Let $A \in M_{ n}(\mathbb{ R})$ be an invertible matrix. Consider the semi-direct product $\mathbb{ R}^{ n} \rtimes \mathbb{ Z}$ where $\mathbb{ Z}$ acts on $\mathbb{ R}^{ n}$ by matrix multiplication. Consider a strongly continuous action $(\alpha,\tau)$ of $\mathbb{ R}^{ n} \rtimes \mathbb{ Z}$ on a $C^{ *}$-algebra $B$ where $\alpha$ is a strongly continuous action of $\mathbb{ R}^{ n}$ and $\tau$ is an automorphism. The map $\tau$ induces a map $\widetilde{ \tau}$ on $B \rtimes_{ \alpha} \mathbb{ R}^{ n}$. 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.
Categories: math.OA math.KT