The Covariant Gromov-Hausdorff Propinquity

TL;DR

This paper introduces a new metric called the covariant Gromov-Hausdorff propinquity for Lipschitz dynamical systems, enabling precise measurement of their convergence and equivalence in quantum metric space settings.

Contribution

It extends the Gromov-Hausdorff propinquity to a covariant setting for quantum metric spaces with monoid actions, establishing conditions for convergence and isometry.

Findings

The metric is zero iff an equivariant full quantum isometry exists.

Provided conditions for Cauchy sequences to converge under the new metric.

Applied framework to convergence of dual actions on fuzzy and quantum tori.

Abstract

We extend the Gromov-Hausdorff propinquity to a metric on Lipschitz dynamical systems, which are given by strongly continuous actions of proper monoids on quantum compact metric spaces via Lipschitz morphisms. We prove that our resulting metric is zero between two Lipschitz dynamical systems if and only if there exists an equivariant full quantum isometry between. We also present sufficient conditions for Cauchy sequences to converge for our new metric, thus exhibiting certain complete classes of Lipschitz dynamical systems. We apply our work to convergence of the dual actions on fuzzy tori to the dual actions on quantum tori. Our framework is general enough to also allow for the study of the convergence of continuous semigroups of positive linear maps and other actions of proper monoids.