CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCJM
Abstract view

$K$-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
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
Format:   LaTeX   MathJax   PDF  

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 $K$-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]
 

© Canadian Mathematical Society, 2014 : http://www.cms.math.ca/