JournalsjncgVol. 7, No. 2pp. 525–546

Bost–Connes systems, Hecke algebras, and induction

  • Marcelo Laca

    University of Victoria, Canada
  • Sergey Neshveyev

    University of Oslo, Norway
  • Mak Trifković

    University of Victoria, Canada
Bost–Connes systems, Hecke algebras, and induction cover

Abstract

We consider a Hecke algebra naturally associated with the affine group with totally positive multiplicative part over an algebraic number field KK and we show that the C*-algebra of the Bost–Connes system for KK 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 KK; in particular, the two algebras coincide if and only if KK has narrow class number one. Passing the known results for the Bost–Connes system for KK to this corner, we obtain a phase transition theorem for our Hecke algebra.

In another application of induction we consider an extension L/KL/K of number fields and we show that the Bost–Connes system for LL embeds into the system obtained from the Bost–Connes system for KK by induction from the group of ideals in KK to the group of ideals in LL. This gives a C*-algebraic correspondence from the Bost–Connes system for KK to that for LL. 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\beta>1 certain extremal KMSβ_\beta-states for LL can be obtained, via induction and rescaling, from KMS[L:K]β_{[L: K]\beta}-states for KK. On the other hand, for 0<β10<\beta\le1 every KMS[L:K]β_{[L: K]\beta}-state for KK 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. 525–546

DOI 10.4171/JNCG/125