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

In recent work, Cuntz, Deninger and Laca have studied the Toeplitz type C*-algebra associated to the affine monoid of algebraic integers in a number field, under a time evolution determined by the absolute norm. The KMS equilibrium states of their system are parametrized by traces on the C*-algebras of the semidirect products $J rtimes O^*$ resulting from the multiplicative action of the units $O^*$ on integral ideals $J$ representing each ideal class. At each fixed inverse temperature $beta > 2$, the extremal equilibrium states correspond to extremal traces of $C^*(Jrtimes O^*)$. Here we undertake the study of these traces using the transposed action of $O^*$ on the duals $hat J$ of the ideals and the recent characterization of traces on transformation group C*-algebras due to Neshveyev. We show that the extremal traces of $C^*(Jrtimes O^*)$ are parametrized by pairs consisting of an ergodic invariant measure for the action of $O^*$ on $hat{J}$ together with a character of the isotropy subgroup associated to the support of this measure. For every ideal the dual group $hat {J}$ is a d-torus on which $O^*$ acts by linear toral automorphisms. Hence, the problem of classifying all extremal traces is a generalized version of Furstenbergs celebrated $times 2$ $times 3$ conjecture. We classify the results for various number fields in terms of ideal class group, degree, and unit rank, and we point along the way the trivial, the intractable, and the conjecturally classifiable cases. At the topological level, it is possible to characterize the number fields for which infinite $O^*$-invariant sets are dense in $hat{J} $, thanks to a theorem of Berend; as an application we give a description of the primitive ideal space of $C^*(Jrtimes O^*)$ for those number fields.