A class of superrigid group von Neumann algebras

Abstract

We prove that for any group $G$ in a fairly large class of generalized wreath product groups, the associated von Neumann algebra $\mathrm{L} G$ completely “remembers” the group $G$. More precisely, if $\mathrm{L} G$ is isomorphic to the von Neumann algebra $\mathrm{L} \Lambda$ of an arbitrary countable group $\Lambda$, then $\Lambda$ must be isomorphic to $G$. This represents the first superrigidity result pertaining to group von Neumann algebras.

  • [Bo09a] Go to document L. Bowen, "Orbit equivalence, coinduced actions and free products," Groups Geom. Dyn., vol. 5, iss. 1, pp. 1-15, 2011.
    @article {Bo09a, MRKEY = {2763777},
      AUTHOR = {Bowen, Lewis},
      TITLE = {Orbit equivalence, coinduced actions and free products},
      JOURNAL = {Groups Geom. Dyn.},
      FJOURNAL = {Groups, Geometry, and Dynamics},
      VOLUME = {5},
      YEAR = {2011},
      NUMBER = {1},
      PAGES = {1--15},
      ISSN = {1661-7207},
      MRCLASS = {37A20},
      MRNUMBER = {2763777},
      MRREVIEWER = {Konstantin Medynets},
      DOI = {10.4171/GGD/114},
      ZBLNUMBER = {05973326},
      }
  • [Bo09b] Go to document L. Bowen, "Stable orbit equivalence of Bernoulli shifts over free groups," Groups Geom. Dyn., vol. 5, iss. 1, pp. 17-38, 2011.
    @article {Bo09b, MRKEY = {2763778},
      AUTHOR = {Bowen, Lewis},
      TITLE = {Stable orbit equivalence of {B}ernoulli shifts over free groups},
      JOURNAL = {Groups Geom. Dyn.},
      FJOURNAL = {Groups, Geometry, and Dynamics},
      VOLUME = {5},
      YEAR = {2011},
      NUMBER = {1},
      PAGES = {17--38},
      ISSN = {1661-7207},
      MRCLASS = {37A20},
      MRNUMBER = {2763778},
      MRREVIEWER = {Konstantin Medynets},
      DOI = {10.4171/GGD/115},
      ZBLNUMBER = {05973327},
      }
  • [CH08] Go to document I. Chifan and C. Houdayer, "Bass-Serre rigidity results in von Neumann algebras," Duke Math. J., vol. 153, iss. 1, pp. 23-54, 2010.
    @article {CH08, MRKEY = {2641939},
      AUTHOR = {Chifan, Ionut and Houdayer, Cyril},
      TITLE = {Bass-{S}erre rigidity results in von {N}eumann algebras},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {153},
      YEAR = {2010},
      NUMBER = {1},
      PAGES = {23--54},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {46L54 (46L55)},
      MRNUMBER = {2641939},
      DOI = {10.1215/00127094-2010-020},
      ZBLNUMBER = {1201.46057},
      }
  • [CI08] Go to document I. Chifan and A. Ioana, "Ergodic subequivalence relations induced by a Bernoulli action," Geom. Funct. Anal., vol. 20, iss. 1, pp. 53-67, 2010.
    @article {CI08, MRKEY = {2647134},
      AUTHOR = {Chifan, Ionut and Ioana, Adrian},
      TITLE = {Ergodic subequivalence relations induced by a {B}ernoulli action},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {20},
      YEAR = {2010},
      NUMBER = {1},
      PAGES = {53--67},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {37A20 (37A15 46L10)},
      MRNUMBER = {2647134},
      MRREVIEWER = {Alain Valette},
      DOI = {10.1007/s00039-010-0058-7},
      ZBLNUMBER = {1211.37006},
      }
  • [Co75] Go to document A. Connes, "Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not=1$," Ann. of Math., vol. 104, iss. 1, pp. 73-115, 1976.
    @article {Co75, MRKEY = {0454659},
      AUTHOR = {Connes, A.},
      TITLE = {Classification of injective factors. {C}ases {$II\sb{1},
     $} {$II\sb{\infty },
     $} {$III\sb{\lambda },
     $} {$\lambda \not=1$}},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {104},
      YEAR = {1976},
      NUMBER = {1},
      PAGES = {73--115},
      ISSN = {0003-486X},
      MRCLASS = {46L10},
      MRNUMBER = {0454659},
      MRREVIEWER = {Francois Combes},
      ZBLNUMBER = {0343.46042},
      DOI = {10.2307/1971057},
     }
  • [Co80a] A. Connes, "A factor of type ${ II}_{1}$ with countable fundamental group," J. Operator Theory, vol. 4, iss. 1, pp. 151-153, 1980.
