Gromov-Hausdorff convergence of metric spaces of UCP maps

TL;DR

This paper demonstrates that under certain conditions, sequences of sets of unital completely positive maps converge in the Gromov-Hausdorff sense, allowing geometric information to be extracted from spectral data even with partial spectral information.

Contribution

It extends van Suijlekom's technique to show Gromov-Hausdorff convergence for operator system spectral triples with partial spectral data.

Findings

Sequences of unital completely positive maps converge in the Gromov-Hausdorff sense.

Partial spectral data can still yield geometric information.

The BW-topology on these maps is metrizable.

Abstract

It is shown that van Suijlekom's technique of imposing a set of conditions on operator system spectral triples ensures Gromov-Hausdorff convergence of sequences of sets of unital completely positive maps (equipped with the BW-topology which is metrizable). This implies that even when only a part of the spectrum of the Dirac operator is available together with a certain truncation of the $C^*$-algebra, information about the geometry can be extracted.