Quantized Gromov–Hausdorff distance

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.