    @article {Co80a, MRKEY = {0587372},
      AUTHOR = {Connes, A.},
      TITLE = {A factor of type {${\rm II}\sb{1}$} with countable fundamental group},
      JOURNAL = {J. Operator Theory},
      FJOURNAL = {Journal of Operator Theory},
      VOLUME = {4},
      YEAR = {1980},
      NUMBER = {1},
      PAGES = {151--153},
      ISSN = {0379-4024},
      MRCLASS = {46L10 (22D35)},
      MRNUMBER = {0587372},
      MRREVIEWER = {Richard I. Loebl},
      ZBLNUMBER = {0455.46056},
      }
  • [Co80b] Go to document A. Connes, "Classification des facteurs," in Operator Algebras and Applications, Part 2, Providence, R.I.: Amer. Math. Soc., 1982, vol. 38, pp. 43-109.
    @incollection {Co80b, MRKEY = {0679497},
      AUTHOR = {Connes, A.},
      TITLE = {Classification des facteurs},
      BOOKTITLE = {Operator Algebras and Applications, {P}art 2},
      VENUE={{K}ingston, {O}nt., 1980},
      SERIES = {Proc. Sympos. Pure Math.},
      VOLUME = {38},
      PAGES = {43--109},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, R.I.},
      YEAR = {1982},
      MRCLASS = {46L35 (22D25 46L55)},
      MRNUMBER = {0679497},
      MRREVIEWER = {Vaughan Jones},
      ZBLNUMBER = {0503.46043},
      DOI = {10.1090/pspum/038.2/679497},
     }
  • [Co94] A. Connes, Noncommutative Geometry, San Diego, CA: Academic Press, 1994.
    @book {Co94, MRKEY = {1303779},
      AUTHOR = {Connes, A.},
      TITLE = {Noncommutative Geometry},
      PUBLISHER = {Academic Press},
      ADDRESS = {San Diego, CA},
      YEAR = {1994},
      PAGES = {xiv+661},
      ISBN = {0-12-185860-X},
      MRCLASS = {46Lxx (19K56 22D25 58B30 58G12 81T13 81V22 81V70)},
      MRNUMBER = {1303779},
      MRREVIEWER = {John Roe},
      ZBLNUMBER = {0818.46076},
      }
  • [CJ83] Go to document A. Connes and V. Jones, "Property $T$ for von Neumann algebras," Bull. London Math. Soc., vol. 17, iss. 1, pp. 57-62, 1985.
    @article {CJ83, MRKEY = {0766450},
      AUTHOR = {Connes, A. and Jones, V.},
      TITLE = {Property {$T$} for von {N}eumann algebras},
      JOURNAL = {Bull. London Math. Soc.},
      FJOURNAL = {The Bulletin of the London Mathematical Society},
      VOLUME = {17},
      YEAR = {1985},
      NUMBER = {1},
      PAGES = {57--62},
      ISSN = {0024-6093},
      CODEN = {LMSBBT},
      MRCLASS = {46L35 (22D25)},
      MRNUMBER = {0766450},
      MRREVIEWER = {Colin E. Sutherland},
      DOI = {10.1112/blms/17.1.57},
      ZBLNUMBER = {1190.46047},
      }
  • [CH88] Go to document M. Cowling and U. Haagerup, "Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one," Invent. Math., vol. 96, iss. 3, pp. 507-549, 1989.
    @article {CH88, MRKEY = {0996553},
      AUTHOR = {Cowling, Michael and Haagerup, Uffe},
      TITLE = {Completely bounded multipliers of the {F}ourier algebra of a simple {L}ie group of real rank one},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {96},
      YEAR = {1989},
      NUMBER = {3},
      PAGES = {507--549},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {22E27 (22E40 43A22 43A80 46J99)},
      MRNUMBER = {0996553},
      MRREVIEWER = {Christopher Meaney},
      DOI = {10.1007/BF01393695},
      ZBLNUMBER = {0681.43012},
      }
  • [Dy52] H. A. Dye, "The unitary structure in finite rings of operators," Duke Math. J., vol. 20, pp. 55-69, 1953.
    @article {Dy52, MRKEY = {0052695},
      AUTHOR = {Dye, H. A.},
      TITLE = {The unitary structure in finite rings of operators},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {20},
      YEAR = {1953},
      PAGES = {55--69},
      ISSN = {0012-7094},
      MRCLASS = {46.3X},
      MRNUMBER = {0052695},
      MRREVIEWER = {F. I. Mautner},
      ZBLNUMBER = {0050.11405},
      }
  • [Dy92b] Go to document K. Dykema, "Free products of hyperfinite von Neumann algebras and free dimension," Duke Math. J., vol. 69, iss. 1, pp. 97-119, 1993.
