Supercritical phase transition on the Toeplitz algebra of $\mathbb N^\times \ltimes \mathbb Z$

Abstract: We study the high-temperature equilibrium for the C*-algebra $\mathcal T(\mathbb N^\times \ltimes \mathbb Z)$ recently considered by an Huef, Laca and Raeburn. We show that the simplex of KMS$_\beta$ states at each inverse temperature $\beta$ in the critical interval $(0,1]$ is a Bauer simplex whose space of extreme points is homeomorphic to $\mathbb N \sqcup{\infty}$. This is in contrast to the uniqueness of equilibrium at high temperature observed in previously considered systems arising from number theory. We also establish a connection between the phase transitions on quotients of our system and the Bost-Connes phase transition.