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

Abstract

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 $\mathrm{KMS}$ 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 ${\mathrm{C}}^{\ast }$-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.

arXiv commentRevised version following the referee's suggestions. To appear in Canadian Journal of Mathematics

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