    @article {Dy92b, MRKEY = {1201693},
      AUTHOR = {Dykema, Ken},
      TITLE = {Free products of hyperfinite von {N}eumann algebras and free dimension},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {69},
      YEAR = {1993},
      NUMBER = {1},
      PAGES = {97--119},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {46L35},
      MRNUMBER = {1201693},
      MRREVIEWER = {Marie Choda},
      DOI = {10.1215/S0012-7094-93-06905-0},
      ZBLNUMBER = {0784.46044},
      }
  • [Dy92a] Go to document K. Dykema, "Interpolated free group factors," Pacific J. Math., vol. 163, iss. 1, pp. 123-135, 1994.
    @article {Dy92a, MRKEY = {1256179},
      AUTHOR = {Dykema, Ken},
      TITLE = {Interpolated free group factors},
      JOURNAL = {Pacific J. Math.},
      FJOURNAL = {Pacific Journal of Mathematics},
      VOLUME = {163},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {123--135},
      ISSN = {0030-8730},
      CODEN = {PJMAAI},
      MRCLASS = {46L35 (46L50)},
      MRNUMBER = {1256179},
      MRREVIEWER = {Marie Choda},
      DOI = {10.2140/pjm.1994.163.123},
      ZBLNUMBER = {0791.46038},
      }
  • [DHR97] Go to document K. Dykema, U. Haagerup, and M. Rørdam, "The stable rank of some free product $C^*$-algebras," Duke Math. J., vol. 90, iss. 1, pp. 95-121, 1997.
    @article {DHR97, MRKEY = {1478545},
      AUTHOR = {Dykema, Ken and Haagerup, Uffe and Rørdam, Mikael},
      TITLE = {The stable rank of some free product {$C\sp *$}-algebras},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {90},
      YEAR = {1997},
      NUMBER = {1},
      PAGES = {95--121},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {46L05 (19B10 19K56 46L35 46L80)},
      MRNUMBER = {1478545},
      MRREVIEWER = {Gustavo Corach},
      DOI = {10.1215/S0012-7094-97-09004-9},
      ZBLNUMBER = {0905.46036},
      }
  • [DR98] Go to document K. J. Dykema and F. Ruadulescu, "Compressions of free products of von Neumann algebras," Math. Ann., vol. 316, iss. 1, pp. 61-82, 2000.
    @article {DR98, MRKEY = {1735079},
      AUTHOR = {Dykema, Kenneth J. and R{\u{a}}dulescu, Florin},
      TITLE = {Compressions of free products of von {N}eumann algebras},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {316},
      YEAR = {2000},
      NUMBER = {1},
      PAGES = {61--82},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {46L54},
      MRNUMBER = {1735079},
      MRREVIEWER = {Dimitri Y. Shlyakhtenko},
      DOI = {10.1007/s002080050004},
      ZBLNUMBER = {1205.90105},
      }
  • [Hi39] Go to document G. Higman, "The units of group-rings," Proc. London Math. Soc., vol. 46, pp. 231-248, 1940.
    @article {Hi39, MRKEY = {0002137},
      AUTHOR = {Higman, Graham},
      TITLE = {The units of group-rings},
      JOURNAL = {Proc. London Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Second Series},
      VOLUME = {46},
      YEAR = {1940},
      PAGES = {231--248},
      ISSN = {0024-6115},
      MRCLASS = {20.0X},
      MRNUMBER = {0002137},
      MRREVIEWER = {M. Hall},
      ZBLNUMBER = {0025.24302},
      DOI = {10.1112/plms/s2-46.1.231},
     }
  • [Io06] Go to document A. Ioana, "Rigidity results for wreath product ${ II}_1$ factors," J. Funct. Anal., vol. 252, iss. 2, pp. 763-791, 2007.
    @article {Io06, MRKEY = {2360936},
      AUTHOR = {Ioana, Adrian},
      TITLE = {Rigidity results for wreath product {${\rm II}\sb 1$} factors},
      JOURNAL = {J. Funct. Anal.},
      FJOURNAL = {Journal of Functional Analysis},
      VOLUME = {252},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {763--791},
      ISSN = {0022-1236},
      CODEN = {JFUAAW},
      MRCLASS = {46L10 (46L55)},
      MRNUMBER = {2360936},
      MRREVIEWER = {Paul Jolissaint},
      DOI = {10.1016/j.jfa.2007.04.005},
      ZBLNUMBER = {1134.46041},
      }
  • [Io10] Go to document A. Ioana, "$W^*$-superrigidity for Bernoulli actions of property (T) groups," J. Amer. Math. Soc., vol. 24, iss. 4, pp. 1175-1226, 2011.
