非线性状态空间
Metrics on State Spaces
摘要：
与紧空间上的通常度量相关的Lipschitz范数不同，我们通过例子表明，可能非交换的紧空间上的Lipschitz范数通常不是由它们在状态空间上定义的度量限制到状态空间的极值点来决定的.我们刻画了由它们在整个状态空间上的度量所决定的Lipschitz范数是下连续的。我们证明了它们的Lipschitz元的区域可以扩大，从而形成一个对偶巴拿赫空间，推广了普通Lipschitz范数的情况。我们给出了由Lipschitz范数得到的状态空间上的度量的一个特征。Kadison的"函数空间"提供了这些结果的自然（更广泛）设置。指出了构造Lipschitz范数的各种方法。
英文摘要：
In contrast to the usual Lipschitz seminorms associated to ordinary metrics on compact spaces, we show by examples that Lipschitz seminorms on possibly non-commutative compact spaces are usually not determined by the restriction of the metric they define on the state space, to the extreme points of the state space. We characterize the Lipschitz norms which are determined by their metric on the whole state space as being those which are lower semicontinuous. We show that their domain of Lipschitz elements can be enlarged so as to form a dual Banach space, which generalizes the situation for ordinary Lipschitz seminorms. We give a characterization of the metrics on state spaces which come from Lipschitz seminorms. The natural (broader) setting for these results is provided by the ``function spaces'' of Kadison. A variety of methods for constructing Lipschitz seminorms is indicated.
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