论文标题
II $ _1 $因素的泊松边界
Poisson boundaries of II$_1$ factors
论文作者
论文摘要
我们引入了II $ _1 $因素的泊松边界，相对于提供痕迹的密度操作员。泊松边界是一个包含II $ _1 $因子的von Neumann代数，并且是Izumi介绍的Unital完全正面图的边界的一个特定示例。在研究将II $ _1 $因子纳入其边界中的，我们发展了许多概念，例如双重性和熵，这些概念可以看作是Furstenberg引入的泊松边界的自然类似物。我们使用开发的技术来回答POPA的问题，表明所有有限因素都满足MV-Property。我们还通过表明特性（t）因子导致熵缝隙来扩展NEVO的结果。
We introduce Poisson boundaries of II$_1$ factors with respect to density operators that give the traces. The Poisson boundary is a von Neumann algebra that contains the II$_1$ factor and is a particular example of the boundary of a unital completely positive map as introduced by Izumi. Studying the inclusion of the II$_1$ factor into its boundary we develop a number of notions, such as double ergodicity and entropy, that can be seen as natural analogues of results regarding the Poisson boundaries introduced by Furstenberg. We use the techniques developed to answer a problem of Popa by showing that all finite factors satisfy the MV-property. We also extend a result of Nevo by showing that property (T) factors give rise to an entropy gap.
