3 Review(s)

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 $\{\theta_n\}$ of parameters converges to a parameter $\theta$, then the sequence $\{A_{\theta_n}\}$ of quantum tori converges in quantum Gromov-Hausdorff distance to $A_\theta$.