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# $K$-theory of Furstenberg Transformation Group $C^*$-algebras

Published:2013-10-09
Printed: Dec 2013
• Kamran Reihani,
Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ 86011, USA
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## Abstract

The 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 $\frak{sl}(2,\mathbb{C})$, 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 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.
 Keywords: $K$-theory, transformation group $C^*$-algebra, Furstenberg transformation, Anzai transformation, minimal homeomorphism, positive cone, minimal homeomorphism
 MSC Classifications: 19K14 - $K_0$ as an ordered group, traces 19K99 - None of the above, but in this section 46L35 - Classifications of $C^*$-algebras 46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22] 05A15 - Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05A16 - Asymptotic enumeration 05A17 - Partitions of integers [See also 11P81, 11P82, 11P83] 15A36 - Matrices of integers (See also 11C20)17B10 - Representations, algebraic theory (weights) 17B20 - Simple, semisimple, reductive (super)algebras 37B05 - Transformations and group actions with special properties (minimality, distality, proximality, etc.) 54H20 - Topological dynamics [See also 28Dxx, 37Bxx]