2004/10/13 by Marcelo Laca, Laca, Marcelo, Machiel van Frankenhuijsen +1
Mathematics · Physics and Astronomy · #46L55 #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #Operator Algebras (math.OA) #Quantum Mechanics and Applications #math.NT #math.OA #msc:46L55
paper · pdf · doi:10.48550/arxiv.math/0410305
arxiv created 2004/10/13 · openalex publication_date 2004/10/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We associate a canonical Hecke pair of semidirect product groups to the ring inclusion of the algebraic integers \oo in a number field \kk, 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 \kk, as given by class-field theory, corresponds to the action of the symmetry group if and only if the number field \kk is \Q.