Topographic Gromov-Hausdorff quantum Hypertopology for Quantum Proper Metric Spaces

TL;DR

This paper develops a new topology for pointed proper quantum metric spaces, extending classical Gromov-Hausdorff topology to the noncommutative setting of quantum locally compact metric spaces.

Contribution

It introduces an infra-metric that generalizes Gromov-Hausdorff distance, enabling a topology that captures quantum proper metric spaces.

Findings

The topology generalizes classical Gromov-Hausdorff and dual propinquity topologies.

The infra-metric is zero only for isometric isomorphisms of quantum spaces.

The framework extends noncommutative metric geometry to locally compact quantum spaces.

Abstract

We construct a topology on the class of pointed proper quantum metric spaces which generalizes the topology of the Gromov-Hausdorff distance on proper metric spaces, and the topology of the dual propinquity on Leibniz quantum compact metric spaces. A pointed proper quantum metric space is a special type of quantum locally compact metric space whose topography is proper, and with properties modeled on Leibniz quantum compact metric spaces, though they are usually not compact and include all the classical proper metric spaces. Our topology is obtained from an infra-metric which is our analogue of the Gromov-Hausdorff distance, and which is null only between isometrically isomorphic pointed proper quantum metric spaces. Thus, we propose a new framework which extends noncommutative metric geometry, and in particular noncommutative Gromov-Hausdorff topology, to the realm of quantum locally compact metric spaces.