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Gromov-Hausdorff Distance for Quantum Metric Spaces/Matrix Algebras Converge to the Sphere for Quantum Gromov-Hausdorff Distance
Marc A. Rieffel, University of California, Berkeley, CA
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Memoirs of the American Mathematical Society
2004; 91 pp; softcover
Volume: 168
ISBN-10: 0-8218-3518-1
ISBN-13: 978-0-8218-3518-0
List Price: US$60 Individual Members: US$36
Institutional Members: US\$48
Order Code: MEMO/168/796

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 $$A_\theta$$.