Phase transitions on C∗-algebras arising from number fields and the generalized Furstenberg conjecture

Abstract

We describe the low-temperature extremal KMS states of the semigroup C∗-algebra of the ax+b semigroup of algebraic integers in a number field in terms of ergodic invariant measures for certain groups of linear toral automorphisms. We classify different behaviours in terms of the ideal class group, the degree, and the unit rank of the field and we also obtain an explicit description of the primitive ideal space of the associated transformation group C∗-algebra for number fields of unit rank at least 2 that are not complex multiplication fields.