# Journal of Operator Theory

Volume 83, Issue 1, Winter 2020  pp. 73-93.

Phase transitions on $C^*$-algebras arising from number fields and the generalized Furstenberg conjecture

Authors:  Marcelo Laca (1), Jacqueline M. Warren (2)
Author institution:(1) Department of Mathematics and Statistics, University of Victoria, Canada
(2) Department of Mathematics, University of California, San Diego, U.S.A.

Summary: 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.

DOI: http://dx.doi.org/10.7900/jot.2018jul13.2201
Keywords: KMS state, phase transition, $ax+b$ semigroup $C^*$-algebra, primitive ideal