Matricial quantum Gromov–Hausdorff distance

We develop a matricial version of Rieffelʹs Gromov–Hausdorff distance for compact quantum metric spaces within the setting of operator systems and unital C∗-algebras. Our approach yields a metric space of “isometric” unital complete order isomorphism classes of metrized operator systems which in many cases exhibits the same convergence properties as those in the quantum metric setting, as for example in Rieffelʹs approximation of the sphere by matrix algebras using Berezin quantization. Within the metric subspace of metrized unital C∗-algebras we establish the convergence of sequences which are Cauchy with respect to a larger Leibniz distance, and we also prove an analogue of the precompactness theorems of Gromov and Rieffel.