Equivariant -correspondences and compact quantum group actions on Pimsner algebras

Abstract

Let G be a compact quantum group. We show that given a G-equivariant $\textrm {C}^*$ -correspondence E, the Pimsner algebra $\mathcal {O}_E$ can be naturally made into a G- $\textrm {C}^*$ -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 KMS 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 $\textrm {C}^*$ -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.