Convergence of Cauchy Sequences for the covariant Gromov-Hausdorff propinquity

TL;DR

This paper establishes conditions for convergence of Cauchy sequences in the covariant Gromov-Hausdorff propinquity, demonstrating the completeness of many classes of quantum dynamical systems under this metric.

Contribution

It provides new sufficient conditions for convergence and proves the completeness of various natural classes of Lipschitz dynamical systems in the quantum setting.

Findings

Several sufficient conditions for convergence of Cauchy sequences.

Many natural classes of dynamical systems are complete under the covariant propinquity.

The covariant Gromov-Hausdorff propinquity is a complete metric for quantum dynamical systems.

Abstract

The covariant Gromov-Hausdorff propinquity is a distance on Lipschitz dynamical systems over quantum compact metric spaces, up to equivariant full quantum isometry. It is built from the dual Gromov-Hausdorff propinquity which, as its classical counterpart, is complete. We prove in this paper several sufficient conditions for convergence of Cauchy sequences for the covariant propinquity and apply it to show that many natural classes of dynamical systems are complete for this metric.