Noncommutative metric geometry, Gromov-Hausdorff convergence, Monge-Kantorovich distance, Quantum Metric Spaces, Lip-norms.
We prove a compactness theorem for the dual Gromov-Hausdorff propinquity as a noncommutative analogue of the Gromov compactness theorem for the Gromov-Hausdorff distance. Our theorem is valid for subclasses of quasi-Leibniz quantum compact metric spaces of the closure of finite dimensional quasi-Leibniz quantum compact metric spaces for the dual propinquity. While finding characterizations of this class proves delicate, we show that all nuclear, quasi-diagonal quasi-Leibniz quantum compact metric spaces are limits of finite dimensional quasi-Leibniz quantum compact metric spaces. This result involves a mild extension of the definition of the dual propinquity to quasi-Leibniz quantum compact metric spaces, which is presented in the first part of this paper
Latrémolière, Frédéric, "A Compactness Theorem for the Dual Gromov-Hausdorff Propinquity" (2015). Mathematics Preprint Series. Paper 7.