    @article {Io10, MRKEY = {2813341},
      AUTHOR = {Ioana, Adrian},
      TITLE = {{$W\sp *$}-superrigidity for {B}ernoulli actions of property ({T}) groups},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {24},
      YEAR = {2011},
      NUMBER = {4},
      PAGES = {1175--1226},
      ISSN = {0894-0347},
      MRCLASS = {46L36 (28D15 37A20)},
      MRNUMBER = {2813341},
      MRREVIEWER = {Stefaan Vaes},
      DOI = {10.1090/S0894-0347-2011-00706-6},
      ZBLNUMBER = {1236.46054},
      }
  • [IPP05] Go to document A. Ioana, J. Peterson, and S. Popa, "Amalgamated free products of weakly rigid factors and calculation of their symmetry groups," Acta Math., vol. 200, iss. 1, pp. 85-153, 2008.
    @article {IPP05, MRKEY = {2386109},
      AUTHOR = {Ioana, Adrian and Peterson, Jesse and Popa, Sorin},
      TITLE = {Amalgamated free products of weakly rigid factors and calculation of their symmetry groups},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {200},
      YEAR = {2008},
      NUMBER = {1},
      PAGES = {85--153},
      ISSN = {0001-5962},
      CODEN = {ACMAA8},
      MRCLASS = {46L10 (20E06 22D25 37A15 37A20 46L09)},
      MRNUMBER = {2386109},
      MRREVIEWER = {Alain Valette},
      DOI = {10.1007/s11511-008-0024-5},
      ZBLNUMBER = {1149.46047},
      }
  • [Ka70] Go to document I. Kaplansky, "“Problems in the theory of rings” revisited," Amer. Math. Monthly, vol. 77, pp. 445-454, 1970.
    @article {Ka70, MRKEY = {0258865},
      AUTHOR = {Kaplansky, Irving},
      TITLE = {``{P}roblems in the theory of rings'' revisited},
      JOURNAL = {Amer. Math. Monthly},
      FJOURNAL = {The American Mathematical Monthly},
      VOLUME = {77},
      YEAR = {1970},
      PAGES = {445--454},
      ISSN = {0002-9890},
      MRCLASS = {16.00},
      MRNUMBER = {0258865},
      DOI = {10.2307/2317376},
     }
  • [McD69] Go to document D. McDuff, "Uncountably many ${ II}_{1}$ factors," Ann. of Math., vol. 90, pp. 372-377, 1969.
    @article {McD69, MRKEY = {0259625},
      AUTHOR = {McDuff, Dusa},
      TITLE = {Uncountably many {${\rm II}\sb{1}$} factors},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {90},
      YEAR = {1969},
      PAGES = {372--377},
      ISSN = {0003-486X},
      MRCLASS = {46.65},
      MRNUMBER = {0259625},
      MRREVIEWER = {Z. Takeda},
      ZBLNUMBER = {0184.16902},
      DOI = {10.2307/1970730},
     }
  • [MvN36] Go to document F. J. Murray and J. von Neumann, "On rings of operators," Ann. of Math., vol. 37, iss. 1, pp. 116-229, 1936.
    @article {MvN36, MRKEY = {1503275},
      AUTHOR = {Murray, F. J. and von Neumann, J.},
      TITLE = {On rings of operators},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {37},
      YEAR = {1936},
      NUMBER = {1},
      PAGES = {116--229},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {Contributed Item},
      MRNUMBER = {1503275},
      DOI = {10.2307/1968693},
      ZBLNUMBER = {0014.16101},
      }
  • [MvN43] Go to document F. J. Murray and J. von Neumann, "On rings of operators. IV," Ann. of Math., vol. 44, pp. 716-808, 1943.
    @article {MvN43, MRKEY = {0009096},
      AUTHOR = {Murray, F. J. and von Neumann, J.},
      TITLE = {On rings of operators. {IV}},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {44},
      YEAR = {1943},
      PAGES = {716--808},
      ISSN = {0003-486X},
      MRCLASS = {46.0X},
      MRNUMBER = {0009096},
      MRREVIEWER = {E. R. Lorch},
      ZBLNUMBER = {0060.26903},
      DOI = {10.2307/1969107},
     }
  • [Oz02] Go to document N. Ozawa, "There is no separable universal $ II_1$-factor," Proc. Amer. Math. Soc., vol. 132, iss. 2, pp. 487-490, 2004.
