AI 中文总结
本文解决了超有限 II$_1$ 因子上流的分类问题，并利用新发现的共振现象证明了所有 III$_1$ 型因子的 Connes 双中心化子猜想。
AI 中文摘要
我们解决了 von Neumann 代数中两个长期悬而未决的问题。首先，我们证明超有限 II$_1$ 因子上具有完全 Connes 谱的每个外部流都具有 Rokhlin 性质。根据 Masuda 和 Tomatsu 的工作，这样的流在余循环共轭意义下是唯一的。这解决了 Takesaki 关于超有限 II$_1$ 型因子上流的分类问题。借鉴 III 型理论，我们为局部紧群的保迹作用发展了双中心化子机制。在 amenable 情形下，我们将双中心化子猜想与 Rokhlin 性质联系起来。对于交换群，我们证明了 Connes-Størmer 传递定理的一个类比，并推广了 Connes-Takesaki 相对交换子定理。我们揭示了一种新的共振现象，这使我们能够解决 R 作用的双中心化子猜想。然后我们回到 III 型世界，利用这种新的共振现象来解决所有 III$_1$ 型因子的 Connes 双中心化子猜想。
英文摘要
We solve two long-standing open problems in von Neumann algebras. First, we classify all flows with full Connes spectrum on the hyperfinite $\mathrm{II}_1$ factor up to cocycle conjugacy. Every such flow is cocycle conjugate to an irrational rotation flow on a noncommutative torus. In particular, there is a unique outer flow with full Connes spectrum up to cocycle conjugacy. This settles Takesaki's classification problem for flows on the hyperfinite $\mathrm{II}_1$ factor. To prove this result, we draw on type $\mathrm{III}$ theory. Notably, we develop bicentralizer machinery for trace-preserving actions of locally compact groups. In the amenable case, we relate the bicentralizer conjecture to the Rokhlin property. For abelian groups, we prove an analog of the Connes--Størmer transitivity theorem and we generalize the Connes--Takesaki relative commutant theorem. A new resonance phenomenon is revealed which allows us to solve the bicentralizer conjecture for actions of $\mathbb{R}$. We then go back to the type $\mathrm{III}$ world and use this new resonance phenomenon to solve Connes' bicentralizer conjecture for all type $\mathrm{III}_1$ factors.
Comments150 pages. The first version did not really settle the classification of flows on the hyperfinite II_1 factor because we did not classify flows with full Connes spectrum that are not outer. We complete this case in this second version. The presentation of various section was also improved
