Unique Cartan decomposition for II_{1}factors arising from arbitrary actions of free groups

Sorin Popa Mathematics Department, University of California, Los Angeles Stefaan Vaes Department of Mathematics, KU Leuven

Functional Analysis Group Theory and Lie Theory Spectral Theory and Operator Algebra mathscidoc:1701.12005

Acta Mathematica, 212, (1), 141-198, 2012.6
We prove that for any free ergodic probability measure-preserving action $${\mathbb{F}_n \curvearrowright (X, \mu)}$$ of a free group on$n$generators $${\mathbb{F}_n, 2\leq n \leq \infty}$$ , the associated group measure space II_{1}factor $${L^\infty (X)\rtimes \mathbb{F}_n}$$ has$L$^{∞}($X$) as its unique Cartan subalgebra, up to unitary conjugacy. We deduce that group measure space II_{1}factors arising from actions of free groups with different number of generators are never isomorphic. We actually prove unique Cartan decomposition results for II_{1}factors arising from arbitrary actions of a much larger family of groups, including all free products of amenable groups and their direct products.
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@inproceedings{sorin2012unique,
  title={Unique Cartan decomposition for II_{1}factors arising from arbitrary actions of free groups},
  author={Sorin Popa, and Stefaan Vaes},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203402567310786},
  booktitle={Acta Mathematica},
  volume={212},
  number={1},
  pages={141-198},
  year={2012},
}
Sorin Popa, and Stefaan Vaes. Unique Cartan decomposition for II_{1}factors arising from arbitrary actions of free groups. 2012. Vol. 212. In Acta Mathematica. pp.141-198. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203402567310786.
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