Linear coactions of discrete quantum groups on the circle

For a (unital) $C^{\ast }$-algebra $\text{cla}$, we construct a $C^{\ast }$-algebraic discrete quantum group (DQG) ${\text{clq}}_{\mathrm{a}\mathrm{u}\mathrm{t}}(\text{cla})$, coacting on $\text{cla}$, which is a quantum generalization of $\text{A}ut(\text{cla})$ in the framework of discrete quantum groups, in the sense that any other coaction of a DQG on $\text{cla}$ factors through the above coaction of ${\text{clq}}_{\mathrm{a}\mathrm{u}\mathrm{t}}(\text{cla})$. We prove by an explicit calculation that if any Kac-type $C^{\ast }$-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 $\{Z,\bar{Z}\}$ invariant, then $\mathcal{Q}$ must be classical, i.e. isomorphic with $C_0(\mathrm{\Gamma })$ for some discrete group $\mathrm{\Gamma }$. 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).