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

HTML articles powered by AMS MathViewer

- by Adrian Ioana
- J. Amer. Math. Soc.
**24**(2011), 1175-1226 - DOI: https://doi.org/10.1090/S0894-0347-2011-00706-6
- Published electronically: June 8, 2011
- PDF | 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

- Bachir Bekka, Pierre de la Harpe, and Alain Valette,
*Kazhdan’s property (T)*, New Mathematical Monographs, vol. 11, Cambridge University Press, Cambridge, 2008. MR**2415834**, DOI 10.1017/CBO9780511542749 - Nathanial P. Brown and Narutaka Ozawa,
*$C^*$-algebras and finite-dimensional approximations*, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008. MR**2391387**, DOI 10.1090/gsm/088 - M. Burger,
*Kazhdan constants for $\textrm {SL}(3,\textbf {Z})$*, J. Reine Angew. Math.**413**(1991), 36–67. MR**1089795**, DOI 10.1515/crll.1991.413.36 - Alain Connes,
*Sur la classification des facteurs de type $\textrm {II}$*, C. R. Acad. Sci. Paris Sér. A-B**281**(1975), no. 1, Aii, A13–A15 (French, with English summary). MR**377534** - A. Connes:
*Correspondences*, handwritten notes, 1980. - A. Connes,
*A factor of type $\textrm {II}_{1}$ with countable fundamental group*, J. Operator Theory**4**(1980), no. 1, 151–153. MR**587372** - A. Connes, J. Feldman, and B. Weiss,
*An amenable equivalence relation is generated by a single transformation*, Ergodic Theory Dynam. Systems**1**(1981), no. 4, 431–450 (1982). MR**662736**, DOI 10.1017/s014338570000136x - Ionut Chifan and Adrian Ioana,
*Ergodic subequivalence relations induced by a Bernoulli action*, Geom. Funct. Anal.**20**(2010), no. 1, 53–67. MR**2647134**, DOI 10.1007/s00039-010-0058-7 - A. Connes and V. Jones,
*A $\textrm {II}_{1}$ factor with two nonconjugate Cartan subalgebras*, Bull. Amer. Math. Soc. (N.S.)**6**(1982), no. 2, 211–212. MR**640947**, DOI 10.1090/S0273-0979-1982-14981-3 - A. Connes and V. Jones,
*Property $T$ for von Neumann algebras*, Bull. London Math. Soc.**17**(1985), no. 1, 57–62. MR**766450**, DOI 10.1112/blms/17.1.57 - Alex Furman,
*Orbit equivalence rigidity*, Ann. of Math. (2)**150**(1999), no. 3, 1083–1108. MR**1740985**, DOI 10.2307/121063 - A. Furman:
*A survey of Measured Group Theory,*Geometry, Rigidity, and Group Actions, 296–374, The University of Chicago Press, Chicago and London, 2011, available at arXiv:0901.0678. - Jacob Feldman and Calvin C. Moore,
*Ergodic equivalence relations, cohomology, and von Neumann algebras. II*, Trans. Amer. Math. Soc.**234**(1977), no. 2, 325–359. MR**578730**, DOI 10.1090/S0002-9947-1977-0578730-2 - Adrian Ioana,
*Rigidity results for wreath product $\textrm {II}_1$ factors*, J. Funct. Anal.**252**(2007), no. 2, 763–791. MR**2360936**, DOI 10.1016/j.jfa.2007.04.005 - A. Ioana:
*Cocycle superrigidity for profinite actions of property (T) groups*, Duke Math. J. Volume**157**, Number 2 (2011), 337–367. - Adrian Ioana,
*Non-orbit equivalent actions of $\Bbb F_n$*, Ann. Sci. Éc. Norm. Supér. (4)**42**(2009), no. 4, 675–696 (English, with English and French summaries). MR**2568879**, DOI 10.24033/asens.2106 - Adrian Ioana, Jesse Peterson, and Sorin Popa,
*Amalgamated free products of weakly rigid factors and calculation of their symmetry groups*, Acta Math.**200**(2008), no. 1, 85–153. MR**2386109**, DOI 10.1007/s11511-008-0024-5 - V. F. R. Jones,
*A $\textrm {II}_{1}$ factor anti-isomorphic to itself but without involutory antiautomorphisms*, Math. Scand.**46**(1980), no. 1, 103–117. MR**585235**, DOI 10.7146/math.scand.a-11855 - Richard V. Kadison and John R. Ringrose,
*Fundamentals of the theory of operator algebras. Vol. II*, Pure and Applied Mathematics, vol. 100, Academic Press, Inc., Orlando, FL, 1986. Advanced theory. MR**859186**, DOI 10.1016/S0079-8169(08)60611-X - D. A. Každan,
*On the connection of the dual space of a group with the structure of its closed subgroups*, Funkcional. Anal. i Priložen.**1**(1967), 71–74 (Russian). MR**0209390** - Yoshikata Kida,
*Measure equivalence rigidity of the mapping class group*, Ann. of Math. (2)**171**(2010), no. 3, 1851–1901. MR**2680399**, DOI 10.4007/annals.2010.171.1851 - Y. Kida:
*Rigidity of amalgamated free products in measure equivalence theory*, to appear in J. Topology, preprint arXiv:0902.2888. - G. A. Margulis,
