$K$ -theory of Furstenberg Transformation Group ${{\text{C}}^{\text{*}}}$ -algebras

Reihani, Kamran

Canadian journal of mathematics, Tome 65 (2013) no. 6, pp. 1287-1319

Résumé

This paper studies the $K$ -theoretic invariants of the crossed product ${{C}^{*}}$ -algebras associated with an important family of homeomorphisms of the tori ${{\mathbb{T}}^{n}}$ called Furstenberg transformations. Using the Pimsner–Voiculescu theorem, we prove that given $n$ , the $K$ -groups of those crossed products whose corresponding $n\,\times \,n$ integer matrices are unipotent of maximal degree always have the same rank ${{a}_{n}}$ . 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 $\mathfrak{s}\mathfrak{l}\left( 2,\,\mathbb{C} \right)$ , 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 Erdős), a simple explicit formula for the asymptotic behavior of the sequence $\{{{a}_{n}}\}$ is given. Finally, we describe the order structure of the ${{K}_{0}}$ -groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips.

Détail

DOI : 10.4153/CJM-2013-022-x

Mots-clés : 19K14, 19K99, 46L35, 46L80, 05A15, 05A16, 05A17, 15A36, 17B10, 17B20, 37B05, 54H20, K-theory, transformation group C*-algebra, Furstenberg transformation, Anzai transformation, minimal homeomorphism, positive cone, minimal homeomorphism