$K$-theory of Furstenberg transformation group $C^*$-algebras

TL;DR

This paper investigates the $K$-theoretic invariants of Furstenberg transformation group $C^*$-algebras, revealing new structural insights, correcting previous misconceptions, and establishing combinatorial and asymptotic properties of their invariants.

Contribution

It provides a detailed analysis of the $K$-groups of these $C^*$-algebras, proves invariance of rank for certain matrices, corrects literature claims on torsion, and links invariants to combinatorial and asymptotic formulas.

Findings

The $K$-groups have a constant rank $a_n$ for unipotent matrices.

A claim about torsion subgroups in the literature is false.

The sequence $a_n$ has a combinatorial interpretation and known asymptotic behavior.

Abstract

The paper studies the $K$-theoretic invariants of the crossed product $C^{*}$-algebras associated with an important family of homeomorphisms of the tori $\Bbb{T}^{n}$ called {\em 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, together with two computing programs - included in an appendix - 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 $\frak{sl}(2,\Bbb{C})$, 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 Erd\"os), 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.