K-theory of Furstenberg Transformation Group C*-algebras

This paper studies the K-theoretic invariants of the crossed product C* -algebras associated with an important family of homeomorphisms of the tori T-n called Furstenberg transformations. Using the Pimsner Voiculescu theorem, we prove that given n, the K-groups of those crossed products whose corresponding n x n integer matrices are unipotent of maximal degree always have the same rank a. 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 51(2, C), we show that, remarkably, a(n) has a combinatorial significance. For example, every a(2n+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 Eras), a simple explicit formula for the asymptotic behavior of the sequence {a(n)} is given. Finally, we describe the order structure of the Ks-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N.C. Phillips.