Isometric actions of compact quantum groups on graph C*-algebras

TL;DR

This paper investigates the isometric actions of compact quantum groups on graph C*-algebras, demonstrating that quantum automorphism groups act isometrically on certain spectral triples, but not on Cuntz algebras.

Contribution

It establishes conditions under which quantum automorphism groups act isometrically on graph C*-algebras and shows the non-isometric nature of certain natural actions.

Findings

Quantum automorphism groups act isometrically on spectral triples of strongly connected graphs.

The natural action of U_{n}^{+} is not isometric on Cuntz algebras.

Provides insight into symmetries of graph C*-algebras in the quantum setting.

Abstract

For a strongly connected directed graph $\Lambda$, it is shown that the quantum automorphism group of $\Lambda$ acts isometrically in the sense of D. Goswami et al. on the spectral triple $(C^{\ast}(\Lambda), L^{2}(\Lambda^{\infty},M),D)$ constructed by Farsi et al.. It is also shown that the natural action of $U_{n}^{+}$ is not isometric on the Cuntz algebra $\mathcal{O}_{n}$.