Harmonic analysis approach to Gromov--Hausdorff convergence for noncommutative tori

TL;DR

This paper demonstrates that higher-dimensional noncommutative tori can be approximated by matrix algebras using Lipschitz norms derived from dynamical systems, extending previous convergence results in noncommutative geometry.

Contribution

It introduces a harmonic analysis approach to establish strong Gromov--Hausdorff convergence of noncommutative tori, generalizing prior work to higher dimensions and operator-valued coefficients.

Findings

Rotation algebras approximate matrix algebras with Lipschitz convergence

Results extend to higher-dimensional noncommutative tori

Uses Lipschitz norms from natural dynamical systems like heat semigroup

Abstract

We show that the rotation algebras are limit 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 Latr\'emoli\`ere we use Lipschitz norms induced by the `carr\'e du champ' of certain natural dynamical systems, including the heat semigroup.