Gromov-Hausdorff Distance for Quantum Metric Spaces

Details Gromov-Hausdorff Distance for Quantum Metric Spaces Gromov-Hausdorff Distance for Quantum Metric Spaces

2000-11-09

arxiv.org/abs/math/0011063v4

Mem. Amer. Math. Soc. 168 (2004) no. 796, 1-65

Mathematics

Operator Algebras

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

Scientific paper

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, $A_{\th}$. We show, for consistently defined ``metrics'', that if a sequence $\{\th_n\}$ of parameters converges to a parameter $\th$, then the sequence $\{A_{\th_n}\}$ of quantum tori converges in quantum Gromov-Hausdorff distance to $A_{\th}$.

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