# MathSciDoc: An Archive for Mathematician ∫

#### Functional AnalysisGroup Theory and Lie TheorySpectral Theory and Operator Algebramathscidoc: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.
@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.