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

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 Latrémolière we use Lipschitz norms induced by the `carré du champ' of certain natural dynamical systems, including the heat semigroup.

81 pages, revised version to appear in CMP

References in corpus (1)

• Subgaussian 1-cocycles on discrete groups

Cited by in corpus (5)

• Sharp estimates of noncommutative Bochner-Riesz means on two-dimensional quantum tori
• Ricci curvature of quantum channels on non-commutative transportation metric spaces
• Continuity of the Spectrum of Dirac Operators of Spectral Triples for the Spectral Propinquity
• Spectral multipliers in group algebras and noncommutative Calderón-Zygmund theory
• Riesz transforms, Hodge-Dirac operators and functional calculus for multipliers I