AbstractThis paper studies the K-theoretic invariants of the crossed product C*-algebras associated with an important family of homeomorphisms of the tori Tn called Furstenberg transformations. Using the Pimsner–Voiculescu theorem, we prove that given n, the K-groups of those crossed products whose corresponding n × n integer matrices are unipotent of maximal degree always have the same rank an. We show using the theory developed here that a claim made in the literature about the torsion subgroups of these K-groups is false. Using the representation theory of the simple Lie algebra sl(2;C), we show that, remarkably, an has a combinatorial significance. For example, every a2n+1 is just the number of ways that 0 can be represented as a sum of integers between–n and n (with no repetitions). By adapting an argument of van Lint (in which he answered a question of Erdős), a simple explicit formula for the asymptotic behavior of the sequence {an} is given. Finally, we describe the order structure of the K0-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips.
Bounds on the torsion subgroups of Néron–Severi groups
