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Classification of regular subalgebras of the hyperfinite II1 factor
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 BR=Z(B) are completely classified (up to conjugacy by an automorphism of R) by the associated discrete measured groupoid G=GBR. We obtain a similar classification result for triple inclusions ABR, 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 (αBR,uBR) of G on B. We in fact prove two very general vanishing cohomology results for free cocycle actions (α,u) of amenable discrete measured groupoids G on arbitrary tracial von Neumann algebras B, resp. Cartan inclusions AB. 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[R=ž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)G=G[R。对于三重包含,我们获得了相似的分类结果一种[R,其中AR中的Cartan子代数,中间von Neumann代数BR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α[Rü[RGB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αü 合适的离散测得的类群 G关于任意族的von Neumann代数B,分别。迦坦包裹体一种。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。

更新日期:2020-02-19
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