Gromov-Hausdorff Distance for Quantum Metric Spaces/Matrix Algebras Converge to the Sphere for Quantum Gromov-Hausdorff Distance

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, $A\theta$. We show, for consistently defined 'metrics', that if a sequence $\{\thetan\}$ of parameters converges to a parameter $\theta$, then the sequence $\{A{\thetan}\}$ of quantum tori converges in quantum Gromov-Hausdorff distance to $A\theta$.