*Finitely-additive invariant measures on Euclidean spaces*, Ergodic Theory Dynam. Systems**2**(1982), no. 3-4, 383–396 (1983). MR**721730**, DOI 10.1017/S014338570000167X - F. J. Murray and J. Von Neumann,
*On rings of operators*, Ann. of Math. (2)**37**(1936), no. 1, 116–229. MR**1503275**, DOI 10.2307/1968693 - F. J. Murray and J. von Neumann,
*On rings of operators. IV*, Ann. of Math. (2)**44**(1943), 716–808. MR**9096**, DOI 10.2307/1969107 - Donald S. Ornstein and Benjamin Weiss,
*Ergodic theory of amenable group actions. I. The Rohlin lemma*, Bull. Amer. Math. Soc. (N.S.)**2**(1980), no. 1, 161–164. MR**551753**, DOI 10.1090/S0273-0979-1980-14702-3 - Narutaka Ozawa,
*Solid von Neumann algebras*, Acta Math.**192**(2004), no. 1, 111–117. MR**2079600**, DOI 10.1007/BF02441087 - Narutaka Ozawa and Sorin Popa,
*On a class of $\textrm {II}_1$ factors with at most one Cartan subalgebra*, Ann. of Math. (2)**172**(2010), no. 1, 713–749. MR**2680430**, DOI 10.4007/annals.2010.172.713 - N. Ozawa:
*Examples of groups which are not weakly amenable*, preprint arXiv:1012.0613. - Jesse Peterson,
*$L^2$-rigidity in von Neumann algebras*, Invent. Math.**175**(2009), no. 2, 417–433. MR**2470111**, DOI 10.1007/s00222-008-0154-6 - J. Peterson:
*Examples of group actions which are virtually W*-superrigid*, preprint arXiv:1002.1745. - S. Popa:
*Correspondences*, INCREST preprint 1986, unpublished. - Sorin Popa,
*Strong rigidity of $\rm II_1$ factors arising from malleable actions of $w$-rigid groups. I*, Invent. Math.**165**(2006), no. 2, 369–408. MR**2231961**, DOI 10.1007/s00222-006-0501-4 - Sorin Popa,
*Strong rigidity of $\rm II_1$ factors arising from malleable actions of $w$-rigid groups. II*, Invent. Math.**165**(2006), no. 2, 409–451. MR**2231962**, DOI 10.1007/s00222-006-0502-3 - Sorin Popa,
*On a class of type $\textrm {II}_1$ factors with Betti numbers invariants*, Ann. of Math. (2)**163**(2006), no. 3, 809–899. MR**2215135**, DOI 10.4007/annals.2006.163.809 - Sorin Popa,
*Some rigidity results for non-commutative Bernoulli shifts*, J. Funct. Anal.**230**(2006), no. 2, 273–328. MR**2186215**, DOI 10.1016/j.jfa.2005.06.017 - Sorin Popa,
*Cocycle and orbit equivalence superrigidity for malleable actions of $w$-rigid groups*, Invent. Math.**170**(2007), no. 2, 243–295. MR**2342637**, DOI 10.1007/s00222-007-0063-0 - Sorin Popa,
*Deformation and rigidity for group actions and von Neumann algebras*, International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Zürich, 2007, pp. 445–477. MR**2334200**, DOI 10.4171/022-1/18 - Sorin Popa,
*On the superrigidity of malleable actions with spectral gap*, J. Amer. Math. Soc.**21**(2008), no. 4, 981–1000. MR**2425177**, DOI 10.1090/S0894-0347-07-00578-4 - Sorin Popa and Stefaan Vaes,
*Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups*, Adv. Math.**217**(2008), no. 2, 833–872. MR**2370283**, DOI 10.1016/j.aim.2007.09.006 - Sorin Popa and Stefaan Vaes,
*On the fundamental group of $\textrm {II}_1$ factors and equivalence relations arising from group actions*, Quanta of maths, Clay Math. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 2010, pp. 519–541. MR**2732063**, DOI 10.1007/s00222-010-0268-5 - Sorin Popa and Stefaan Vaes,
*Group measure space decomposition of $\textrm {II}_1$ factors and $W^\ast$-superrigidity*, Invent. Math.**182**(2010), no. 2, 371–417. MR**2729271**, DOI 10.1007/s00222-010-0268-5 - I. M. Singer,
*Automorphisms of finite factors*, Amer. J. Math.**77**(1955), 117–133. MR**66567**, DOI 10.2307/2372424 - Stefaan Vaes,
*Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa)*, Astérisque**311**(2007), Exp. No. 961, viii, 237–294. Séminaire Bourbaki. Vol. 2005/2006. MR**2359046** - Stefaan Vaes,
*Explicit computations of all finite index bimodules for a family of $\textrm {II}_1$ factors*, Ann. Sci. Éc. Norm. Supér. (4)**41**(2008), no. 5, 743–788 (English, with English and French summaries). MR**2504433**, DOI 10.24033/asens.2081 - Stefaan Vaes,
*Factors of type II$_1$ without non-trivial finite index subfactors*, Trans. Amer. Math. Soc.**361**(2009), no. 5, 2587–2606. MR**2471930**, DOI 10.1090/S0002-9947-08-04585-6

## Bibliographic 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