Abstract
Motivated by Popa’s seminal work Popa (Invent Math 165:409-45, 2006), in this paper, we provide a fairly large class of examples of group actions \(\Gamma \curvearrowright X\) satisfying the extended Neshveyev–Størmer rigidity phenomenon Neshveyev and Størmer (J Funct Anal 195(2):239-261, 2002): whenever \(\Lambda \curvearrowright Y\) is a free ergodic pmp action and there is a \(*\)-isomorphism \(\Theta :L^\infty (X)\rtimes \Gamma {\rightarrow }L^\infty (Y)\rtimes \Lambda \) such that \(\Theta (L(\Gamma ))=L(\Lambda )\) then the actions \(\Gamma \curvearrowright X\) and \(\Lambda \curvearrowright Y\) are conjugate (in a way compatible with \(\Theta \)). We also obtain a complete description of the intermediate subalgebras of all (possibly non-free) compact extensions of group actions in the same spirit as the recent results of Suzuki (Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems, To appear in Comm Math Phy. Ar**v Preprint: ar**v:1805.02077, 2020). This yields new consequences to the study of rigidity for crossed product von Neumann algebras and to the classification of subfactors of finite Jones index.
Similar content being viewed by others
Notes
For every diffuse \(A\subseteq L(\Gamma )\) the relative commutant \(A'\cap L(\Gamma )\) is amenable
For every diffuse amenable \(A\subseteq L(\Gamma )\) the normalizer \(\mathcal N_{L(\Gamma )}(A)''\) is amenable
References
Alekseev, V., Brugger, R.: A rigidity result for normalized subfactors, Preprint ar**v:1903.04895
Bratteli, O., Jørgensen, P.E.T., Kishimoto, A., Werner, R.F.: Pure states on \(O_d\). J. Operator Theory 43, 97–143 (2000)
Bhattacharjee, M.: Constructing finitely presented infinite nearly simple groups. Comm. Algebra 22, 4561–4589 (1994)
Boutonnet, R., Carderi, A.: Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups. Geom. Funct. Anal. 25, 1688–1705 (2015)
Boutonnet, R., Ioana, A., Peterson, J.: Properly proximal groups and their von Neumann algebras, Preprint ar**v:1809.01881
Boutonnet, R.: \(W^*\)-superrigidity of mixing Gaussian actions of rigid groups. Adv. Math. 244, 69–90 (2013)
Burger, M., Mozes, S.: Lattices in products of trees. Inst. Hautes Ètudes Sci. Pub. Sér. I Math. 92, 151–194 (2001)
Camm, R.: Simple free products. J. Lond. Math. Soc. 28, 66–76 (1953)
Chifan, I., Das, S.: A remark on the ultrapower algebra of the hyperfinite factor. Proc. Amer. Math. Soc. 146, 5289–5294 (2018)
Chifan, I., de Santiago, R., Sucpikarnon, W.: Tensor product decompositions of \(\text{II}_1\) factors arising from extensions of amalgamated free product groups , Comm. Math. Phy., 364(218) issue 3, 1163–1194
Chifan, I., Kida, Y.: OE and \(W^{\ast }\) superrigidity results for actions by surface braid groups. Proc. Lond. Math. Soc. 111(6), 1431–1470 (2015)
Chifan, I., Ioana, A., Kida, Y.: \(W^*\)-superrigidity for arbitrary actions of central quotients of braid groups. Math. Ann. 361(3–4), 563–582 (2015)
Chifan, I., Kida, Y., Pant, S.: Primeness results for von Neumann algebas associated with surface braid groups. Int. Math. Res. Not. 16, 4807–4848 (2016)
Chifan, I., Peterson, J.: Some unique group measure space decomposition results. Duke Math. J. 162(11), 1923–1966 (2013)
Chifan, I., Peterson, J.: On approximation properties for probability measure preserving actions, Preprint (2011)
Chifan, I., Popa, S., Sizemore, O.: Some OE- and \(W^{\ast }\)-rigidity results for actions by wreath product groups. J. Funct. Anal. 263(11), 3422–3448 (2012)
Chifan, I., Sinclair, T.: On the structural theory of \(\text{ II}_1\)factors of negatively curved groups. Ann. Sci. Éc. Norm. Sup. 46, 1–33 (2013)
Chifan, I., Sinclair, T., Udrea, B.: On the structural theory of \(\text{ II}_1\) factors of negatively curved groups, II. Actions by product groups. Adv. Math. 245, 208–236 (2013)
Chifan, I., Sinclair, T., Udrea, B.: Inner amenability for groups and central sequences in factors. Ergodic Theory Dyn. Syst. 36, 1106–1129 (2016)
Creutz, D., Peterson, J.: Character rigidity for lattices and commensurators, preprint ar**v:1311.4513
Choda, H.: A Galois correspondence in a von Neumann algebra. Tohuku Math. J. 30, 491–504 (1978)
Connes, A.: Classification of injective factors. Ann. Math. 104, 73–115 (1976)
