Ground states of groupoid C*-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 $C^{\ast }(P_K^{+},P_O^{+})$ has a natural time evolution $\sigma$, and we describe the corresponding phase transition for KMS${}_{\beta }$-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 $C^{\ast }(P_K^{+},P_O^{+})$ to a corner in the Bost-Connes system established by Laca, Neshveyev and Trifkovic, we obtain an arithmetic subalgebra of $C^{\ast }(P_K^{+},P_O^{+})$ on which ground states exhibit the `fabulous' property with respect to an action of the Galois group $Gal(K^{ab}/H_{+}(K))$, where $H_{+}(K)$ is the narrow Hilbert class field. In order to characterize the ground states of the $C^{\ast }$-dynamical system $(C^{\ast }(P_K^{+},P_O^{+}),\sigma )$, we obtain first a characterization of the ground states of a groupoid $C^{\ast }$-algebra, refining earlier work of Renault. This is independent from number theoretic considerations, and may be of interest by itself in other situations.

21 pages; v2: minor changes and corrections