The modular Gromov–Hausdorff propinquity

Abstract

The dual modular propinquity is a complete metric, up to full modular quantum isometry, on the class metrized quantum vector bundles, i.e. of Hilbert modules endowed with a type of densely defined norm, called a D-norm, which generalize the operator norm given by a connection on a Riemannian manifold. The dual modular propinquity is weaker than the modular propinquity yet it is complete, which is the main purpose of its introduction. Moreover, we show that the modular propinquity can be extended to a larger class of objects which involve quantum compact metric spaces acting on metrized quantum vector bundles.

Sampling and periodization of generators of Heisenberg modules

“…In addition, one is often interested in the behaviour of finitely generated projective modules over A n and if it in the "limit" it converges to a finitely generated projective module over B. The definition of a distance between modules over two C * -algebras has been recently addressed in [19,27] and Latremoliere has worked out the case of Heisenberg modules over noncommutative 2-tori in [20,21]. Here we complement these results by pointing out that if one uses that Heisenberg modules are finitely generated and projective over noncommutative tori, then one can study the behaviour of sequences of Heisenberg modules converging to a Heisenberg modules by understanding what is going on for the generators.…”

Metrized Quantum Vector Bundles over Quantum Tori Built from Riemannian Metrics and Rosenberg's Levi-Civita Connections

“…Example 4.6. In [6], Latrémolière constructed metrized quantum vector bundles from actual vector bundles over compact Riemannian manifolds that provide motivating examples for Definition 4.5. In the same paper, he also showed that free Hilbert modules over the underlying unital C * -algebra of a quasi-Leibniz quantum compact metric space can be turned into metrized quantum vector bundles.…”

“…In the same paper, he also showed that free Hilbert modules over the underlying unital C * -algebra of a quasi-Leibniz quantum compact metric space can be turned into metrized quantum vector bundles. The tools developed in [6] are then applied in [7] to prove the convergence of Heisenberg modules over quantum 2-tori with respect to the modular Gromov-Hausdorff propinquity.…”

“…This paper was inspired by an apparent connection between Jonathan Rosenberg's work on Riemannian metrics on a generically transcendental quantum torus and Levi-Civita connections for these metrics [14], and Frédéric Latrémolière's work on metrized quantum vector bundles and the modular Gromov-Hausdorff propinquity [6].…”

“…Recently, Latrémolière was able to generalize the Gromov-Hausdorff propinquity to Hilbert C * -modules over a quantum compact metric space endowed with a special metric structure [6]. He calls these objects metrized quantum vector bundles, regarding them to be a noncommutative arXiv:1803.04036v2 [math.OA] 29 Jul 2018 generalization of vector bundles over a compact Riemannian manifold endowed with a metric.…”