We consider a Hecke algebra naturally associated with the affine group with totally positive multiplicative part over an algebraic number field and we show that the C*-algebra of the Bost–Connes system for 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 ; in particular, the two algebras coincide if and only if has narrow class number one. Passing the known results for the Bost–Connes system for to this corner, we obtain a phase transition theorem for our Hecke algebra.
In another application of induction we consider an extension of number fields and we show that the Bost–Connes system for embeds into the system obtained from the Bost–Connes system for by induction from the group of ideals in to the group of ideals in . This gives a C*-algebraic correspondence from the Bost–Connes system for to that for . 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 certain extremal KMS-states for can be obtained, via induction and rescaling, from KMS-states for . On the other hand, for every KMS-state for induces to an infinite weight.
Cite this article
Marcelo Laca, Sergey Neshveyev, Mak Trifković, Bost–Connes systems, Hecke algebras, and induction. J. Noncommut. Geom. 7 (2013), no. 2, pp. 525–546DOI 10.4171/JNCG/125