Subquotients of Hecke C * -algebras

In [2], Bost and Connes studied a particular Hecke C-algebra CQ arising in number theory. The algebra CQ can be realised as a semigroup crossed product C (Q/Z)⋊αN ∗ by an endomorphic action α of the multiplicative semigroup N on the group Calgebra C(Q/Z) [7], and this realisation has provided useful insight into the analysis of CQ [5, 13]. Since individual elements of Q/Z and N ∗ involve only finitely many primes, C(Q/Z)⋊α N ∗ is the direct limit of subalgebras C(GF )⋊α N F , where F is a finite set of primes, GF is the subgroup of Q/Z in which the denominators have all prime factors in F , and N acts through the embedding (np) 7→ ∏ p∈F p np of N in N (see Section 1). One can therefore hope to understand the Hecke algebra CQ in terms of the finite-prime analogues C(GF )⋊α N F . Our goal is to analyse the structure of these finite-prime analogues of the BostConnes algebra. We started this analysis in [10], where we described a composition series for the two-prime analogue and identified the subquotients in familiar terms: there is a large type I ideal, a commutative quotient isomorphic to C(T), and the intermediate subquotient is isomorphic to a direct sum of Bunce-Deddens algebras. Here we describe a composition series for C(GF )⋊αN F . Again there are 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 Z for S ⊂ F . 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. We begin with a short section in which we describe the algebras we intend to study. In §2, we describe our composition series for the semigroup crossed product C(GF )⋊αN F . It has |F |+1 subquotients, and all but two of them are direct sums of algebras stably isomorphic to ordinary crossed products of the form C(U(ZF\S))⋊Z S , where S ⊂ F and U(ZF\S) is the group of units in the ring ∏ p∈F\S Zp. Our main tools are the analysis of invariant ideals in semigroup crossed products from [9] and some technical lemmas on sums and intersections of ideals in C-algebras. We also use the general results of [16] to see that the simple summands are classifiable.