We will introduce the concept of ergodicity of states with respect to some group of transformations on a von Neumann algebra and its properties are studied. A connection between the ergodic states on the von Neumann algebra and the representations of von Neumann algebras associated to them will be described. We also study the properties of ultraproducts of von Neumann algebras with ergodic states and corresponding representations. Here we use ultraproducts of von Neumann algebras by Groh (J. Operator Theory 11 (2), 395–404 1984 ) and Raynaud (J. Operator Theory 48 (1), 41–68 2002 ). In particular, we will show that the ultraproduct of irreducibles representations isn’t, generally speaking, irreducible.

中文翻译：

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冯诺依曼代数和超积的表示

我们将介绍关于冯诺依曼代数上的一组变换的状态遍历性的概念，并研究其性质。将描述 von Neumann 代数上的遍历态与与它们相关联的 von Neumann 代数的表示之间的联系。我们还研究了具有遍历状态和相应表示的冯诺依曼代数的超积的性质。在这里，我们使用 Groh (J. Operator Theory 11 (2), 395–404 1984 ) 和 Raynaud (J. Operator Theory 48 (1), 41–68 2002 ) 的冯诺依曼代数的超积。特别是，我们将证明不可约表示的超积一般来说不是不可约的。