Gromov--Hausdorff Convergence of Spectral Truncations for Tori

We consider operator systems associated to spectral truncations of tori. We show that their state spaces, when equipped with the Connes distance function, converge in the Gromov--Hausdorff sense to the space of all Borel probability measures on the torus equipped with the Monge--Kantorovich distance. A crucial role will be played by the relationship between Schur and Fourier multipliers. Along the way, we introduce the spectral Fej\'er kernel and show that it is a good kernel. This allows to make the estimates sufficient to prove the desired convergence of state spaces. We conclude with some structure analysis of the pertinent operator systems, including the C*-envelope and the propagation number, and with an observation about the dual operator system.