A Computation with the Connes–Thom Isomorphism

Abstract

Let $A\,\in \,{{M}_{n}}\left( \mathbb{R} \right)$ be an invertible matrix. Consider the semi-direct product ${{\mathbb{R}}^{n}}\rtimes \,\mathbb{Z}$ where the action of $\mathbb{Z}$ on ${{\mathbb{R}}^{n}}$ is induced by the left multiplication by $A$. Let $\left( \alpha ,\,\tau \right)$ be a strongly continuous action 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 $\widetilde{\tau }\,\text{on}\,\text{B}\,{{\rtimes }_{\alpha }}\,{{\mathbb{R}}^{n}}$. We show that, at the $K$-theory level, $\tau$ commutes with the Connes–Thom map if $\det \left( A \right)\,>\,0$ and anticommutes if $\det \left( A \right)\,>\,0$. As an application, we recompute the $K$-groups of the Cuntz–Li algebra associated with an integer dilation matrix.