Unbounded derivations, free dilations, and indecomposability results for II$_1$ factors
HTML articles powered by AMS MathViewer
- by Yoann Dabrowski and Adrian Ioana PDF
- Trans. Amer. Math. Soc. 368 (2016), 4525-4560 Request permission
Abstract:
We give sufficient conditions, in terms of the existence of unbounded derivations satisfying certain regularity properties, which ensure that a II$_1$ factor $M$ is prime or has at most one Cartan subalgebra. For instance, we prove that if there exists a real closable unbounded densely defined derivation $\delta :M\rightarrow L^2(M)\bar {\otimes }L^2(M)$ whose domain contains a non-amenability set, then $M$ is prime. If $\delta$ is moreover “algebraic” (i.e. its domain $M_0$ is finitely generated, $\delta (M_0)\subset M_0\otimes M_0$ and $\delta ^*(1\otimes 1)\in M_0$), then we show that $M$ has no Cartan subalgebra. We also give several applications to examples from free probability. Finally, we provide a class of countable groups $\Gamma$, defined through the existence of an unbounded cocycle $b:\Gamma \rightarrow \mathbb C(\Gamma /\Lambda )$, for certain subgroups $\Lambda <\Gamma$, such that the II$_1$ factor $L^{\infty }(X)\rtimes \Gamma$ has a unique Cartan subalgebra, up to unitary conjugacy, for any free ergodic probability measure preserving (pmp) action $\Gamma \curvearrowright (X,\mu )$.References
- C. Anantharaman-Delaroche, Amenable correspondences and approximation properties for von Neumann algebras, Pacific J. Math. 171 (1995), no. 2, 309–341. MR 1372231
- P. Biane, M. Capitaine, and A. Guionnet, Large deviation bounds for matrix Brownian motion, Invent. Math. 152 (2003), no. 2, 433–459. MR 1975007, DOI 10.1007/s00222-002-0281-4
- Mohammed E. B. Bekka and Alain Valette, Group cohomology, harmonic functions and the first $L^2$-Betti number, Potential Anal. 6 (1997), no. 4, 313–326. MR 1452785, DOI 10.1023/A:1017974406074
- Serban Teodor Belinschi, The Lebesgue decomposition of the free additive convolution of two probability distributions, Probab. Theory Related Fields 142 (2008), no. 1-2, 125–150. MR 2413268, DOI 10.1007/s00440-007-0100-3
- 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
- Ionut Chifan and Thomas Sinclair, On the structural theory of $\textrm {II}_1$ factors of negatively curved groups, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 1, 1–33 (2013) (English, with English and French summaries). MR 3087388, DOI 10.24033/asens.2183
- A. Connes, Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not =1$, Ann. of Math. (2) 104 (1976), no. 1, 73–115. MR 454659, DOI 10.2307/1971057
- Yoann Dabrowski, A note about proving non-$\Gamma$ under a finite non-microstates free Fisher information assumption, J. Funct. Anal. 258 (2010), no. 11, 3662–3674. MR 2606868, DOI 10.1016/j.jfa.2010.02.010
- Yoann Dabrowski, A non-commutative path space approach to stationary free stochastic differential equations, preprint arXiv:1006.4351.
- Yoann Dabrowski, A free stochastic partial differential equation, Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), no. 4, 1404–1455. MR 3270000, DOI 10.1214/13-AIHP548
- E. Brian Davies and J. Martin Lindsay, Noncommutative symmetric Markov semigroups, Math. Z. 210 (1992), no. 3, 379–411. MR 1171180, DOI 10.1007/BF02571804
- Pierre Fima and Stefaan Vaes, HNN extensions and unique group measure space decomposition of $\rm II_1$ factors, Trans. Amer. Math. Soc. 364 (2012), no. 5, 2601–2617. MR 2888221, DOI 10.1090/S0002-9947-2012-05415-8
- Liming Ge, Applications of free entropy to finite von Neumann algebras. II, Ann. of Math. (2) 147 (1998), no. 1, 143–157. MR 1609522, DOI 10.2307/120985
- Cyril Houdayer, Structure of $\mathrm {II}_1$ factors arising from free Bogoljubov actions of arbitrary groups, Adv. Math. 260 (2014), 414–457. MR 3209358, DOI 10.1016/j.aim.2014.04.010
- Cyril Houdayer, Sorin Popa, and Stefaan Vaes, A class of groups for which every action is $\mathrm {W}^*$-superrigid, Groups Geom. Dyn. 7 (2013), no. 3, 577–590. MR 3095710, DOI 10.4171/GGD/198
- Adrian Ioana, Classification and rigidity for von Neumann algebras, preprint arXiv:1212.0453, to appear in Proceedings of the 6th European Congress of Mathematics (Krakow, 2012), European Mathematical Society Publishing House.
- Adrian Ioana, Cartan subalgebras of amalgamated free product $\textrm {II}_1$ factors, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 1, 71–130 (English, with English and French summaries). With an appendix by Ioana and Stefaan Vaes. MR 3335839, DOI 10.24033/asens.2239
- 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
- Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
- Nicolas Monod and Sorin Popa, On co-amenability for groups and von Neumann algebras, C. R. Math. Acad. Sci. Soc. R. Can. 25 (2003), no. 3, 82–87 (English, with French summary). MR 1999183
- 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
- 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
- Narutaka Ozawa, Solid von Neumann algebras, Acta Math. 192 (2004), no. 1, 111-117.
