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On cocycle superrigidity for Gaussian actions

Published online by Cambridge University Press:  10 June 2011

JESSE PETERSON  and
THOMAS SINCLAIR
JESSE PETERSON
Affiliation:
Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, USA (email: [email protected], [email protected])
THOMAS SINCLAIR
Affiliation:
Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, USA (email: [email protected], [email protected])
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Abstract

We present a general setting to investigate 𝒰fin-cocycle superrigidity for Gaussian actions in terms of closable derivations on von Neumann algebras. In this setting we give new proofs to some 𝒰fin-cocycle superrigidity results of S. Popa and we produce new examples of this phenomenon. We also use a result of K. R. Parthasarathy and K. Schmidt to give a necessary cohomological condition on a group representation in order for the resulting Gaussian action to be 𝒰fin-cocycle superrigid.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 1 , February 2012 , pp. 249 - 272
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Bekka, B.. Amenable unitary representations of locally compact groups. Invent. Math. 100(2) (1990), 383–401.CrossRefGoogle Scholar
[2]Bekka, B., de la Harpe, P. and Valette, A.. Kazhdan’s Property (T) (New Mathematical Monographs, 11). Cambridge University Press, Cambridge, 2008.CrossRefGoogle Scholar
[3]Bekka, M. E. B. and Valette, A.. Group cohomology, harmonic functions and the first L2-Betti number. Potential Anal. 6(4) (1997), 313–326.CrossRefGoogle Scholar
[4]Bridson, M. R., Tweedale, M. and Wilton, H.. Limit groups, positive-genus towers and measure-equivalence. Ergod. Th. & Dynam. Sys. 27(3) (2007), 703–712.CrossRefGoogle Scholar
[5]Cheeger, J. and Gromov, M.. L 2-cohomology and group cohomology. Topology 25(2) (1986), 189–215.CrossRefGoogle Scholar
[6]Chifan, I. and Ioana, A.. Ergodic subequivalence relations induced by a Bernoulli action. Geom. Funct. Anal. 20(1) (2010), 53–57.CrossRefGoogle Scholar
[7]Chifan, I. and Ioana, A.. On relative property (T) and Haagerup’s property. Trans. Amer. Math. Soc. to appear. Preprint, 2009, arXiv:0906.5363.Google Scholar
[8]Connes, A.. Noncommutative Geometry. Academic Press, New York, 1994.Google Scholar
[9]Connes, A., Feldman, J. and Weiss, B.. An amenable equivalence relation is generated by a single transformation. Ergod. Th. & Dynam. Sys. 1(4) (1981), 431–450.CrossRefGoogle Scholar
[10]Connes, A. and Weiss, B.. Property (T) and asymptotically invariant sequences. Israel J. Math. 37(3) (1980), 209–210.CrossRefGoogle Scholar
[11]Cowling, M. and Zimmer, R. J.. Actions of lattices in Sp(1,n). Ergod. Th. & Dynam. Sys. 9(2) (1989), 221–237.CrossRefGoogle Scholar
[12]Davies, E. B. and Lindsay, J. M.. Non-commutative symmetric Markov semigroups. Math. Z. 210 (1992), 379–411.CrossRefGoogle Scholar
[13]Dye, H. A.. On groups of measure preserving transformations. I. Amer. J. Math. 81 (1959), 119–159.CrossRefGoogle Scholar
[14]Dye, H. A.. On groups of measure preserving transformations. II. Amer. J. Math. 85 (1963), 551–576.CrossRefGoogle Scholar
[15]Furman, A.. On Popa’s cocycle superrigidity theorem. Int. Math. Res. Not. IMRN 19 (2007), 46.Google Scholar
