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Gromov-Hausdorff distance for quantum metric spaces (2001)
| Venue: | MEM. AMER. MATH. SOC |
| Citations: | 54 - 6 self |
BibTeX
@ARTICLE{Rieffel01gromov-hausdorffdistance,
author = {Marc A. Rieffel},
title = {Gromov-Hausdorff distance for quantum metric spaces},
journal = {MEM. AMER. MATH. SOC},
year = {2001},
pages = {1--65}
}
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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 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, Aθ. We show, for consistently defined “metrics”, that if a sequence {θn} of parameters converges to a parameter θ, then the sequence {Aθn} of quantum tori converges in quantum Gromov–Hausdorff distance to Aθ.
Keyphrases
quantum metric space gromov-hausdorff distance parameter converges quantum torus converges quantum torus natural quantum analogue order-unit space compact quantum usual lipschitz seminorm classical theory basic theorem main example quantum gromov hausdorff distance gromov hausdorff distance
