论文标题
正确地近端von Neumann代数
Properly Proximal von Neumann Algebras
论文作者
论文摘要
我们介绍了有限的von Neumann代数的适当近端概念，该代数自然扩展了群体适当近端的概念。除了适当近端组的von Neumann代数外，我们还提供了许多其他示例，包括在免费产品，交叉产品和紧凑型量子组的设置中的示例。使用这个概念，我们通过表明一个不可统的内部木木组的von Neumann代数可以回答POPA的问题。我们还引入了适当的近端概念，以衡量概率保存动作，这为轨道等效关系提供了不变的概念。这提供了一种新的方法来建立强大的性质类型属性，我们将其用于高斯动作的设置，以扩大奇法和Ioana首先建立的固体千古化结果，后来由Boutonnet概括。开发的技术还使我们能够回答Anantharaman-Delaroche在1995年留下的问题，它通过显示Haagerup属性与II $ _1 $因子的紧凑型近似属性之间的等价性。
We introduce the notion of proper proximality for finite von Neumann algebras, which naturally extends the notion of proper proximality for groups. Apart from the group von Neumann algebras of properly proximal groups, we provide a number of additional examples, including examples in the settings of free products, crossed products, and compact quantum groups. Using this notion, we answer a question of Popa by showing that the group von Neumann algebra of a nonamenable inner amenable group cannot embed into a free group factor. We also introduce a notion of proper proximality for probability measure preserving actions, which gives an invariant for the orbit equivalence relation. This gives a new approach for establishing strong ergodicity type properties, and we use this in the setting of Gaussian actions to expand on solid ergodicity results first established by Chifan and Ioana, and later generalized by Boutonnet. The techniques developed also allow us to answer a problem left open by Anantharaman-Delaroche in 1995, by showing the equivalence between the Haagerup property and the compact approximation property for II$_1$ factors.