[16]Furman, A.. Orbit equivalence rigidity. Ann. of Math. (2) 150(3) (1999), 1083–1108.CrossRefGoogle Scholar
[17]Gaboriau, D.. Examples of groups that are measure equivalent to the free group. Ergod. Th. & Dynam. Sys. 25(6) (2005), 1809–1827.CrossRefGoogle Scholar
[18]Guichardet, A.. Symmetric Hilbert Spaces and Related Topics (Lecture Notes in Mathematics, 261). Springer, Berlin, 1972.CrossRefGoogle Scholar
[19]Ioana, A.. Rigidity results for wreath product II1 factors. J. Funct. Anal. 252(2) (2007), 763–791.CrossRefGoogle Scholar
[20]Ioana, A., Peterson, J. and Popa, S.. Amalgamated free products of weakly rigid factors and calculation of their symmetry groups. Acta Math. 200(1) (2008), 85–153.CrossRefGoogle Scholar
[21]Murray, F. J. and von Neumann, J.. On rings of operators. Ann. of Math. (2) 37(1) (1936), 116–229.CrossRefGoogle Scholar
[22]Murray, F. J. and von Neumann, J.. On rings of operators. IV. Ann. of Math. (2) 44 (1943), 716–808.CrossRefGoogle Scholar
[23]Ornstein, D. S. and Weiss, B.. Ergodic theory of amenable group actions. I. The Rohlin lemma. Bull. Amer. Math. Soc. (N.S.) 2(1) (1980), 161–164.CrossRefGoogle Scholar
[24]Ozawa, N.. Solid von Neumann algebras. Acta Math. 192(1) (2004), 111–117.CrossRefGoogle Scholar
[25]Ozawa, N. and Popa, S.. On a class of II1 factors with at most one Cartan subalgebra I. Ann. of Math. (2) 172(1) (2010), 713–749.CrossRefGoogle Scholar
[26]Ozawa, N. and Popa, S.. On a class of II1 factors with at most one Cartan subalgebra II. Amer. J. Math. 132(3) (2010), 841–866.CrossRefGoogle Scholar
[27]Parthasarathy, K. R. and Schmidt, K.. Infinitely divisible projective representations, cocycles and Levy–Khinchine formula on locally compact groups. Preprint, 1970, unpublished.Google Scholar
[28]Peterson, J.. A 1-cohomology characterization of property (T) in von Neumann algebras. Pacific J. Math. 243(1) (2009), 181–199.CrossRefGoogle Scholar
[29]Peterson, J.. L 2-rigidity in von Neumann algebras. Invent. Math. 175 (2009), 417–433.CrossRefGoogle Scholar
[30]Peterson, J.. Examples of group actions which are virtually W *E-superrigid. Preprint, 2009, arXiv:1005.0810.Google Scholar
[31]Peterson, J. and Thom, A.. Group cocycles and the ring of affiliated operators. Preprint, 2007, arXiv:0708.4327. Invent. Math. to appear.Google Scholar
[32]Popa, S.. Correspondences. INCREST preprint, 1986, unpublished.Google Scholar
[33]Popa, S.. Some rigidity results for non-commutative Bernoulli shifts. J. Funct. Anal. 230(2) (2006), 273–328.CrossRefGoogle Scholar
[34]Popa, S.. On a class of type II1 factors with Betti numbers invariants. Ann. of Math. (2) 163(3) (2006), 809–899.CrossRefGoogle Scholar
[35]Popa, S.. Some computations of 1-cohomology groups and construction of non-orbit-equivalent actions. J. Inst. Math. Jussieu 5(2) (2006), 309–332.CrossRefGoogle Scholar
[36]Popa, S.. Strong rigidity of II1 factors arising from malleable actions of w-rigid groups. I. Invent. Math. 165(2) (2006), 369–408.CrossRefGoogle Scholar
[37]Popa, S.. Strong rigidity of II1 factors arising from malleable actions of w-rigid groups. II. Invent. Math. 165(2) (2006), 409–451.CrossRefGoogle Scholar
[38]Popa, S.. Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups. Invent. Math. 170(2) (2007), 243–295.CrossRefGoogle Scholar
