Random Operators and Crossed Products
Random Operators and Crossed Products
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随机算子和交叉积
DOI：
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发表时间：
1999
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影响因子：
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通讯作者：
D. Lenz
中科院分区：
文献类型：
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作者：
D. Lenz
本文讨论了交叉积及其在随机算子中的应用。我们研究了动力系统的冯诺依曼代数使用的基础希尔伯特代数结构。这给出了一个特别容易的方法来引入这个冯诺依曼代数上的迹。本文回顾了这一迹的几个公式，说明了它是如何作为Connes的非对易积分理论的一个应用而出现的，并讨论了Shubin的迹公式。然后，我们限制自己的情况下，一个行动的一组一组，并包括新的证明的一些定理Bellissard和Testard类似的经典Plancherel定理。我们证明了积分态密度在周期情况下是一个谱测度，从而推广了Kaminker和Xia的一个结果。最后讨论了对偶结果，并应用Gordon等人的方法建立了Z.
This article is concerned with crossed products and their applications to random operators. We study the von Neumann algebra of a dynamical system using the underlying Hilbert algebra structure. This gives a particularly easy way to introduce a trace on this von Neumann algebra. We review several formulas for this trace, show how it comes as an application of Connes" noncommutative integration theory and discuss Shubin"s trace formula. We then restrict ourselves to the case of an action of a group on a group and include new proofs for some theorems of Bellissard and Testard on an analogue of the classical Plancherel theorem. We show that the integrated density of states is a spectral measure in the periodic case, thereby generalizing a result of Kaminker and Xia. Finally, we discuss duality results and apply a method of Gordon et al. to establish a duality result for crossed products by Z.