IUMJ

Title: Purely infinite simple $C^*$-algebras associated to integer dilation matrices

Authors: Ruy Exel, Astrid an Huef and Iain Raeburn

Issue: Volume 60 (2011), Issue 3, 1033-1058

Abstract:

Given an $n \times n$ integer matrix $A$ whose eigenvalues are strictly greater than $1$ in absolute value, let $\sigma_A$ be the transformation of the $n$-torus $\mathbb{T}^n = \mathbb{R}^n/\mathbb{Z}^n$ defined by $\sigma_A(e^{2\pi ix}) = e^{2\pi iAx}$ for $x \in \mathbb{R}^n$. We study the associated crossed-product $C^{*}$-algebra, which is defined using a certain transfer operator for $\sigma_A$, proving it to be simple and purely infinite and computing its $K$-theory groups.