Gromov-Hausdorff Distance for Quantum Metric Spaces/Matrix Algebras Converge to the Sphere for Quantum Gromov-Hausdorff Distance
- ISBN-10
- 0821835181
- ISBN-13
- 9780821835180
- Category
- Global differential geometry
- Pages
- 91
- Language
- English
- Published
- 2004
- Publisher
- American Mathematical Soc.
- Author
- Marc Aristide Rieffel
Description
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 $A_\theta$.
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