Gromov-Hausdorff distance for quantum metric spaces

Abstract

By a quantum metric space we mean a C * -algebra (or more generally an order-unit space) equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to an ordinary metric.We develop for compact quantum metric spaces a version of Gromov-Hausdorff distance.We show that the basic theorems of the classical theory have natural quantum analogues.Our main example involves the quantum tori, A θ .We show, for consistently defined "metrics", that if a sequence {θ n } of parameters converges to a parameter θ, then the sequence {A θn } of quantum tori converges in quantum Gromov-Hausdorff distance to A θ .