Supercritical phase transition on the Toeplitz algebra of ${\mathbb{N}}^{\times }⋉\mathbb{Z}$

Abstract

We study the high-temperature equilibrium for the C*-algebra $\mathcal{T}({\mathbb{N}}^{\times }⋉\mathbb{N})$ 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 $\mathbb{N}\bigsqcup \{\infty \}$. This is in contrast to the uniqueness of equilibrium at high temperature observed in previously considered systems arising from number theory. We also show that quotients of our system exhibit spontaneous symmetry-breaking by finite cyclotomic Galois groups and establish their connection to the Bost-Connes phase transition.

Introduction has been improved for motivation and focus; old appendix B is now included as Section 6; section numbers have changed after that