to the equilibrium states of Bost and Connes.