2011/09/21 by Kamran Reihani, Reihani, Kamran
Mathematics · #05A16 #05A17 #15A36 #17B10 #17B20 #19K99 #37B05 #46L35 #46L80 #54H20 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Geometric and Algebraic Topology #Operator Algebras (math.OA) #Primary: 19K14 #Secondary: 05A15
paper · pdf · doi:10.48550/arxiv.1109.4473
openalex publication_date 2011/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The paper studies the K-theoretic invariants of the crossed product C*-algebras associated with an important family of homeomorphisms of the tori \BbbTn called \em 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, together with two computing programs - included in an appendix - 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 \fraksl(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 Erdös), 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.