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Classification of amenable subfactors of type II

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References

  • [A]Avez, A., Entropie des groupes de type fini.C. R. Acad. Sci. Paris Sér. I Math., 275 (1972), 1363–1366.

    MATH  MathSciNet  Google Scholar 

  • [AD]Anantharam-Delaroche, C., Correspondences and related properties of subfactors. Preprint, 1992.

  • [BDH]Baillet, M., Denizeau, Y. &Havet, J., Indice d'une esperance conditionelle.Compositio Math., 66 (1988), 199–236.

    MathSciNet  MATH  Google Scholar 

  • [Bi1]Bisch, D., Entropy of groups and subfactors.J. Funct. Anal., 103 (1992), 190–208.

    Article  MATH  MathSciNet  Google Scholar 

  • [Bi2] —, On the existence of central sequences in subfactors.Trans. Amer. Math. Soc., 321 (1990), 117–128.

    Article  MATH  MathSciNet  Google Scholar 

  • [Bi3]Bisch, D., A note on intermediate subfactors. To appear inPacific J. Math.

  • [BN]Bion-Nadal, J., Subfactors associated toE 6.J. Operator Theory, 28 (1992), 27–50.

    MATH  MathSciNet  Google Scholar 

  • [C1]Connes, A., Une classification des facteurs de type III.Ann. Sci. École Norm. Sup. (4), 6 (1973), 133–252.

    MATH  MathSciNet  Google Scholar 

  • [C2] —, Outer conjugacy classes of automorphisms of factors.Ann. Sci. École Norm. Sup. (4), 8 (1975), 383–419.

    MATH  MathSciNet  Google Scholar 

  • [C3] —, Classification of injective factors.Ann. of Math., 104 (1976), 73–115.

    Article  MATH  MathSciNet  Google Scholar 

  • [C4] —, Classification des facteurs.Proc. Sympos. Pure Math., 38:2 (1982), 43–109.

    MATH  MathSciNet  Google Scholar 

  • [C5] —, Periodic automorphisms of the hyperfinite type II1 factors.Acta. Sci. Math. (Szeged), 39 (1977), 39–66.

    MATH  MathSciNet  Google Scholar 

  • [C6] —, Compact metric spaces, Fredholm modules and hyperfiniteness.Pacific J. Math., 137 (1989), 120–135.

    MathSciNet  Google Scholar 

  • [Cho]Choda, M., The conjugacy classes of subfactors and the outer conjugacy classes of the automorphism group.Math. Japon., 32 (1987), 379–388.

    MATH  MathSciNet  Google Scholar 

  • [Chr]Christensen, E., Subalgebras of finite algebras.Math. Ann., 243 (1979), 17–29.

    Article  MATH  MathSciNet  Google Scholar 

  • [CS]Connes, A. &Størmer, E., Entropy for automorphisms of II1 von Neumann algebras.Acta Math., 134 (1975), 288–306.

    Google Scholar 

  • [CT]Connes, A. &Takesaki, M., The flow of weights on a factor of type III.Tôhoku Math. J., 29 (1977), 473–575.

    MathSciNet  MATH  Google Scholar 

  • [D]Dixmier, J.,Les algèbres d'opérateurs sur l'espace hilbertien (algèbres de von Neumann). Gauthier-Villars, Paris, 1957, 1969.

    Google Scholar 

  • [DHR]Doplicher, S., Haag, R. &Roberts, J., Local observables and particle statistics.Comm. Math. Phys., 23 (1971), 199–230.

    Article  MathSciNet  Google Scholar 

  • [E]Effros, E., Amenability and virtual diagonals for von Neumann algebras.J. Funct. Anal., 78 (1988), 137–154.

    Article  MATH  MathSciNet  Google Scholar 

  • [EvKa]Evans, D. & Kawahigashi, Y., Orbifold subfactors from Hecke algebras. Preprint, 1992.

  • [FeM]Feldman, J. &Moore, C. C., Ergodic equivalence relations, cohomology and von Neumann algebras I and II.Trans. Amer. Math. Soc., 236 (1977), 289–361.

    Article  MathSciNet  Google Scholar 

  • [Fr]Frölich, J., Statistics of fields, the Yang-Baxter equation and the theory of knots and links, inNon-Perturbative Quantum Field Theory. 1987 Cargese Lectures. Plenum Press, New York, 1988.

    Google Scholar 

  • [FRS]Fredenhagen, K., Rehren, K. &Schroer B., Superselection sectors, with braid group statistics and exchange algebras I: General theory.Comm. Math. Phys., 125 (1989), 201–230.

    Article  MathSciNet  MATH  Google Scholar 

  • [GHJ]Goodman, F., De la Harpe, P. & Jones, V. F. R.,Coxeter Graphs and Towers of Algebras. Math. Sci. Res. Inst. Publ. 14. Springer-Verlag, 1989.

  • [H]Haagerup, U., In preparation.

  • [HaG]Handelman D. & Gould, J., Positive multiplications in difference equations. Preprint, 1991.

  • [Hi]Hiai, F., Minimizing indices of conditional expectations onto a subfactor.Publ. Res. Inst. Math. Sci., 24 (1988), 673–678.

    MATH  MathSciNet  Google Scholar 

  • [HSc]Haagerup, U. & Schou, J., Some new subfactors of the hyperfinite type II1 factor. Preprint.

