Ground states of groupoid [formula omitted]-algebras, phase transitions and arithmetic subalgebras for Hecke algebras.

Abstract We consider the Hecke pair consisting of the group P K + of affine transformations of a number field K that preserve the orientation in every real embedding and the subgroup P O + consisting of transformations with algebraic integer coefficients. The associated Hecke algebra C r ∗ (P K + , P O +) has a natural time evolution σ , and we describe the corresponding phase transition for KMS β -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 C r ∗ (P K + , P O +) to a corner in the Bost–Connes system established by Laca, Neshveyev and Trifković, we obtain an arithmetic subalgebra of C r ∗ (P K + , P O +) on which ground states exhibit the 'fabulous' property with respect to an action of the Galois group G (K ab ∕ H + (K)) , where H + (K) is the narrow Hilbert class field. In order to characterize the ground states of the C ∗ -dynamical system (C r ∗ (P K + , P O +) , 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. [ABSTRACT FROM AUTHOR]