Bost–Connes systems, Hecke algebras and induction
Journal of Noncommutative Geometry
We consider a Hecke algebra naturally associated with the affine group with totally positive multiplicative part over an algebraic number field 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 field, in the Bost-Connes C * -algebra of K; in particular, the
... 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 fields and we show that the Bost-Connes system for L embeds into the system obtained from the Bost-Connes system for K by induction from the group of ideals in K to the group of ideals in L. 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 fields to C * -dynamical systems with equivariant correspondences as morphisms. We use this correspondence to induce KMS-states and we show that for β > 1 certain extremal KMS β -states for L can be obtained, via induction and rescaling, from KMS [L:K]β -states for K. On the other hand, for 0 < β ≤ 1 every KMS [L:K]β -state for K induces to an infinite weight.