On Gromov-Hausdorff convergence for operator metric spaces

TL;DR

This paper develops a framework for operator Gromov-Hausdorff convergence, establishing its properties, and explores its implications for quantum metric spaces, including continuity results and nonseparability of certain classes.

Contribution

It introduces an operator Gromov-Hausdorff distance, proves its equivalence to existing metrics, and applies it to various quantum spaces, advancing the theory of operator metric convergence.

Findings

Established a complete theory of operator Gromov-Hausdorff convergence.

Proved continuity of quantum tori, Berezin-Toeplitz quantizations, and theta-deformations.

Showed nonseparability of isometry classes of certain operator systems.

Abstract

We introduce an analogue for Lip-normed operator systems of the second author's order-unit quantum Gromov-Hausdorff distance and prove that it is equal to the first author's complete distance. This enables us to consolidate the basic theory of what might be called operator Gromov-Hausdorff convergence. In particular we establish a completeness theorem and deduce continuity in quantum tori, Berezin-Toeplitz quantizations, and theta-deformations from work of the second author. We show that approximability by Lip-normed matrix algebras is equivalent to 1-exactness of the underlying operator space and, by applying a result of Junge and Pisier, that for n greater than or equal to 7 the set of isometry classes of n-dimensional Lip-normed operator systems is nonseparable. We also treat the question of generic complete order structure.