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Search: MSC category 37B05 ( Transformations and group actions with special properties (minimality, distality, proximality, etc.) )

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1. CJM 2013 (vol 65 pp. 1287)

Reihani, Kamran
$K$-theory of Furstenberg Transformation Group $C^*$-algebras
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
Categories:19K14, 19K99, 46L35, 46L80, , 05A15, 05A16, 05A17, 15A36, 17B10, 17B20, 37B05, 54H20

2. CJM 2007 (vol 59 pp. 596)

Itzá-Ortiz, Benjamín A.
Eigenvalues, $K$-theory and Minimal Flows
Let $(Y,T)$ be a minimal suspension flow built over a dynamical system $(X,S)$ and with (strictly positive, continuous) ceiling function $f\colon X\to\R$. We show that the eigenvalues of $(Y,T)$ are contained in the range of a trace on the $K_0$-group of $(X,S)$. Moreover, a trace gives an order isomorphism of a subgroup of $K_0(\cprod{C(X)}{S})$ with the group of eigenvalues of $(Y,T)$. Using this result, we relate the values of $t$ for which the time-$t$ map on the minimal suspension flow is minimal with the $K$-theory of the base of this suspension.

Categories:37A55, 37B05

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