Quantized Gromov–Hausdorff distance
Quantized Gromov–Hausdorff distance
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DOI：
10.1016/j.jfa.2005.02.017
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发表时间：
2005-03
影响因子：
1.7
通讯作者：
Wei Wu
中科院分区：
文献类型：
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作者：
Wei Wu
量子化度量空间是一个矩阵阶单位空间，具有Rieffel's lip -范数的算子空间版本。我们在量子化度量空间中建立了量子Gromov-Hausdorff距离的算子空间版本。我们证明了两个量子化度量空间是完全等距的当且仅当它们的量子化Gromov-Hausdorff距离为零。我们建立了完备性定理。作为应用，我们证明了具有1-精确底层矩阵阶单位空间的量子化度量空间是矩阵代数对量子化Gromov-Hausdorff距离的极限，并且对于量子化Gromov-Hausdorff距离，矩阵代数自然收敛于球。
A quantized metric space is a matrix order unit space equipped with an operator space version of Rieffel's Lip-norm. We develop for quantized metric spaces an operator space version of quantum Gromov–Hausdorff distance. We show that two quantized metric spaces are completely isometric if and only if their quantized Gromov–Hausdorff distance is zero. We establish a completeness theorem. As applications, we show that a quantized metric space with 1-exact underlying matrix order unit space is a limit of matrix algebras with respect to quantized Gromov–Hausdorff distance, and that matrix algebras converge naturally to the sphere for quantized Gromov–Hausdorff distance.