Abstract view
$K$theory of Furstenberg Transformation Group $C^*$algebras


Published:20131009
Printed: Dec 2013
Kamran Reihani,
Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ 86011, USA
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 PimsnerVoiculescu 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 ErdÅ‘s), 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
$K$theory, transformation group $C^*$algebra, Furstenberg transformation, Anzai transformation, minimal homeomorphism, positive cone, minimal homeomorphism

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