1-bounded entropy and regularity problems in von Neumann algebras
1-bounded entropy and regularity problems in von Neumann algebras
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冯·诺依曼代数中的 1 有界熵和正则性问题
DOI：
10.1093/imrn/rnw237
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Ben Hayes
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文献类型：
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作者：
Ben Hayes
研究迹vonNeumann代数包含的奇异子空间。奇异子空间是L^{2}（M）的一个标准N-N子双模，它包含Popa引入的拟正规化子、Fang-Gao-Smith引入的单侧拟正规化子和Galatan-Popa（在Ioana-Peterson-Popa和Popa的工作基础上）引入的wq-正规化子。然后，我们通过要求M中N的奇异子空间生成M，得到了一个弱的正则性概念（称为谱正则性）。通过抽象Voiculescu关于不存在Cartan子代数的原始证明，证明了L（\FF_{n}）不可能存在谱正则的扩散超有限子代数.我们的技术足够强大，可以通过超限归纳重复这个过程，并排除从扩散超有限代数开始并以L（\FF_{n}）结束的代数的谱正则包含链。利用这一点，我们证明了Galatan-Popa在研究II_{1}-因子的光滑上同调时所作的一些结论.我们的结果可以看作是对II_{1}-因子的“好”上同调理论存在可能性的一致性检验。最后，我们得到了q-变形自由群因子在Bogoliubov作用下的交叉积以及q-变形自由Araki-Woods代数的连续核的非同构结果.这扩展了Houdayer-Shlyakhtenko和Shlyakhtenko的工作。
We investigate the singular subspace of an inclusion of tracial von Neumann algebras. The singular subspace is a canonical N-N subbimodule of L^{2}(M) and it contains the quasinormalizer introduced by Popa, one-sided quasinormalizer introduced by Fang-Gao-Smith, and wq-normalizer introduced in Galatan-Popa (following upon work in Ioana-Peterson-Popa and Popa). We then obtain a weak notion of regularity (called spectral regularity) by demanding that the singular subspace of N in M generates M. By abstracting Voiculescu's original proof of absence of Cartan subalgebras, we show that there can be no diffuse, hyperfinite subalgebra of L(\FF_{n}) which is spectrally regular. Our techniques are robust enough to repeat this process by transfinite induction and rule out chains of spectrally regular inclusions of algebras starting from a diffuse, hyperfinite algebra and ending in L(\FF_{n}). We use this to prove some conjectures made by Galatan-Popa in their study of smooth cohomology of II_{1}-factors. Our results may be regarded as a consistency check for the possibility of existence of a "good" cohomology theory of II_{1}-factors. Lastly, we deduce nonisomorphism results for crossed products of q-deformed free group factors by Bogoliubov actions, as well as for the continuous core of q-deformed Free Araki-Woods algebras. This extends work of Houdayer-Shlyakhtenko as well as Shlyakhtenko.