Bost–Connes systems, Hecke algebras, and induction

Abstract

We consider a Hecke algebra naturally associated with the ane group with totally positive multiplicative part over an algebraic number eld K and we show that the C -algebra of the Bost-Connes system for K can be obtained from our Hecke algebra by induction, from the group of totally positive principal ideals to the whole group of ideals. Our Hecke algebra is therefore a full corner, corresponding to the narrow Hilbert class eld, in the Bost-Connes C -algebra of K; in particular, the two algebras coincide if and only if K has narrow class number one. Passing the known results for the Bost-Connes system for K to this corner, we obtain a phase transition theorem for our Hecke algebra. In another application of induction we consider an extension L=K of number elds and we show that the Bost-Connes system for L embeds into the system obtained from the Bost-Connes system forK by induction from the group of ideals inK to the group of ideals inL. This gives a C -algebraic correspondence from the Bost-Connes system for K to that for L. Therefore the construction of Bost-Connes systems can be extended to a functor from number elds to C -dynamical systems with equivariant correspondences as morphisms. We use this correspondence to induce KMS-states and we show that for > 1 certain extremal KMS -states for L can be obtained, via induction and rescaling, from KMS(L:K) -states for K. On the other hand, for 0 < 1 every KMS(L:K) -state for K induces to an innite weight.