[39]Popa, S.. Deformation and rigidity for group actions and von Neumann algebras. International Congress of Mathematicians, Vol. I. European Mathematical Society, Zürich, 2007, pp. 445–477.Google Scholar
[40]Popa, S.. On the superrigidity of malleable actions with spectral gap. J. Amer. Math. Soc. 21(4) (2008), 981–1000.CrossRefGoogle Scholar
[41]Popa, S. and Sasyk, R.. On the cohomology of Bernoulli actions. Ergod. Th. & Dynam. Sys. 27(1) (2007), 241–251.CrossRefGoogle Scholar
[42]Popa, S. and Vaes, S.. Cocycle and orbit superrigidity for lattices in SL (n,R) acting on homogeneous spaces. Preprint, 2008, arXiv:0810.3630, to appear in Proc. Conf. In Honor of Bob Zimmer’s 60th Birthday, Geometry, Rigidity and Group Actions, University of Chicago, 6–9 September 2007.Google Scholar
[43]Popa, S. and Vaes, S.. Group measure space decomposition of II1 factors and W *-superrigidity. Invent. Math. 182(2) (2010), 371–417.CrossRefGoogle Scholar
[44]Sako, H.. The class S as an ME invariant. Int. Math. Res. Not. IMRN 15 (2009), 2749–2759.Google Scholar
[45]Sauvageot, J.-L.. Tangent bimodules and locality for dissipative operators on C *-algebras. Quantum Probability and Applications, IV (Lecture Notes in Mathematics, 1396). Springer, Berlin, 1989,pp. 322–338.CrossRefGoogle Scholar
[46]Sauvageot, J.-L.. Quantum Dirichlet forms, differential calculus and semigroups. Quantum Probability and Applications, V (Lecture Notes in Mathematics, 1442). Springer, Berlin, 1990, pp. 334–346.CrossRefGoogle Scholar
[47]Schmidt, K.. Amenability, Kazhdan’s property (T), strong ergodicity and invariant means for ergodic groups actions. Ergod. Th. & Dynam. Sys. 1 (1981), 223–236.CrossRefGoogle Scholar
[48]Schmidt, K.. From infinitely divisible representations to cohomological rigidity. Analysis, Geometry and Probability (Texts and Readings in Mathematics, 10). Hindustan Book Agency, Delhi, 1996, pp. 173–197.CrossRefGoogle Scholar
[49]Shalom, Y.. Measurable group theory. European Congress of Mathematics (Stockholm, 2004). Ed. Laptev, A.. European Mathematical Society, Zürich, 2005, pp. 391–423.Google Scholar
[50]Singer, I. M.. Automorphisms of finite factors. Amer. J. Math. 77 (1955), 117–133.CrossRefGoogle Scholar
[51]Vaes, S.. Rigidity results for Bernoulli shifts and their von Neumann algebras (after Sorin Popa). Astèrisque 311 (2007), 237–294 [Sém. Bourbaki, Exp. No. 961].Google Scholar
[52]Voiculescu, D.. Symmetries of some reduced free product C *-algebras. Operator Algebras and their Connections with Topology and Ergodic Theory (Buşteni, 1983) (Lecture Notes in Mathematics, 1132). Springer, Berlin, 1985, pp. 556–588.Google Scholar
[53]Voiculescu, D.. The analogues of entropy and of Fisher’s information measure in free probability theory V: noncommutative Hilbert transforms. Invent. Math. 132(1) (1998), 189–227.CrossRefGoogle Scholar
[54]Voiculescu, D.. The analogues of entropy and of Fisher’s information measure in free probability theory. VI. Liberation and mutual free information. Adv. Math. 146(2) (1999), 101–166.CrossRefGoogle Scholar
[55]Zimmer, R. J.. Ergodic Theory and Semisimple Groups (Monographs in Mathematics, 81). Birkhäuser, Basel, 1984, x+209 pp.CrossRefGoogle Scholar