Journal de Mathématiques Pures et Appliquées ( IF 2.3 ) Pub Date : 2020-02-19 , DOI: 10.1016/j.matpur.2020.02.009 Sorin Popa , Dimitri Shlyakhtenko , Stefaan Vaes
We prove that the regular von Neumann subalgebras B of the hyperfinite II1 factor R satisfying the condition are completely classified (up to conjugacy by an automorphism of R) by the associated discrete measured groupoid . We obtain a similar classification result for triple inclusions , where A is a Cartan subalgebra in R and the intermediate von Neumann algebra B is regular in R. A key step in proving these results is to show the vanishing cohomology for the associated cocycle actions of on B. We in fact prove two very general vanishing cohomology results for free cocycle actions of amenable discrete measured groupoids on arbitrary tracial von Neumann algebras B, resp. Cartan inclusions . Our work provides a unified approach and generalizations to many known vanishing cohomology and classification results [6], [25], [35], [3], [10], [29], etc.
中文翻译:
超有限II 1因子的正则子代数的分类
我们证明超有限II 1因子R的正规von Neumann子代数B满足条件通过相关的离散测得的类群被完全分类(通过R的同构性达到共轭)。对于三重包含,我们获得了相似的分类结果,其中A是R中的Cartan子代数,中间von Neumann代数B在R中是规则的。证明这些结果的关键步骤是显示相关联的循环运动消失的同调性 的 在B上。实际上,我们证明了两个免费乘车行为的两个非常普遍的同调结果 合适的离散测得的类群 关于任意族的von Neumann代数B,分别。迦坦包裹体。我们的工作为许多已知的消失的同调性和分类结果[6],[25],[35],[3],[10],[29]等提供了统一的方法和概括。