A Compactness Theorem for The Dual Gromov-Hausdorff Propinquity

TL;DR

This paper establishes a compactness theorem for the dual Gromov-Hausdorff propinquity, extending the classical Gromov compactness theorem into the noncommutative setting of quantum metric spaces.

Contribution

It introduces a compactness theorem for the dual Gromov-Hausdorff propinquity applicable to subclasses of quasi-Leibniz compact quantum metric spaces, including limits of finite-dimensional spaces.

Findings

Nuclear, quasi-diagonal quasi-Leibniz spaces are limits of finite-dimensional spaces.

Extension of dual propinquity to quasi-Leibniz spaces is developed.

The theorem generalizes classical compactness results to noncommutative geometry.

Abstract

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 compact quantum metric spaces of the closure of finite dimensional quasi-Leibniz compact quantum metric spaces for the dual propinquity. While finding characterizations of this class proves delicate, we show that all nuclear, quasi-diagonal quasi-Leibniz compact quantum metric spaces are limits of finite dimensional quasi-Leibniz compact quantum metric spaces. This result involves a mild extension of the definition of the dual propinquity to quasi-Leibniz compact quantum metric spaces, which is presented in the first part of this paper.