Linear coactions of discrete quantum groups on the circle

For a (unital) $C^*$-algebra $\cla$, we construct a $C^*$-algebraic discrete quantum group (DQG) $\clq_{\rm aut}(\cla)$, coacting on $\cla$, which is a quantum generalization of ${\text Aut}(\cla)$ in the framework of discrete quantum groups, in the sense that any other coaction of a DQG on $\cla$ factors through the above coaction of $\clq_{\rm aut}(\cla)$. We prove by an explicit calculation that if any Kac-type $C^*$-algebraic discrete quantum group $\mathcal{Q}$ has a `weakly faithful' coaction on $C(S^1)$ which is `linear' in the sense that it leaves the space spanned by $\{ Z, \overline{Z} \}$ invariant, then $\mathcal{Q}$ must be classical, i.e. isomorphic with $C_0(\Gamma)$ for some discrete group $\Gamma$. This parallels the well-known result of non-existence of genuine compact quantum group symmetry obtained by the first author and his collaborators ([GB16] and the references therein).