Bost–Connes systems, Hecke algebras, and induction

Abstract

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.