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Noncommutative metric geometry, Quantum Gromov-Hausdorff distance, Monge-Kantorovich distance, Quantum Metric Spaces, Lip-norms, Compact C*-metric spaces, Leibniz seminorms, Quantum Tori, Finite dimensional approximations


We introduce the quantum Gromov-Hausdorff propinquity, a new distance between quantum compact metric spaces, which extends the GromovHausdorff distance to noncommutative geometry and strengthens Rieffel’s quantum Gromov-Hausdorff distance and Rieffel’s proximity by making *-isomorphism a necessary condition for distance zero, while being well adapted to Leibniz seminorms. This work offers a natural solution to the long-standing problem of finding a framework for the development of a theory of Leibniz Lip-norms over C*- algebras.


First published in Transactions of the Amer. Math. Soc. 368 (2016), published by the American Mathematical Society. © 2016 American Mathematical Society.

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