Equivariant ${\mathrm{C}}^{\ast }$-correspondences and compact quantum group actions on Pimsner algebras

Let $G$ be a compact quantum group. We show that given a $G$-equivariant ${\mathrm{C}}^{\ast }$-correspondence $E$, the Pimsner algebra ${\mathcal{O}}_E$ can be naturally made into a $G$-${\mathrm{C}}^{\ast }$-algebra. We also provide sufficient conditions under which it is guaranteed that a $G$-action on the Pimsner algebra ${\mathcal{O}}_E$ arises in this way, in a suitable precise sense. When $G$ is of Kac type, a $\mathrm{K}\mathrm{M}\mathrm{S}$ state on the Pimsner algebra, arising from a quasi-free dynamics, is $G$-equivariant if and only if the tracial state obtained from restricting it to the coefficient algebra is $G$-equivariant, under a natural condition. We apply these results to the situation when the ${\mathrm{C}}^{\ast }$-correspondence is obtained from a finite, directed graph and draw various conclusions on the quantum automorphism groups of such graphs, both in the sense of Banica and Bichon.