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

TL;DR

This paper explores how compact quantum group actions can be incorporated into Pimsner algebras derived from $C^*$-correspondences, establishing conditions for equivariance and analyzing implications for quantum automorphism groups of graphs.

Contribution

It introduces a framework for making Pimsner algebras $G$-equivariant under quantum group actions and characterizes when such actions arise naturally, especially for Kac type quantum groups.

Findings

Pimsner algebra can be naturally endowed with a $G$-action from a $G$-equivariant correspondence.

Conditions are provided for when a $G$-action on the Pimsner algebra is induced by the correspondence.

The results are applied to graph $C^*$-algebras, impacting the understanding of quantum automorphism groups.

Abstract

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