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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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W$^{*}$–superrigidity for Bernoulli actions of property (T) groups
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by Adrian Ioana PDF
J. Amer. Math. Soc. 24 (2011), 1175-1226 Request permission

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 $M$.

References
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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
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2813341