## Acta Mathematica

### Unique Cartan decomposition for II1 factors arising from arbitrary actions of free groups

#### Abstract

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 II1 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 II1 factors arising from actions of free groups with different number of generators are never isomorphic. We actually prove unique Cartan decomposition results for II1 factors arising from arbitrary actions of a much larger family of groups, including all free products of amenable groups and their direct products.

#### Note

S.P. was supported in part by NSF Grant DMS-1101718. S.V. was supported by ERC Starting Grant VNALG-200749, Research Programme G.0639.11 of the Research Foundation – Flanders (FWO) and KU Leuven BOF research grant OT/08/032.

#### Article information

Source
Acta Math., Volume 212, Number 1 (2014), 141-198.

Dates
First available in Project Euclid: 30 January 2017

https://projecteuclid.org/euclid.acta/1485801727

Digital Object Identifier
doi:10.1007/s11511-014-0110-9

Mathematical Reviews number (MathSciNet)
MR3179609

Zentralblatt MATH identifier
1307.46047

Rights

#### Citation

Popa, Sorin; Vaes, Stefaan. Unique Cartan decomposition for II 1 factors arising from arbitrary actions of free groups. Acta Math. 212 (2014), no. 1, 141--198. doi:10.1007/s11511-014-0110-9. https://projecteuclid.org/euclid.acta/1485801727

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