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

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 $C^*(P^+_K,P^+_O)$ 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 $C^*(P^+_K,P^+_O)$ to a corner in the Bost-Connes system established by Laca, Neshveyev and Trifkovic, we obtain an arithmetic subalgebra of $C^*(P^+_K,P^+_O)$ 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 $C^*$-dynamical system $(C^*(P^+_K,P^+_O),\sigma)$, 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.Comment: 21 pages; v2: minor changes and correction

A note on repetition in Spanish: volver a + VInf, re-prefixation, and adverbs of repetition.

In this paper, we describe the semantics of three types of expressions of repetition that constitute presupposition triggers in Spanish, a verbal periphrasis, adverbs of repetition, and prefixation. We show that the main difficulty in their semantic analysis is that of formulating the minimal presupposition they trigger, and we assume that, in the case of the periphrasis, Ramchand's first-phase syntax templates are better predictors of this minimal presupposition than analyses relying on less fine-grained structures for the VP (such as those merely distinguishing adjuncts from arguments and internal from external arguments). As for the differences between the three types of expression, the periphrasis shows a clearly temporal nature, since it both modifies temporal structure and relies exclusively on temporal orders. Adverbs of repetition do not modify temporal structure and may rely on other types of order. Re1-prefixation in Spanish is confined to expressing the restitution (with or without modification) of a previously holding state