Dahmani, F., Guirardel, V., Osin, D.: (2017) Hyperbolically embedded subgroups and rotating families in groups acting on HYperbolic spaces. Memoirs Amer. Math. Soc. 245 (1156)
Drimbe, D., Hoff, D., Ioana, A.: Prime \(\text{ II}_1\) factors arising from irreducible lattices in products of rank one simple Lie groups, J. Reine. Angew. Math. (to appear), preprint, ar**v:1611.02209
Drimbe, D.: \(W^*\) superrigidity for coinduced actions. Int. J. Math. 29, 1850033 (2018)
Evans, D., Kawahigashi, Y.: Quantum symmetries on operator algebras. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, (1998). xvi+829 pp. ISBN: 0-19-851175-2
Furman, A.: Gromov’s measure equivalence and rigidity of higher rank lattices. Ann. of Math. (2) 150, 1059–1081 (1999)
Furstenberg, H.: Ergodic behaviour of diagonal measures and a theorem of Szemeredi on arithmetic progressions. J. Anal. Math. 31, 204–256 (1977)
Fima, P., Vaes, S.: HNN extensions and unique group measure space decomposition of \(\text{ II}_1\) factors. Trans. Amer. Math. Soc. 354, 2601–2617 (2012)
Gaboriau, D.: Orbit equivalence and measured group theory, Proceedings of the International Congress of Mathematicians (Hyderabad, India, 2010), Vol. III, 1501-1527, Hindustan Book Agency, New Delhi, (2010)
Ge, L.: On maximal injective subalgebras of factors. Adv. Math. 118, 34–70 (1996)
Gaboriau, D., Ioana, A., Tucker-Drob, R.: Cocycle superrigidity for translation actions of product groups, Amer. Journal of Math. (to appear), preprint, ar**v:1603.07616
Ge, L., Kadison, R.: On tensor products of von Neumann algebras. Inventiones Math. 123, 453–466 (1996)
Hull, M., Osin, D.: Induced quasi-cocycles on groups with hyperbolically embedded subgroups. Alg. Geom. Topol. 13, 2635–2665 (2013)
Houdayer, C., Popa, S., Vaes, S.: A class of groups for which every action is \(W^*\)-superrigid. Groups Geom. Dyn. 7, 577–590 (2013)
Ioana, A.: Cocycle superrigidity for profinite actions of property (T) groups, Ar**v preprint version, ar**v:0805.2998v1
Ioana, A.: Cocycle superrigidity for profinite actions of property (T) groups. Duke Math J. 157, 337–367 (2011)
Ioana, A.: \(W^*\)-superrigidity for Bernoulli actions of property (T) groups. J. Amer. Math. Soc. 24, 1175–1226 (2011)
Ioana, A.: Cartan subalgebras of amalgamated free product \(\text{ II}_1\) factors. Ann. Sci. Éc. Norm. Sup. 48, 71–130 (2015)
Ioana, A.: Uniqueness of the group measure space decomposition for Popa’s \(\cal{H}\cal{T}\) factors. Geom. Funct. Anal. 22(3), 699–732 (2012)
Ioana, A.: Rigidity for von Neumann algebras, Submitted to Proceedings ICM (2018). Preprint ar**v:1712.00151v1
Ioana, A., Popa, S., Vaes, S.: A class of superrigid group von Neumann algebras. Ann. of Math. (2) 178, 231–286 (2013)
Izumi, M., Longo, R., Popa, S.: Automorphisms of von Neumann Algebras with a Generalization to Kac Algebras. J. Functional Anal. 155(1), 25–63 (1998)
Jones, V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983)
Jones, V.F.R.: von Neumann algebras in mathematics and physics, Proceedings of the International Congress of Mathematicians, vol I, II (Kyoto 1990), 121–138, Math Soc. Japan, Tokyo (1991)
Jones, V.F.R.: Subfactor and Knots, CBMS Regional Conference Series in Mathematics 80, AMS (1991)
Jones, V.F.R.: On the origin and development of subfactors and quantum topology. Bull. Amer. Math. Soc. 46, 309–326 (2009)
Jiang, Y., Skalski, A.: Maximal subgroups and von Neumann subalgebras with the Haagerup property, (2019). ar** class group. Ann. of Math. (2) 171, 1851–1901 (2010)
Krogager, A., Vaes, S.: A class of \(\text{ II}_1\) factors with exactly two group measure space decompositions. J. Math. Pures et Appl. 108, 88–110 (2017)
Monod, N., Shalom, Y.: Orbit equivalence rigidity and bounded cohomology. Ann. Math. (2) 164, 825–878 (2006)
Neshveyev, S., Størmer, E.: Ergodic theory and maximal abelian subalgebras of the hyperfinite factor. J. Funct. Anal. 195(2), 239–261 (2002)
Nielsen, O.A.: Maximal abelian subalgebras of hyperfinite factors II. J. Funct. Anal. 6, 192–202 (1970)
Osin, D.: Acylindrically Hyperbolic groups. Trans. Amer. Math. Soc. 368, 851–888 (2016)
Ozawa, N., Popa, S.: On a class of \(\text{ II}_1\) factors with at most one Cartan subalgebra. Ann. Math. 172, 713–749 (2010)
Ozawa, N., Popa, S.: On a class of \(\text{ II}_1\) factors with at most one Cartan subalgebra II. Am. J. Math. 132(3), 841–866 (2010)
