SearchPublisher and Institutes
You are here: Home :: Area NEM :: Mathematics
Marcelo Laca, Machiel van Frankenhuijsen
Phase transitions on Hecke C*-algebras and class-field theory over ℚ
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.
Journal fur die reine und angewandte Mathematik (Crelles Journal), Walter de Gruyter
Print ISSN: 0075-4102
Volume: 2006, 06/2006
Pages: 25 - 53