Isometric actions of compact quantum groups on graph $C^{\ast }$-algebras

Abstract

For a strongly connected directed graph $\Lambda$, it is shown that the quantum automorphism group of $\Lambda$ acts isometrically in the sense of J. Bhowmick and D. Goswami (2009) 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 $U_n^{+}$ on the Cuntz algebra ${\mathcal{O}}_n$ is not isometric.