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In an earlier paper of mine relating vector bundles and Gromov-Hausdorff distance for ordinary compact metric spaces, it was crucial that the Lipschitz seminorms from the metrics satisfy a strong Leibniz property.
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by same author, Order-unit quantum Gromov-Hausdorff distance , J. Funct. Anal. 231
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by same author, Lipschitz extension constants equal projection constants , Operator theory, operator algebras, and applications, Contemp. Math., vol. 414, Amer. Math. Soc., Providence, RI, 2006, arXiv:math.FA/0508097, pp. 147–162. MR 2277209
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2002
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David Kerr, Matricial quantum Gromov-Hausdorff distance , J. Funct. Anal. 205
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Cristina Antonescu and Erik Christensen, Metrics on group C ∗ C^{*} -algebras and a non-commutative Arzelà-Ascoli theorem , J. Funct. Anal. 214
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by same author, Compact quantum metric spaces , Operator algebras, quantization, and noncommutative geometry, Contemp. Math., vol. 365, Amer. Math. Soc., Providence, RI, 2004, arXiv:math.OA/0308207, pp. 315–330. MR 2106826 (2005h:46099)
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Hanfeng Li, θ \theta -deformations as compact quantum metric spaces , Comm. Math. Phys. 256
2005
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Frédéric Latrémolière Bounded-Lipschitz distances on the state space of a C ∗ C^{*} -algebra , Taiwanese J. Math. 11
2007
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2008
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2009
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