A computation with the Connes-Thom isomorphism

Abstract

Let $A\in M_n(\mathbb{R})$ be an invertible matrix. Consider the semi-direct product ${\mathbb{R}}^n⋊\mathbb{Z}$ where $\mathbb{Z}$ acts on ${\mathbb{R}}^n$ by matrix multiplication. Consider a strongly continuous action $(\alpha ,\tau )$ of ${\mathbb{R}}^n⋊\mathbb{Z}$ on a $C^{\ast }$-algebra $B$ where $\alpha$ is a strongly continuous action of ${\mathbb{R}}^n$ and $\tau$ is an automorphism. The map $\tau$ induces a map $\tilde{\tau }$ on $B{⋊}_{\alpha }{\mathbb{R}}^n$. We show that, at the $K$-theory level, $\tau$ commutes with the Connes-Thom map if $\det (A)>0$ and anticommutes if $\det (A)<0$. As an application, we recompute the $K$-groups of the Cuntz-Li algebra associated to an integer dilation matrix.