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W$ ^{*}$-superrigidity for Bernoulli actions of property (T) groups


Author: Adrian Ioana
Journal: J. Amer. Math. Soc. 24 (2011), 1175-1226
MSC (2010): Primary 46L36; Secondary 28D15, 37A20
DOI: https://doi.org/10.1090/S0894-0347-2011-00706-6
Published electronically: June 8, 2011
MathSciNet review: 2813341
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Abstract: We consider group measure space II$ _{1}$ factors $ M=L^{\infty }(X)\rtimes \Gamma $ arising from Bernoulli actions of ICC property (T) groups $ \Gamma $ (more generally, of groups $ \Gamma $ containing an infinite normal subgroup with the relative property (T)) and prove a rigidity result for $ *$-homomorphisms $ \theta:M\rightarrow M\overline{\otimes }M$.

We deduce that the action $ \Gamma \curvearrowright X$ is W$ ^{*}$-superrigid, i.e. if $ \Lambda \curvearrowright Y$ is any free, ergodic, measure preserving action such that the factors $ M=L^{\infty }(X)\rtimes \Gamma $ and $ L^{\infty }(Y)\rtimes \Lambda $ are isomorphic, then the actions $ \Gamma \curvearrowright X$ and $ \Lambda \curvearrowright Y$ must be conjugate.

Moreover, we show that if $ p\in M\setminus \{1\}$ is a projection, then $ pMp$ does not admit a group measure space decomposition nor a group von Neumann algebra decomposition (the latter under the additional assumption that $ \Gamma $ is torsion free).

We also prove a rigidity result for $ *$-homomorphisms $ \theta :M\rightarrow M$, this time for $ \Gamma $ in a larger class of groups than above, now including products of non-amenable groups. For certain groups $ \Gamma $, e.g. $ \Gamma =\mathbb{F}_{2}\times \mathbb{F}_{2}$, we deduce that $ M$ does not embed into $ pMp$, for any projection $ p\in M\setminus \{1\}$, and obtain a description of the endomorphism semigroup of $ M$.


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Additional Information

Adrian Ioana
Affiliation: Department of Mathematics, UCLA, Los Angeles, California 91125 and IMAR, 21 Calea Grivitei Street, 010702 Bucharest, Romania
Email: adiioana@math.ucla.edu

DOI: https://doi.org/10.1090/S0894-0347-2011-00706-6
Received by editor(s): November 30, 2010
Received by editor(s) in revised form: April 20, 2011
Published electronically: June 8, 2011
Additional Notes: The author was supported by a Clay Research Fellowship
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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