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Gromov-Hausdorff Distance for Quantum Metric Spaces

Marc A. Rieffel
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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 $A_þ$.

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BibTeXRIS

Marc A. Rieffel. 2003-02-20. Gromov-Hausdorff Distance for Quantum Metric Spaces. https://arxiv.org/abs/math/0011063

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