The Bost-Connes phase transition and unitary representations
The celebrated construction due to \textit{J.-B. Bost} and \textit{A. Connes} [Sel. Math., New Ser. 1, No. 3, 411--457 (1995; Zbl 0842.46040)] of a quantum statistical mechanical system from number theory has been the source of many developments. In the present paper, the author constructs a family of unitary representations of the \(ax+b\) group \(\mathbb{Q}\rtimes \mathbb{Q}^{\times_+}\) and shows that, in an appropriate sense, this family exhibits a phase transition. More precisely, this family is parametrised by the half-plane \(\text{Re}z\geq 0\), and the following hold: at \(\text{Re}z= 0\), the representations are one-dimensional and mutually inequivalent; for \(0<\text{Re}z\leq 1\), the representations are mutually inequivalent, irreducible and infinite-dimensional; when \(\text{Re}z>1\), the representations are mutually equivalent and reducible, and their irreducible components are parametrised by the Galois group \(\text{Gal}(\mathbb{Q}^{\text{ab}}, \mathbb{Q})\). Using \textit{M. Laca}'s analysis of KMS states [J. Funct. Anal. 152, No. 2, 330--378 (1998; Zbl 0957.46039)], the author describes relations between these representations and the KMS states of the Bost-Connes system.
scientific article; zbMATH DE number 6155612