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 Lipschitz seminorm on functions which is defined by 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, $A_{þ}$. We show, for consistently defined ``metrics'', that if a sequence $\{{þ}_n\}$ of parameters converges to a parameter $þ$, then the sequence $\{A_{{þ}_n}\}$ of quantum tori converges in quantum Gromov-Hausdorff distance to $A_{þ}$.

81 pages. Several minor improvements and several references added. To appear Memoirs Amer. Math. Soc