Metrics on state spaces
Metrics on state spaces
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DOI：
10.4171/dm/68
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发表时间：
1999-06
影响因子：
0.9
通讯作者：
M. Rieffel
中科院分区：
文献类型：
--
作者：
M. Rieffel
与紧致空间上的普通度量相关的通常 Lipschitz 半范数相反，我们通过例子表明，可能非交换紧致空间上的 Lipschitz 半范数通常不是由它们在状态空间上定义的度量的限制来确定的，即状态空间的极值点。我们将由整个状态空间上的度量确定的利普希茨范数描述为下半连续范数。我们证明了它们的 Lipschitz 元素的域可以扩大以形成对偶 Banach 空间，这概括了普通 Lipschitz 半范数的情况。我们给出了来自 Lipschitz 半范数的状态空间度量的表征。这些结果的自然（更广泛）设置是由 Kadison 的“功能空间”提供的。指出了构建 Lipschitz 半范数的多种方法。
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.