Phase transitions on Hecke C*-algebras and class-field theory over Q
| dc.creator | Laca, Marcelo | |
| dc.creator | van Frankenhuijsen, Machiel | |
| dc.date | 2004-10-13 | |
| dc.date.accessioned | 2026-07-07T05:13:14Z | |
| dc.date.available | 2026-07-07T05:13:14Z | |
| dc.description | 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 KMS$_β$ 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$_\infty$ 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$_\infty$ 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 $\Q$. | |
| dc.identifier | https://arxiv.org/abs/math/0410305 | |
| dc.identifier | http://arxiv.org/abs/math/0410305 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/72870 | |
| dc.subject | Operator Algebras | |
| dc.subject | Number Theory | |
| dc.subject | 46L55 | |
| dc.title | Phase transitions on Hecke C*-algebras and class-field theory over Q | |
| dc.type | text |