Bi-Exact Groups, Strongly Ergodic Actions and Group Measure Space Type III Factors with No Central Sequence
Bi-Exact Groups, Strongly Ergodic Actions and Group Measure Space Type III Factors with No Central Sequence
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DOI：
10.1007/s00220-016-2634-7
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发表时间：
2015-10
影响因子：
2.4
通讯作者：
Cyril Houdayer;Yusuke Isono
中科院分区：
文献类型：
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作者：
Cyril Houdayer;Yusuke Isono
我们研究了（可能是 III 型）交叉积冯诺依曼代数的渐近结构，该结构源自双精确离散群（例如自由群）对服从的冯诺依曼代数的任意作用。我们证明了具有正态期望的任何不可命名的冯诺依曼子代数的中心序列代数的谱间隙刚性结果。我们使用这个结果来表明，对于标准概率空间上任何双精确可数离散群的任何强遍历本质上自由的非奇异作用，相应的群测度空间因子没有非平凡的中心序列。使用 Boutonnet 等人的最新结果。 （简单李群和应用中的局部谱间隙，2015），我们为每个标准概率空间上的自由群构造一个类型强遍历本质上自由的非奇异作用，以便根据我们的主要结果，相应的群测度空间类型因子没有非平凡的中心序列。特别是，我们获得了第一个没有非平凡中心序列的群测度空间类型因子的例子。
We investigate the asymptotic structure of (possibly type III) crossed product von Neumann algebrasarising from arbitrary actionsof bi-exact discrete groups (e.g. free groups) on amenable von Neumann algebras. We prove a spectral gap rigidity result for the central sequence algebraof any nonamenable von Neumann subalgebra with normal expectation. We use this result to show that for any strongly ergodic essentially free nonsingular actionof any bi-exact countable discrete group on a standard probability space, the corresponding group measure space factorhas no nontrivial central sequence. Using recent results of Boutonnet et al. (Local spectral gap in simple Lie groups and applications, 2015), we construct, for every, a typestrongly ergodic essentially free nonsingular actionof the free groupon a standard probability space so that the corresponding group measure space typefactorhas no nontrivial central sequence by our main result. In particular, we obtain the first examples of group measure space typefactors with no nontrivial central sequence.