Quantum Metric Spaces and the Gromov-Hausdorff Propinquity

TL;DR

This paper surveys the dual Gromov-Hausdorff propinquity, a noncommutative metric framework for C*-algebras, including new results on metric perturbations and examples of quantum metric spaces.

Contribution

It introduces the dual Gromov-Hausdorff propinquity for noncommutative spaces and presents new results on metric perturbations within this framework.

Findings

Development of the dual Gromov-Hausdorff propinquity framework

Examples of quantum locally compact metric spaces

New results on perturbations of metrics on Leibniz quantum compact metric spaces

Abstract

We present a survey of the dual Gromov-Hausdorff propinquity, a noncommutative analogue of the Gromov-Hausdorff distance which we introduced to provide a framework for the study of the noncommutative metric properties of C*-algebras. We first review the notions of quantum locally compact metric spaces, and present various examples of such structures. We then explain the construction of the dual Gromov-Hausdorff propinquity, first in the context of quasi-Leibniz quantum compact metric spaces, and then in the context of pointed quantum proper metric spaces. We include a few new result concerning perturbations of the metrics on Leibniz quantum compact metric spaces in relation with the dual Gromov-Hausdorff propinquity.