A computation with the Connes-Thom isomorphism

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 $(α,τ)$ of $\mathbb{R}^{n} \rtimes \mathbb{Z}$ on a $C^{*}$-algebra $B$ where $α$ is a strongly continuous action of $\mathbb{R}^{n}$ and $τ$ is an automorphism. The map $τ$ induces a map $\widetildeτ$ on $B \rtimes_α \mathbb{R}^{n}$. We show that, at the $K$-theory level, $τ$ 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.

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