  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 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 beta>1
certain extremal KMS_beta-states for L can be obtained, via induction and
rescaling, from KMS_{[L:K]beta}-states for K. On the other hand, for 0<beta\le1
every KMS_{[L:K]\beta}-state for K induces to an infinite weight.
