Subquotients of Hecke C^*-algebras

Abstract

We realize the Hecke C^*-algebra $\mathcal{C}_{\mathbb{Q}}$ of Bost and Connes as a direct limit of Hecke C^*-algebras which are semigroup crossed products by $\mathbb{N}^F$, for F a finite set of primes. For each approximating Hecke C^*-algebra we describe a composition series of ideals. In all cases there is a large type I ideal and a commutative quotient, and the intermediate subquotients are direct sums of simple C^*-algebras. We can describe the simple summands as ordinary crossed products by actions of $\mathbb{Z}^S$ for S a finite set of primes. When $\vert S\vert =1$, these actions are odometers and the crossed products are Bunce–Deddens algebras; when $\vert S\vert >1$, the actions are an apparently new class of higher-rank odometer actions, and the crossed products are an apparently new class of classifiable AT-algebras.