Properly proximal groups and their von Neumann algebras
Properly proximal groups and their von Neumann algebras
复制标题
DOI：
10.24033/asens.2462
复制
发表时间：
2018-09
期刊：
影响因子：
--
通讯作者：
R. Boutonnet;A. Ioana;J. Peterson
中科院分区：
文献类型：
--
作者：
R. Boutonnet;A. Ioana;J. Peterson
我们引入了一类广泛的可数群，称为适当的近端，其中包含所有非顺从的双确切群，所有非初等收敛群，和所有格在非紧半单李群，但不包括所有内部顺从群。我们证明了这类群的自由遍历概率测度保持作用所产生的交叉积II$_1$因子至多有一个弱紧Cartan子代数，直到酉共轭。作为应用，我们得到了SL_d（\mathbb Z）$对d \geq 3$的紧作用的第一个W^*$-强刚性结果.
We introduce a wide class of countable groups, called properly proximal, which contains all non-amenable bi-exact groups, all non-elementary convergence groups, and all lattices in non-compact semi-simple Lie groups, but excludes all inner amenable groups. We show that crossed product II$_1$ factors arising from free ergodic probability measure preserving actions of groups in this class have at most one weakly compact Cartan subalgebra, up to unitary conjugacy. As an application, we obtain the first $W^*$-strong rigidity results for compact actions of $SL_d(\mathbb Z)$ for $d \geq 3$.