- Jesse Peterson, A 1-cohomology characterization of property (T) in von Neumann algebras, Pacific J. Math. 243 (2009), no. 1, 181–199. MR 2550142, DOI 10.2140/pjm.2009.243.181
- 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
- Jesse Peterson and Thomas Sinclair, On cocycle superrigidity for Gaussian actions, Ergodic Theory Dynam. Systems 32 (2012), no. 1, 249–272. MR 2873170, DOI 10.1017/S0143385710000751
- Sorin Popa, Orthogonal pairs of $\ast$-subalgebras in finite von Neumann algebras, J. Operator Theory 9 (1983), no. 2, 253–268. MR 703810
- S. Popa, Correspondences, INCREST Preprint 1986.
- Sorin Popa, Markov traces on universal Jones algebras and subfactors of finite index, Invent. Math. 111 (1993), no. 2, 375–405. MR 1198815, DOI 10.1007/BF01231293
- 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, 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, On Ozawa’s property for free group factors, Int. Math. Res. Not. IMRN 11 (2007), Art. ID rnm036, 10. MR 2344271, DOI 10.1093/imrn/rnm036
- 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, 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
- Sorin Popa and Stefaan Vaes, Unique Cartan decomposition for $\rm II_1$ factors arising from arbitrary actions of free groups, Acta Math. 212 (2014), no. 1, 141–198. MR 3179609, DOI 10.1007/s11511-014-0110-9
- Sorin Popa and Stefaan Vaes, Unique Cartan decomposition for $\rm II_{1}$ factors arising from arbitrary actions of hyperbolic groups, J. Reine Angew. Math. 694 (2014), 215–239. MR 3259044, DOI 10.1515/crelle-2012-0104
- J.-L. Sauvageot, Tangent bimodule and locality for dissipative operators on $C^*$-algebras, Quantum probability and applications, IV (Rome, 1987) Lecture Notes in Math., vol. 1396, Springer, Berlin, 1989, pp. 322–338. MR 1019584, DOI 10.1007/BFb0083561
- Dimitri Shlyakhtenko, Free entropy with respect to a completely positive map, Amer. J. Math. 122 (2000), no. 1, 45–81. MR 1737257
- Dimitri Shlyakhtenko, Lower estimates on microstates free entropy dimension, Anal. PDE 2 (2009), no. 2, 119–146. MR 2547131, DOI 10.2140/apde.2009.2.119
- Yoshimichi Ueda, HNN extensions of von Neumann algebras, J. Funct. Anal. 225 (2005), no. 2, 383–426. MR 2152505, DOI 10.1016/j.jfa.2005.01.004
- 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, Rigidity for von Neumann algebras and their invariants, Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi, 2010, pp. 1624–1650. MR 2827858
- Stefaan Vaes, An inner amenable group whose von Neumann algebra does not have property Gamma, Acta Math. 208 (2012), no. 2, 389–394. MR 2931384, DOI 10.1007/s11511-012-0079-1
- Dan Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. I, Comm. Math. Phys. 155 (1993), no. 1, 71–92. MR 1228526
- Dan Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. II, Invent. Math. 118 (1994), no. 3, 411–440. MR 1296352, DOI 10.1007/BF01231539
- D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. III. The absence of Cartan subalgebras, Geom. Funct. Anal. 6 (1996), no. 1, 172–199. MR 1371236, DOI 10.1007/BF02246772
- Dan Voiculescu, A strengthened asymptotic freeness result for random matrices with applications to free entropy, Internat. Math. Res. Notices 1 (1998), 41–63. MR 1601878, DOI 10.1155/S107379289800004X
- Dan Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. V. Noncommutative Hilbert transforms, Invent. Math. 132 (1998), no. 1, 189–227. MR 1618636, DOI 10.1007/s002220050222
- Dan Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. VI. Liberation and mutual free information, Adv. Math. 146 (1999), no. 2, 101–166. MR 1711843, DOI 10.1006/aima.1998.1819
- Dan Voiculescu, Cyclomorphy, Int. Math. Res. Not. 6 (2002), 299–332. MR 1877005, DOI 10.1155/S1073792802105046
- D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. MR 1217253, DOI 10.1090/crmm/001
Additional Information
- Yoann Dabrowski
- Affiliation: Université de Lyon, Université Lyon 1, Institut Camille Jordan UMR 5208, 43 Boulevard du 11 novembre 1918, F-69622 Villeurbanne cedex, France
- Email: dabrowski@math.univ-lyon1.fr
- Adrian Ioana
- Affiliation: Department of Mathematics, University of California, San Diego, California 90095-1555
- Email: aioana@ucsd.edu
- Received by editor(s): May 4, 2013
- Received by editor(s) in revised form: May 4, 2014
- Published electronically: October 28, 2015
- Additional Notes: The first author was partially supported by ANR grant NEUMANN
The second author was partially supported by NSF Grant DMS #1161047, NSF Career Grant DMS #1253402, and a Sloan Foundation Fellowship - © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 4525-4560
- MSC (2010): Primary 46L36; Secondary 28D15, 37A20
- DOI: https://doi.org/10.1090/tran/6470
- MathSciNet review: 3456153