Spectral truncations of SU(2) and the spectral localizer

We study operator systems arising from Peter-Weyl and Dirac spectral truncations of $\mathrm{S}\mathrm{U}(2)$ in the sense of Connes-Van Suijlekom. We determine operator system invariants for these systems, including the $C^{\ast }$-envelope and uniform bounds on the propagation number. For Peter-Weyl truncations, we find $\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}\le 2$ for all connected compact Lie groups, generalizing known results for the torus. We also carry out an analysis of the spectral localizer by Loring and Schulz-Baldes on $\mathrm{S}\mathrm{U}(2)$. In this analysis, we find numerical evidence that an index map for operator systems previously proposed by Van Suijlekom only captures the right information in a suitable interval of the parameter $\kappa$, instead of the limit $\kappa \downarrow 0$.