Phase transitions on Hecke C*-algebras and class-field theory over ℚ
-
Marcelo Laca
and
Machiel van Frankenhuijsen
Abstract
We associate a canonical Hecke pair of semidirect product groups to the ring inclusion of the algebraic integers 𝒪 in a number field 𝒦, and we construct a C*-dynamical system on the corresponding Hecke C*-algebra, analogous to the one constructed by Bost and Connes for the inclusion of the integers in the rational numbers. We describe the structure of the resulting Hecke C*-algebra as a semigroup crossed product and then, in the case of class number one, analyze the equilibrium (KMS) states of the dynamical system. The extreme KMSβ states at low-temperature exhibit a phase transition with symmetry breaking that strongly suggests a connection with class field theory. Indeed, for purely imaginary fields of class number one, the group of symmetries, which acts freely and transitively on the extreme KMS∞ states, is isomorphic to the Galois group of the maximal abelian extension over the field. However, the Galois action on the restrictions of extreme KMS∞ states to the (arithmetic) Hecke algebra over 𝒦, as given by class-field theory, corresponds to the action of the symmetry group if and only if the number field 𝒦 is ℚ.
© Walter de Gruyter
Articles in the same Issue
- Extension of structure groups of principal bundles in positive characteristic
- Phase transitions on Hecke C*-algebras and class-field theory over ℚ
- Existence results for a class of rate-independent material models with nonconvex elastic energies
- Endotrivial modules for finite groups of Lie type
- Solution to the inverse problem for upper asymptotic density
- Entire spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space
- Cancellation in totally definite quaternion algebras
- Mesures et équidistribution sur les espaces de Berkovich
Articles in the same Issue
- Extension of structure groups of principal bundles in positive characteristic
- Phase transitions on Hecke C*-algebras and class-field theory over ℚ
- Existence results for a class of rate-independent material models with nonconvex elastic energies
- Endotrivial modules for finite groups of Lie type
- Solution to the inverse problem for upper asymptotic density
- Entire spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space
- Cancellation in totally definite quaternion algebras
- Mesures et équidistribution sur les espaces de Berkovich
