Quantized Gromov-Hausdorff distance
Abstract
A quantized metric space is a matrix order unit space equipped with an operator space version of Rieffel's Lip-norm. We develop for quantized metric spaces an operator space version of quantum Gromov-Hausdorff distance. We show that two quantized metric spaces are completely isometric if and only if their quantized Gromov-Hausdorff distance is zero. We establish a completeness theorem. As applications, we show that a quantized metric space with 1-exact underlying matrix order unit space is a limit of matrix algebras with respect to quantized Gromov-Hausdorff distance, and that matrix algebras converge naturally to the sphere for quantized Gromov-Hausdorff distance.
Cite
@article{arxiv.math/0503344,
title = {Quantized Gromov-Hausdorff distance},
author = {Wei Wu},
journal= {arXiv preprint arXiv:math/0503344},
year = {2009}
}
Comments
34 pages. An oversight appeared in Proposition 4.9 of Version 1. This proposition has been deleted. Also some type errors have been corrected