Harmonic Analysis Approach to Gromov–Hausdorff Convergence for Noncommutative Tori

We show that the rotation algebras are limits of matrix algebras in a very strong sense of convergence for algebras with additional Lipschitz structure. Our results generalize to higher dimensional noncommutative tori and operator valued coefficients. In contrast to previous results by Rieffel, Li, Kerr, and Latrémolière, we use Lipschitz norms induced by the ‘carré du champ’ of certain natural dynamical systems, including the heat semigroup.