Phase transitions with spontaneous symmetry breaking on Hecke C*-algebras from number fields

Abstract

When one attempts to generalize the results of Bost and Connes [BC] to algebraic number fields, one has to face sooner or later the fact that in a number field there is no unique factorization in terms of primes. As is well known, this failure is twofold: the ring of integers has nontrivial units, and even if one considers integers modulo units, (equivalently the principal integral ideals), it turns out that factorization in terms of these can fail too, essentially because ‘irreducible’ does not mean ‘prime’. The first difficulty, with the units, already arises in the situation of [BC], Remark 33.b], but is easily dealt with by considering elements fixed by a symmetry corresponding to complex conjugation. In the existing generalizations the lack of unique factorization has been dealt with in various ways. It has been eliminated, through replacing the integers by a principal ring that generates the same field [HLe], it has been sidestepped, by basing the construction of the dynamical system on the additive integral adeles with multiplication by (a section of) the integral ideles [Coh], and it has been ignored, by considering an almost normal subgroup that makes no reference to multiplication [ALR]. These simplifications make the construction and analysis of interesting dynamical systems possible, but they come at a price. Indeed, the noncanonical choices introduced in [HLe] and [Coh] lead to phase transitions with groups of symmetries that are not obviously isomorphic to actual Galois groups of maximal abelian extensions, and have slightly perturbed zeta functions in the case of [HLe], while the units not included in the almost normal subgroup in [ALR], reappear as a (possibly infinite) group of symmetries under which KMS states have to be invariant, which causes severe difficulties in their computation.