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

Suvrajit Bhattacharjee, Soumalya Joardar

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.

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

Abstract

Let be a compact quantum group. We show that given a -equivariant -correspondence , the Pimsner algebra can be naturally made into a --algebra. We also provide sufficient conditions under which it is guaranteed that a -action on the Pimsner algebra arises in this way, in a suitable precise sense. When is of Kac type, a state on the Pimsner algebra, arising from a quasi-free dynamics, is -equivariant if and only if the tracial state obtained from restricting it to the coefficient algebra is -equivariant, under a natural condition. We apply these results to the situation when the -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.
Paper Structure (7 sections, 25 theorems, 175 equations)

This paper contains 7 sections, 25 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Compact quantum groups and their actions
  3. correspondences and Pimsner algebras
  4. KMS states on the Pimsner algebras
  5. Applications to quantum symmetries of graphs I: Generalities
  6. Applications to quantum symmetries of graphs II: The case of simple graphs
  7. Applications to quantum symmetries of graphs III: The case of multigraphs

Key Result

Theorem 1.1

Let $G$ be a compact quantum group, $(A,\alpha)$ be a unital $G$-$\mathrm{C}^*$-algebra and $(E,\phi,\lambda)$ be a $G$-equivariant $\mathrm{C}^*$-correspondence over the $G$-$\mathrm{C}^*$-algebra $(A,\alpha)$. Assume further that the Hilbert $A$-module $E$ is finitely generated and projective. The such that Moreover, the pair $(\mathcal{O}_E,\omega)$ is a $G$-$\mathrm{C}^*$-algebra. Here, $\mat

Theorems & Definitions (73)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 63 more