Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T22:10:56.076Z Has data issue: false hasContentIssue false
  • English
  • Français

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])
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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