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

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

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, $A_ heta$. We show, for consistently defined "e;metrics"e;, that if a sequence ${ heta_n}$ of parameters converges to a parameter $ heta$, then the sequence ${A_{ heta_n}}$ of quantum tori converges in quantum Gromov-Hausdorff distance to $A_ heta$.