“矩阵代数收敛于球面”的莱布尼茨定理
Leibniz seminorms for "Matrix algebras converge to the sphere"
摘要：
在我早期的一篇关于普通紧度量空间的向量丛和Gromov-Hausdorff距离的论文中，度量的Lipschitz范数满足强莱布尼茨性质是至关重要的。本文针对矩阵代数收敛到球面（或其它空间）的量子Gromov-Hausdorff距离的非交换情形，给出了如何构造满足强Leibniz性质的矩阵代数.这是为了准备在高能物理学文献中做出精确的某些陈述，这些陈述涉及矩阵代数上的“向量丛“，而矩阵代数“对应”于球面上的“向量丛”。我们表明，一个相当普遍的来源，满足强莱布尼茨性质的导子到赋范双模。对于矩阵代数，我们的主要技术工具是相干态和Berezin符号。
英文摘要：
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. In the present paper, for the now non-commutative situation of matrix algebras converging to the sphere (or to other spaces) for quantum Gromov-Hausdorff distance, we show how to construct suitable seminorms that also satisfy the strong Leibniz property. This is in preparation for making precise certain statements in the literature of high-energy physics concerning "vector bundles" over matrix algebras that "correspond" to monopole bundles over the sphere. We show that a fairly general source of seminorms that satisfy the strong Leibniz property consists of derivations into normed bimodules. For matrix algebras our main technical tools are coherent states and Berezin symbols.
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