The Quantum Gromov-Hausdorff Propinquity

TL;DR

This paper introduces the quantum Gromov-Hausdorff propinquity, a new metric for quantum compact metric spaces that improves upon previous distances by incorporating *-isomorphism and Leibniz seminorms, advancing noncommutative geometry.

Contribution

It defines a novel distance measure for quantum metric spaces that extends and strengthens existing concepts, addressing a key open problem in noncommutative metric geometry.

Findings

Defines the quantum Gromov-Hausdorff propinquity as a new distance measure.

Ensures *-isomorphism is necessary for zero distance, strengthening previous metrics.

Provides a framework compatible with Leibniz Lip-norms over C*-algebras.

Abstract

We introduce the quantum Gromov-Hausdorff propinquity, a new distance between quantum compact metric spaces, which extends the Gromov-Hausdorff distance to noncommutative geometry and strengthens Rieffel's quantum Gromov-Hausdorff distance and Rieffel's proximity by making *-isomorphism a necessary condition for distance zero, while being well adapted to Leibniz seminorms. This work offers a natural solution to the long-standing problem of finding a framework for the development of a theory of Leibniz Lip-norms over C*-algebras.