Subquotients of Hecke C*-algebras

We realize the Hecke C*-algebra CQ of Bost and Connes as a direct limit of Hecke C*-algebras which are semigroup crossed products by NF, 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 ZS for S a finite set of primes. When |S|=1, these actions are odometers and the crossed products are Bunce–Deddens algebras; when |S|>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.