Abstract
We provide a unified and self-contained treatment of several of the recent uniqueness theorems for the group measure space decomposition of a II1 factor. We single out a large class of groups Γ, characterized by a one-cohomology property, and prove that for every free ergodic probability measure preserving action of Γ the associated II1 factor has a unique group measure space Cartan subalgebra up to unitary conjugacy. Our methods follow closely a recent article of Chifan–Peterson, but we replace the usage of Peterson’s unbounded derivations by Thomas Sinclair’s dilation into a malleable deformation by a one-parameter group of automorphisms.
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References
Brown, N.P., Ozawa, N.: C*-algebras and finite-dimensional approximations. Graduate Studies in Mathematics, vol. 88. American Mathematical Society, Providence (2008)
Chifan, I., Peterson, J.: Some unique group-measure space decomposition results. Preprint. arXiv:1010.5194 (2011)
Connes A., Feldman J., Weiss B.: An amenable equivalence relation is generated by a single transformation. Ergodic Theory Dynam. Syst. 1, 431–450 (1981)
de Cornulier Y., Tessera R., Valette A.: Isometric group actions on Banach spaces and representations vanishing at infinity. Transform. Groups 13, 125–147 (2008)
de Cornulier Y., Stalder Y., Valette A.: Proper actions of lamplighter groups associated with free groups. C.R. Acad. Sci. Paris Ser. I 346, 173–176 (2008)
Fima P., Vaes S.: HNN extensions and unique group measure space decomposition of II1 factors. Trans. Amer. Math. Soc. 364, 2601–2617 (2012)
Houdayer, C., Popa, S., Vaes, S.: A class of groups for which every action is W*-superrigid. Groups Geom. Dyn. arXiv:1010.5077 (2011, to appear)
Ioana, A., Popa, S., Vaes, S.: A class of superrigid group von Neumann algebras. Preprint. arXiv:1007.1412 (2010)
Kida Y.: Rigidity of amalgamated free products in measure equivalence theory. J. Topol. 4, 687–735 (2011)
Kida, Y.: Examples of amalgamated free products and coupling rigidity. Ergodic Theory Dynam. Syst. arXiv:1007.1529 (2011, to appear)
Monod N., Shalom Y.: Orbit equivalence rigidity and bounded cohomology. Ann. Math. 164, 825–878 (2006)
Ozawa N.: Solid von Neumann algebras. Acta Math. 192, 111–117 (2004)
Ozawa, N.: A Kurosh-type theorem for type II1 factors. Int. Math. Res. Not., Art. ID 97560 (2006)
Ozawa N., Popa S.: On a class of II1 factors with at most one Cartan subalgebra. Ann. Math. 172, 713–749 (2010)
Ozawa N., Popa S.: On a class of II1 factors with at most one Cartan subalgebra, II. Am. J. Math. 132, 841–866 (2010)
Peterson J.: L 2-rigidity in von Neumann algebras. Invent. Math. 175, 417–433 (2009)
Peterson, J.: Examples of group actions which are virtually W*-superrigid. Preprint. arXiv:1002.1745 (2011)
Popa S.: On a class of type II1 factors with Betti numbers invariants. Ann. Math. 163, 809–899 (2006)
Popa S.: Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, I. Invent. Math. 165, 369–408 (2006)
Popa S.: Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups. Invent. Math. 170, 243–295 (2007)
Popa, S.: Deformation and rigidity for group actions and von Neumann algebras. In: Proceedings of the International Congress of Mathematicians, Madrid (2006), vol. I, European Mathematical Society Publishing House, pp. 445–477 (2007)
Popa, S.: On Ozawa’s property for free group factors. Int. Math. Res. Not. Article ID rnm036 (2007)
Popa S.: On the superrigidity of malleable actions with spectral gap. J. Amer. Math. Soc. 21, 981–1000 (2008)
Popa S., Vaes S.: Group measure space decomposition of II1 factors and W*-superrigidity. Invent. Math. 182, 371–417 (2010)
Sinclair T.: Strong solidity of group factors from lattices in SO(n, 1) and SU(n, 1). J. Funct. Anal. 260, 3209–3221 (2011)
Vaes S.: Explicit computations of all finite index bimodules for a family of II1 factors. Ann. Sci. École Norm. Sup. 41, 743–788 (2008)
Vaes, S.: Rigidity for von Neumann algebras and their invariants. In: Proceedings of the International Congress of Mathematicians, Hyderabad (2010), vol. III, Hindustan Book Agency, pp. 1624–1650 (2010)
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Partially supported by ERC Starting Grant VNALG-200749, Research Programme G.0639.11 of the Research Foundation—Flanders (FWO) and K.U.Leuven BOF research grant OT/08/032.
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Vaes, S. One-cohomology and the uniqueness of the group measure space decomposition of a II1 factor. Math. Ann. 355, 661–696 (2013). https://doi.org/10.1007/s00208-012-0797-x
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DOI: https://doi.org/10.1007/s00208-012-0797-x