    @article {Oz02, MRKEY = {2022373},
      AUTHOR = {Ozawa, Narutaka},
      TITLE = {There is no separable universal {$\rm II\sb 1$}-factor},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {132},
      YEAR = {2004},
      NUMBER = {2},
      PAGES = {487--490},
      ISSN = {0002-9939},
      CODEN = {PAMYAR},
      MRCLASS = {46L10 (20F65)},
      MRNUMBER = {2022373},
      MRREVIEWER = {Paul Jolissaint},
      DOI = {10.1090/S0002-9939-03-07127-2},
      ZBLNUMBER = {1041.46045},
      }
  • [Oz04] Go to document N. Ozawa, "A Kurosh-type theorem for type $ II_1$ factors," Int. Math. Res. Not., vol. 2006, p. I, 2006.
    @article {Oz04, MRKEY = {2211141},
      AUTHOR = {Ozawa, Narutaka},
      TITLE = {A {K}urosh-type theorem for type {$\rm II\sb 1$} factors},
      JOURNAL = {Int. Math. Res. Not.},
      FJOURNAL = {International Mathematics Research Notices},
      YEAR = {2006},
      PAGES = {Art. ID 97560},
      ISSN = {1073-7928},
      MRCLASS = {46L10 (20F67 46L09 46L35 46L55)},
      MRNUMBER = {2211141},
      MRREVIEWER = {Claire Anantharaman-Delaroche},
      DOI = {10.1155/IMRN/2006/97560},
      VOLUME = {2006},
      ZBLNUMBER = {1114.46041},
     }
  • [OP07] Go to document N. Ozawa and S. Popa, "On a class of ${ II}_1$ factors with at most one Cartan subalgebra," Ann. of Math., vol. 172, iss. 1, pp. 713-749, 2010.
    @article {OP07, MRKEY = {2680430},
      AUTHOR = {Ozawa, Narutaka and Popa, Sorin},
      TITLE = {On a class of {${\rm II}\sb 1$} factors with at most one {C}artan subalgebra},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {172},
      YEAR = {2010},
      NUMBER = {1},
      PAGES = {713--749},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {46L10 (37A20)},
      MRNUMBER = {2680430},
      MRREVIEWER = {Stuart A. White},
      DOI = {10.4007/annals.2010.172.713},
      ZBLNUMBER = {1201.46054},
      }
  • [Pe06] Go to document J. Peterson, "$L^2$-rigidity in von Neumann algebras," Invent. Math., vol. 175, iss. 2, pp. 417-433, 2009.
    @article {Pe06, MRKEY = {2470111},
      AUTHOR = {Peterson, Jesse},
      TITLE = {{$L\sp 2$}-rigidity in von {N}eumann algebras},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {175},
      YEAR = {2009},
      NUMBER = {2},
      PAGES = {417--433},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {46L10 (46L54 46L55 46L57)},
      MRNUMBER = {2470111},
      MRREVIEWER = {Narutaka Ozawa},
      DOI = {10.1007/s00222-008-0154-6},
      ZBLNUMBER = {1170.46053},
      }
  • [Po01a] Go to document S. Popa, "Some rigidity results for non-commutative Bernoulli shifts," J. Funct. Anal., vol. 230, iss. 2, pp. 273-328, 2006.
    @article {Po01a, MRKEY = {2186215},
      AUTHOR = {Popa, Sorin},
      TITLE = {Some rigidity results for non-commutative {B}ernoulli shifts},
      JOURNAL = {J. Funct. Anal.},
      FJOURNAL = {Journal of Functional Analysis},
      VOLUME = {230},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {273--328},
      ISSN = {0022-1236},
      CODEN = {JFUAAW},
      MRCLASS = {46L10 (37A20 46L40 46L55)},
      MRNUMBER = {2186215},
      MRREVIEWER = {Stefaan Vaes},
      DOI = {10.1016/j.jfa.2005.06.017},
      ZBLNUMBER = {1097.46045},
      }
  • [Po01b] Go to document S. Popa, "On a class of type ${ II}_1$ factors with Betti numbers invariants," Ann. of Math., vol. 163, iss. 3, pp. 809-899, 2006.
    @article {Po01b, MRKEY = {2215135},
      AUTHOR = {Popa, Sorin},
      TITLE = {On a class of type {${\rm II}\sb 1$} factors with {B}etti numbers invariants},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {163},
      YEAR = {2006},
      NUMBER = {3},
      PAGES = {809--899},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {46L10 (37A20)},
      MRNUMBER = {2215135},
      MRREVIEWER = {Paul Jolissaint},
      DOI = {10.4007/annals.2006.163.809},
      ZBLNUMBER = {1120.46045},
      }
  • [Po03] Go to document S. Popa, "Strong rigidity of $ II_1$ factors arising from malleable actions of $w$-rigid groups. I," Invent. Math., vol. 165, iss. 2, pp. 369-408, 2006.