Packer, J.: Point spectrum of ergodic abelian group actions and the corresponding group-measure space factors. Pacific J. Math 201, 421–428 (2001)
Peterson, J.: Examples of group actions which are virtually \(W^*\)-superrigid, preprint, ar**v:1002.1745
Peterson, J.: Character rigidity for lattices in higher-rank groups, (2014). preprint, www.math.vanderbilt.edu/~peters10/rigidity.pdf
Peterson, J., Thom, A.: Group cocycles and the ring of affiliated operators. Invent. Math. 185, 561–592 (2011)
Pimsner, M., Popa, S.: Entropy and index for subfactors. Ann. Sci. École Norm. Sup. 19, 57–106 (1986)
Popa, S.: Notes on Cartan subalgebras in type \(\text{ II}_1\) factors. Math. Scand. 57, 171–188 (1985)
Popa, S.: Some properties of the symmetric envelo** algebra of a factor, with applications to amenability and property (T), 4 Doc. Math. 665–744, (1999)
Popa, S.: On a class of type \(\text{ II}_1\) factors with Betti numbers invariants. Ann. Math. 163, 809–899 (2006)
Popa, S.: Universal construction of subfactors. J. Reine Angew. Math. 543, 39–81 (2002)
Popa, S.: Strong rigidity of \(\text{ II}_1\) factors arising from malleable actions of \(w\)-rigid groups I. Invent. Math. 165, 369–408 (2006)
Popa, S.: Strong rigidity of \(\text{ II}_1\) factors arising from malleable actions of \(w\)-rigid groups II. Invent. Math. 165, 409–451 (2006)
Popa, S.: Deformation and rigidity for group actions and von Neumann algebras, International Congress of Mathematicians. Vol. I, 445–477, Eur. Math. Soc., Zürich, (2007)
Popa, S.: On the superrigidity of malleable actions with spectral gap. J. Amer. Math. Soc. 21, 981–1000 (2008)
Popa, S.: On the inductive limits of \(\text{ II}_1\) factors with spectral gap. Trans. Amer. Math. Soc. 364, 2987–3000 (2012)
Popa, S., Vaes, S.: Group measure space decomposition of \(\text{ II}_1\) factors and \(W^*\)-superrigidity. Invent. Math. 182, 371–417 (2010)
Popa, S., Vaes, S.: Unique Cartan decomposition for \(\text{ II}_1\) factors arising from arbitrary actions of free groups. Acta Math. 212, 141–198 (2014)
Popa, S., Vaes, S.: Unique Cartan decomposition for \(\text{ II}_1\) factors arising from arbitrary actions of hyperbolic groups. J. Reine Angew. Math. 694, 215–239 (2014)
Singer, I.M.: Automorphisms of finite factors. Amer. J. Math. 77, 117–133 (1955)
Suzuki, Y.: Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems, To appear in Comm. Math. Phy., Ar**v Preprint, ar**v:1805.02077
Smith, R., White, S., Wiggins, A.: Normalizers of irreducibe subfactors. J. Math. Anal. Appl. 352, 684–695 (2009)
Vaes, S.: Rigidity for von Neumann algebras and their invariants, Proceedings of the International Congress of Mathematicians(Hyderabad, India, 2010) Vol III, Hindustan Book Agency, 1624–1650, (2010)
Vaes, S.: One-cohomology and the uniqueness of the group measure space decomposition of a \(\text{ II}_1\) factor. Math. Ann. 355, 661–696 (2013)
Zimmer, R.J.: Ergodic actions with generalized discrete spectrum. Illinois J. Math. 20(4), 555–588 (1976)
Acknowledgements
The authors are grateful to Adrian Ioana and Jesse Peterson for many helpful discussions related to this project. The authors are extremely grateful to Yuhei Suzuki for carefully reading a first draft of this paper, for his helpful comments and suggestions, and for correcting numerous typos and minor inaccuracies. The authors are also grateful to Rahel Brugger for her helpful comments on our paper, and for correcting a few typos. The second author would like to thank Vaughan Jones for several suggestions and comments regarding the results of this paper. The second author would also like to thank Krishnendu Khan and Pieter Spaas for stimulating conversations regarding the contents of this paper. The first author was partially supported by NSF Grant DMS #1600688. Finally, the authors would like to thank the anonymous referee for many helpful comments and suggestions that greatly improved the exposition of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Andreas Thom.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Chifan, I., Das, S. Rigidity results for von Neumann algebras arising from mixing extensions of profinite actions of groups on probability spaces. Math. Ann. 378, 907–950 (2020). https://doi.org/10.1007/s00208-020-02064-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-020-02064-8