Details

Autor(en) / Beteiligte

• Laca, Marcelo
• Larsen, Nadia S.
• Neshveyev, Sergey

Titel

Ground states of groupoid C∗-algebras, phase transitions and arithmetic subalgebras for Hecke algebras

Ist Teil von

• Journal of geometry and physics, 2019-02, Vol.136, p.268-283

Ort / Verlag

Elsevier B.V

Erscheinungsjahr

2019

Link zum Volltext

Quelle

Elsevier ScienceDirect Journals Complete

Beschreibungen/Notizen

• We consider the Hecke pair consisting of the group PK+ of affine transformations of a number field K that preserve the orientation in every real embedding and the subgroup PO+ consisting of transformations with algebraic integer coefficients. The associated Hecke algebra Cr∗(PK+,PO+) has a natural time evolution σ, and we describe the corresponding phase transition for KMSβ-states and for ground states. From work of Yalkinoglu and Neshveyev it is known that a Bost–Connes type system associated to K has an essentially unique arithmetic subalgebra. When we import this subalgebra through the isomorphism of Cr∗(PK+,PO+) to a corner in the Bost–Connes system established by Laca, Neshveyev and Trifković, we obtain an arithmetic subalgebra of Cr∗(PK+,PO+) on which ground states exhibit the ‘fabulous’ property with respect to an action of the Galois group G(Kab∕H+(K)), where H+(K) is the narrow Hilbert class field. In order to characterize the ground states of the C∗-dynamical system (Cr∗(PK+,PO+),σ), we obtain first a characterization of the ground states of a groupoid C∗-algebra, refining earlier work of Renault. This is independent from number theoretic considerations, and may be of interest by itself in other situations.