    @article {Po03, MRKEY = {2231961},
      AUTHOR = {Popa, Sorin},
      TITLE = {Strong rigidity of {$\rm II\sb 1$} factors arising from malleable actions of {$w$}-rigid groups. {I}},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {165},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {369--408},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {46L10 (22D25 37A20 46L55)},
      MRNUMBER = {2231961},
      MRREVIEWER = {Alain Valette},
      DOI = {10.1007/s00222-006-0501-4},
      ZBLNUMBER = {1120.46043},
      }
  • [Po04] Go to document S. Popa, "Strong rigidity of $ II_1$ factors arising from malleable actions of $w$-rigid groups. II," Invent. Math., vol. 165, iss. 2, pp. 409-451, 2006.
    @article {Po04, MRKEY = {2231962},
      AUTHOR = {Popa, Sorin},
      TITLE = {Strong rigidity of {$\rm II\sb 1$} factors arising from malleable actions of {$w$}-rigid groups. {II}},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {165},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {409--451},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {46L55 (22D25 37A20 37A55 46L10)},
      MRNUMBER = {2231962},
      MRREVIEWER = {Alain Valette},
      DOI = {10.1007/s00222-006-0502-3},
      ZBLNUMBER = {1120.46044},
      }
  • [Po05] Go to document S. Popa, "Cocycle and orbit equivalence superrigidity for malleable actions of $w$-rigid groups," Invent. Math., vol. 170, iss. 2, pp. 243-295, 2007.
    @article {Po05, MRKEY = {2342637},
      AUTHOR = {Popa, Sorin},
      TITLE = {Cocycle and orbit equivalence superrigidity for malleable actions of {$w$}-rigid groups},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {170},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {243--295},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {37A20 (28D15 37A15 37A35 46L55)},
      MRNUMBER = {2342637},
      MRREVIEWER = {Alain Valette},
      DOI = {10.1007/s00222-007-0063-0},
      ZBLNUMBER = {1131.46040},
      }
  • [Po06a] Go to document S. Popa, "On the superrigidity of malleable actions with spectral gap," J. Amer. Math. Soc., vol. 21, iss. 4, pp. 981-1000, 2008.
    @article {Po06a, MRKEY = {2425177},
      AUTHOR = {Popa, Sorin},
      TITLE = {On the superrigidity of malleable actions with spectral gap},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {21},
      YEAR = {2008},
      NUMBER = {4},
      PAGES = {981--1000},
      ISSN = {0894-0347},
      MRCLASS = {46L35 (22D25 22D40 28D15 37A20)},
      MRNUMBER = {2425177},
      MRREVIEWER = {Claire Anantharaman-Delaroche},
      DOI = {10.1090/S0894-0347-07-00578-4},
      ZBLNUMBER = {1222.46048},
      }
  • [Po06b] Go to document S. Popa, "Deformation and rigidity for group actions and von Neumann algebras," in International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Zürich, 2007, pp. 445-477.
    @incollection {Po06b, MRKEY = {2334200},
      AUTHOR = {Popa, Sorin},
      TITLE = {Deformation and rigidity for group actions and von {N}eumann algebras},
      BOOKTITLE = {International {C}ongress of {M}athematicians. {V}ol. {I}},
      PAGES = {445--477},
      PUBLISHER = {Eur. Math. Soc., Zürich},
      YEAR = {2007},
      MRCLASS = {46L35 (22D25 28D15 37A20 37A55 46L55)},
      MRNUMBER = {2334200},
      MRREVIEWER = {Bachir Bekka},
      DOI = {10.4171/022-1/18},
      ZBLNUMBER = {1132.46038},
      }
  • [PV81] M. Pimsner and D. Voiculescu, "$K$-groups of reduced crossed products by free groups," J. Operator Theory, vol. 8, iss. 1, pp. 131-156, 1982.
    @article {PV81, MRKEY = {0670181},
      AUTHOR = {Pimsner, M. and Voiculescu, D.},
      TITLE = {{$K$}-groups of reduced crossed products by free groups},
      JOURNAL = {J. Operator Theory},
      FJOURNAL = {Journal of Operator Theory},
      VOLUME = {8},
      YEAR = {1982},
      NUMBER = {1},
      PAGES = {131--156},
      ISSN = {0379-4024},
      MRCLASS = {46L05 (16A54 46M20)},
      MRNUMBER = {0670181},
      MRREVIEWER = {Vern Paulsen},
      ZBLNUMBER = {0533.46045},
      }
  • [PV06] Go to document S. Popa and S. Vaes, "Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups," Adv. Math., vol. 217, iss. 2, pp. 833-872, 2008.
