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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

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.