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Subquotients of Hecke \(C^*\)-algebras. (English) Zbl 1093.46027

Summary: 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 \(|S|=1\), 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.

MSC:

46L05 General theory of \(C^*\)-algebras
46L55 Noncommutative dynamical systems
46L35 Classifications of \(C^*\)-algebras
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