    @article {PV06, MRKEY = {2370283},
      AUTHOR = {Popa, Sorin and Vaes, Stefaan},
      TITLE = {Strong rigidity of generalized {B}ernoulli actions and computations of their symmetry groups},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {217},
      YEAR = {2008},
      NUMBER = {2},
      PAGES = {833--872},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {37A15 (22F10 46L10 46L55)},
      MRNUMBER = {2370283},
      MRREVIEWER = {Alain Valette},
      DOI = {10.1016/j.aim.2007.09.006},
      ZBLNUMBER = {1137.37003},
      }
  • [PV08] S. Popa and S. Vaes, "Cocycle and orbit superrigidity for lattices in ${ SL}(n,\Bbb R)$ acting on homogeneous spaces," in Geometry, Rigidity, and Group Actions, Farb, B. and Fisher, D., Eds., Chicago, IL: Univ. Chicago Press, 2011, pp. 419-451.
    @incollection {PV08, MRKEY = {2807839},
      AUTHOR = {Popa, Sorin and Vaes, Stefaan},
      TITLE = {Cocycle and orbit superrigidity for lattices in {${\rm SL}(n,\Bbb R)$} acting on homogeneous spaces},
      BOOKTITLE = {Geometry, Rigidity, and Group Actions},
      SERIES = {Chicago Lectures in Math.},
      EDITOR={Farb, B. and Fisher, D.},
      PAGES = {419--451},
      PUBLISHER = {Univ. Chicago Press},
      ADDRESS = {Chicago, IL},
      YEAR = {2011},
      MRCLASS = {37A20 (22D10 22F10 46L36)},
      MRNUMBER = {2807839},
      MRREVIEWER = {Alain Valette},
      ZBLNUMBER = {06077154},
      }
  • [PV09] Go to document S. Popa and S. Vaes, "Group measure space decomposition of ${ II}_1$ factors and $W^\ast$-superrigidity," Invent. Math., vol. 182, iss. 2, pp. 371-417, 2010.
    @article {PV09, MRKEY = {2729271},
      AUTHOR = {Popa, Sorin and Vaes, Stefaan},
      TITLE = {Group measure space decomposition of {${\rm II}\sb 1$} factors and {$W\sp \ast$}-superrigidity},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {182},
      YEAR = {2010},
      NUMBER = {2},
      PAGES = {371--417},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {46L10 (37A25 37A55)},
      MRNUMBER = {2729271},
      MRREVIEWER = {Gabriel Picioroaga},
      DOI = {10.1007/s00222-010-0268-5},
      ZBLNUMBER = {1238.46052},
      }
  • [Ri87] M. A. Rieffel, "The homotopy groups of the unitary groups of noncommutative tori," J. Operator Theory, vol. 17, iss. 2, pp. 237-254, 1987.
    @article {Ri87, MRKEY = {0887221},
      AUTHOR = {Rieffel, Marc A.},
      TITLE = {The homotopy groups of the unitary groups of noncommutative tori},
      JOURNAL = {J. Operator Theory},
      FJOURNAL = {Journal of Operator Theory},
      VOLUME = {17},
      YEAR = {1987},
      NUMBER = {2},
      PAGES = {237--254},
      ISSN = {0379-4024},
      MRCLASS = {22D25 (46L55 46L80)},
      MRNUMBER = {0887221},
      MRREVIEWER = {Gustavo Corach},
      ZBLNUMBER = {0656.46056},
     }
  • [Sc63] Go to document J. Schwartz, "Two finite, non-hyperfinite, non-isomorphic factors," Comm. Pure Appl. Math., vol. 16, pp. 19-26, 1963.
    @article {Sc63, MRKEY = {0149322},
      AUTHOR = {Schwartz, J.},
      TITLE = {Two finite, non-hyperfinite, non-isomorphic factors},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {16},
      YEAR = {1963},
      PAGES = {19--26},
      ISSN = {0010-3640},
      MRCLASS = {46.65},
      MRNUMBER = {0149322},
      MRREVIEWER = {E. L. Griffin, Jr.},
      ZBLNUMBER = {0131.33201},
      DOI = {10.1002/cpa.3160160104},
     }
  • [Sh00] Go to document Y. Shalom, "Rigidity of commensurators and irreducible lattices," Invent. Math., vol. 141, iss. 1, pp. 1-54, 2000.
