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]β}-state for K induces to an infinite weight.

16 pages, AMS-LaTeX; minor corrections; to appear in JNCG

References in corpus (2)

• Quantum Statistical Mechanics, L-series and Anabelian Geometry
• Von Neumann algebras arising from Bost-Connes type systems

Cited by in corpus (10)

• KMS states on the C*-algebras of non-principal groupoids
• C*-algebras of Toeplitz type associated with algebraic number fields
• Ground states of groupoid C*-algebras, phase transitions and arithmetic subalgebras for Hecke algebras
• Phase transitions on C*-algebras from actions of congruence monoids on rings of algebraic integers
• Irreducible Representations of Bost-Connes systems
• The Riemann-Roch strategy, Complex lift of the Scaling Site
• Exel and Stacey crossed products, and Cuntz-Pimsner algebras
• Stacey crossed products associated to Exel systems
• On arithmetic models and functoriality of Bost-Connes systems. With an appendix by Sergey Neshveyev
• Bost-Connes systems for local fields of characteristic zero