Abstract

This paper deals with properties of the ultraproducts for various structures. We introduce and study the concept of the ergodic action of a group with respect to a normal state on an abelian von Neumann algebra. In particular, we provide an example showing that the ultraproduct of ergodic states, generally speaking, is not ergodic. The ultraproduct of the Radon measures on a compact convex subset of a locally convex space is also investigated in the paper. As is well-known, the study of the extreme points in the state set for a \(C^{*}-\)algebra is a very interesting problem in itself. Considering the ultraproducts of \(C^{*}\)-algebras and the states on these algebras, we get quite nontrivial results.

中文翻译：

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$$ \ boldsymbol {C} ^ {\ boldsymbol {*}} $$的状态空间的超积-代数和Radon测度

摘要

本文讨论了各种结构的超产品的性能。我们介绍并研究了在阿贝尔冯·冯·诺依曼代数上的相对于正常状态的群体的遍历动作的概念。特别是，我们提供了一个示例，表明遍历状态的超产品通常不是遍历的。还研究了局部凸空间的紧凸子集上Radon测度的超积。众所周知，对于\（C ^ {*}-\）代数的状态集的极点的研究本身就是一个非常有趣的问题。考虑到\（C ^ {*} \）-代数的超积以及这些代数上的状态，我们得到了非常重要的结果。