    @article {Sh00, MRKEY = {1767270},
      AUTHOR = {Shalom, Yehuda},
      TITLE = {Rigidity of commensurators and irreducible lattices},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {141},
      YEAR = {2000},
      NUMBER = {1},
      PAGES = {1--54},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {22E40 (20E08 20F65 20G10)},
      MRNUMBER = {1767270},
      MRREVIEWER = {Alain Valette},
      DOI = {10.1007/s002220000064},
      ZBLNUMBER = {0978.22010},
      }
  • [Va06] S. Vaes, "Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa)," in Séminaire Bourbaki. Vol. 2005/2006, , 2007, vol. 311, p. exp. no. 961, viii, 237-294.
    @incollection{Va06, MRKEY = {2359046},
      AUTHOR = {Vaes, Stefaan},
      TITLE = {Rigidity results for {B}ernoulli actions and their von {N}eumann algebras (after {S}orin {P}opa)},
      BOOKTITLE = {S{é}minaire Bourbaki. Vol. 2005/2006},
      SERIES = {Astérisque},
      FJOURNAL = {Astérisque},
      VOLUME = {311},
      YEAR = {2007},
      PAGES = {Exp. No. 961, viii, 237--294},
      ISSN = {0303-1179},
      ISBN = {978-2-85629-230-3},
      MRCLASS = {46L35 (28D15 37A20 37A55 37L10)},
      MRNUMBER = {2359046},
      MRREVIEWER = {Narutaka Ozawa},
      ZBLNUMBER = {1194.46085},
     }
  • [Va07] Go to document S. Vaes, "Explicit computations of all finite index bimodules for a family of ${ II}_1$ factors," Ann. Sci. Éc. Norm. Supér., vol. 41, iss. 5, pp. 743-788, 2008.
    @article {Va07, MRKEY = {2504433},
      AUTHOR = {Vaes, Stefaan},
      TITLE = {Explicit computations of all finite index bimodules for a family of {${\rm II}\sb 1$} factors},
      JOURNAL = {Ann. Sci. Éc. Norm. Supér.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {41},
      YEAR = {2008},
      NUMBER = {5},
      PAGES = {743--788},
      ISSN = {0012-9593},
      MRCLASS = {46L10 (46L37 46L40)},
      MRNUMBER = {2504433},
      MRREVIEWER = {Mart{\'ı}n Argerami},
      ZBLNUMBER = {1194.46086},
      URL = {http://smf4.emath.fr/en/Publications/AnnalesENS/4_41/html/ens_ann-sc_41_743-788.html},
     }
  • [Va10] Go to document S. Vaes, "One-cohomology and the uniqueness of the group measure space decomposition of a ${ II}_1$ factor," Math. Ann., vol. 355, iss. 2, pp. 661-696, 2013.
    @article {Va10, MRKEY = {3010143},
      AUTHOR = {Vaes, Stefaan},
      TITLE = {One-cohomology and the uniqueness of the group measure space decomposition of a {${\rm II}\sb 1$} factor},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {355},
      YEAR = {2013},
      NUMBER = {2},
      PAGES = {661--696},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {Preliminary Data},
      MRNUMBER = {3010143},
      DOI = {10.1007/s00208-012-0797-x},
      ZBLNUMBER = {06133904},
      }
  • [Va11] Go to document S. Vaes, $W^\ast$-superrigidity for Bernoulli actions and wreath product group von~Neumann algebras.
    @misc{Va11, KEY={Vae11},
      SORTYEAR={2011},
      AUTHOR={Vaes, Stefaan},
      TITLE={$W^\ast$-superrigidity for {B}ernoulli actions and wreath product group von~{N}eumann algebras},
      NOTE={Lecture notes for a course at the Institut Henri Poincaré,
      Paris, May 2011, during the program on Von Neumann Algebras and Ergodic Theory of Group Actions},
      URL={www.wis.kuleuven.be/analyse/stefaan},
     }
  • [Vo89] D. Voiculescu, "Circular and semicircular systems and free product factors," in Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, Boston, MA: Birkhäuser, 1990, vol. 92, pp. 45-60.
    @incollection {Vo89, MRKEY = {1103585},
      AUTHOR = {Voiculescu, Dan},
      TITLE = {Circular and semicircular systems and free product factors},
      BOOKTITLE = {Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory},
      VENUE={{P}aris, 1989},
      SERIES = {Progr. Math.},
      VOLUME = {92},
      PAGES = {45--60},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {1990},
      MRCLASS = {46L35},
      MRNUMBER = {1103585},
      MRREVIEWER = {Marie Choda},
      ZBLNUMBER = {0744.46055},
      }

Authors

Adrian Ioana

Department of Mathematics
University of California, San Diego
La Jolla, CA 92093

Sorin Popa

Department of Mathematics
University of California at Los Angeles
Los Angeles, CA 90095-1555

Stefaan Vaes

Department of Mathematics
KU Leuven
Celestijnenlaan 200B
B-3001 Leuven
Belgium