Convergence of spectral smoothings from compact quantum group actions

In this note, we study complete and quantum Gromov-Hausdorff convergence of operator systems obtained using smoothings of C*-algebras, where our smoothings arise from the action of a compact quantum group on the C*-algebra. Specifically, we show how if $\mathbb{G}$ is a co-amenable compact quantum group acting on unital C*-algebra $\mathcal{A}$, and both are equipped with suitable quantum metrics, bounded approximate identities in $L^1(\mathbb{G})$ may be used for approximations of the original space $\mathcal{A}$ under the complete and quantum Gromov-Hausdorff distance. We finish by presenting some applications to the convergence of spectral truncations of compact groups and connections to previous results.