Abstract

The seminal work of Bost and Connes has inspired a great deal of interest in the KMS equilibrium states of C*-dynamical systems of number theoretic origin. One such system is the Toeplitz algebra of the (right) ax+b monoid over the natural numbers N with natural dynamics, first considered by an Heuf, Laca and Raeburn; they classified the KMS states of this system at low temperatures and showed that the extremal states are parametrized by points on the circle. I will present on my joint work with Marcelo Laca, in which we complete the classification of KMS states and demonstrate that the extremal KMS states at high temperatures are parametrized by the 1-point compactification of N (equiv., by closed subgroups of the circle). 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.