The quantum Gromov-Hausdorff Hypertopology on the class of pointed Proper Quantum Metric Spaces

TL;DR

This paper develops a new hypertopology for pointed proper quantum metric spaces, extending noncommutative metric geometry to locally compact quantum spaces and establishing convergence properties analogous to classical Gromov-Hausdorff topology.

Contribution

It introduces a hypertopology on pointed proper quantum metric spaces that generalizes the Gromov-Hausdorff distance to the noncommutative setting, including noncompact cases.

Findings

The hypertopology is compatible with the Gromov-Hausdorff propinquity on quantum compact metric spaces.

Convergence in classical Gromov-Hausdorff implies convergence in the new quantum hypertopology.

Provides examples of noncompact quantum metric spaces as limits of finite-dimensional spaces.

Abstract

We introduce a hypertopology, induced by an inframetric up to full quantum isometry, on the class of pointed proper quantum metric spaces, which are separable, possibly non-unital, C*-algebras endowed with an analogue of the Lipschitz seminorm, with a distinguished state, and with a particular type of approximate units. Our hypertopology provides an analogue of the Gromov-Hausdorff distance on proper metric spaces, and in fact, convergence in the latter implies convergence in the former. Moreover, when restricted to the class of quantum compact metric spaces, our new topology is compatible with the topology of the Gromov-Hausdorff propinquity. We include new examples of noncompact, noncommutative pointed proper quantum metric spaces which are limits, for our new topology, of finite dimensional quantum compact metric spaces. This article thus provides a first answer to the challenging question of how to extend noncommutative metric geometry to the locally compact quantum space realm.