  • [Iz]Izumi, M., Application of fusion rules to classification of subfactors.Publ. Res. Inst. Math. Sci., 27 (1991), 953–994.

    MATH  MathSciNet  Google Scholar 

  • [IzKa]Izumi, M. &Kawahigashi, Y., Classification of subfactors with principal graphD (1)n .J. Funct. Anal., 112 (1993), 257–286.

    Article  MathSciNet  MATH  Google Scholar 

  • [J1]Jones, V. F. R.,Actions of Finite Groups on the Hyperfinite Type II 1 Factor. Mem. Amer. Math. Soc., 237. Providence, RI, 1980.

  • [J2] — Index for subfactors.Invent. Math., 72 (1983), 1–25.

    Article  MATH  MathSciNet  Google Scholar 

  • [J3]Jones, V. F. R., Subfactors of II1 factors and related topics, inProceedings of the International Congress of Mathematicians, Berkeley, 1986, pp. 939–947.

  • [J4] — Hecke algebra representation of braid groups and link polynomials.Ann. of Math., 126 (1987), 335–388.

    Article  MATH  MathSciNet  Google Scholar 

  • [K]Kadison, R. V., Normality in operator algebras.Duke Math. J., 29 (1962), 459–464.

    Article  MATH  MathSciNet  Google Scholar 

  • [Ka]Kawahigashi, Y., On flatness of Ocneanu's connection on the Dynkin diagrams and classification of subfactors. Preprint, 1990.

  • [KaiVe]Kaimanovitch, V. &Vershik, A., Random walks on discrete groups: boundary and entropy.Ann. Probab., 11 (1983), 457–490.

    MathSciNet  Google Scholar 

  • [Ke]Kesten, H., Symmetric random walks on groups.Trans. Amer. Math. Soc., 92 (1959), 336–354.

    Article  MATH  MathSciNet  Google Scholar 

  • [Ko]Kosaki, H., Extension of Jones theory on index to arbitrary factors.J. Funct. Anal., 66 (1986), 123–140.

    Article  MATH  MathSciNet  Google Scholar 

  • [KoL]Kosaki, H. & Longo, R., A remark on the minimal index for subfactors. Preprint, 1991.

  • [L1]Longo, R., Index of subfactors and statistics of quantum fields I.Comm. Math. Phys., 126 (1989), 217–247.

    Article  MATH  MathSciNet  Google Scholar 

  • [L2] —, Index of subfactors and statistics of quantum fields II.Comm. Math. Phys., 130 (1990), 285–309.

    MATH  MathSciNet  Google Scholar 

  • [L3] —, Minimal index and braided subfactors.J. Funct. Anal., 109 (1992), 98–112.

    Article  MATH  MathSciNet  Google Scholar 

  • [Lo]Loi, P., On the theory of index and type III factors. Thesis, 1988

  • [MvN]Murray, F. &von Neumann, J., Rings of operators IV.Ann. of Math., 44 (1943), 716–808.

    Article  MathSciNet  Google Scholar 

  • [Oc1]Ocneanu, A.,Actions of Discrete Amenable Groups on von Neumann Algebras. Lecture Notes in Math., 1138. Springer-Verlag, Berlin-Heidelberg-New York, 1985.

    Google Scholar 

  • [Oc2]Ocneanu, A., Quantized groups, string algebras, and Galois theory for von Neumann algebras, inOperator Algebras and Applications. London Math. Soc. Lecture Note Ser., 136, pp. 119–172. Cambridge Univ. Press, 1988.

  • [Ocetal]Ocneanu, A., Freyd, P., Yetter, D., Hoste, J., Likorish, W. &Millet, K., A new polynomial invariant of knots and links.Bull. Amer. Math. Soc., 12 (1985), 183–189.

    Article  MathSciNet  Google Scholar 

  • [Ok]Okamoto, S., Invariants for subfactors arizing from Coxeter graphs. Thesis.

  • [Or]Orey, S.,Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand Reinhold Math. Studies, 34. London, 1971.

  • [PiPo1]Pimsner, M. &Popa, S., Entropy and index for subfactors.Ann. Sci. École Norm. Sup. (4), 19 (1986), 57–106.

    MathSciNet  MATH  Google Scholar 

  • [PiPo2] —, Iterating the basic construction.Trans. Amer. Math. Soc., 310 (1988), 127–133.

    Article  MathSciNet  MATH  Google Scholar 

  • [PiPo3] —, Finite dimensional approximation for pairs of algebras and obstructions for the index.J. Funct. Anal., 98 (1991), 270–291.

    Article  MathSciNet  MATH  Google Scholar 

  • [Po1]Popa, S., On a problem of R. V. Kadison on maximal abelian *-subalgebras in factors.Invent. Math., 65 (1981), 269–281.

    Article  MATH  MathSciNet  Google Scholar 

  • [Po2] —, Singular maximal abelian *-subalgebras in continuous von Neumann algebras.J. Funct. Anal., 50 (1983), 151–166.

    Article  MATH  MathSciNet  Google Scholar 

  • [Po3] —, Maximal injective subalgebras in factors associated with free groups.Adv. In Math., 50 (1983), 27–48.

    Article  MATH  MathSciNet  Google Scholar 

  • [Po4]Popa, S., Correspondences. Preprint, 1986.

  • [Po5] —, Classification of subfactors: reduction to commuting squares.Invent. Math., 101 (1990), 19–43.

    Article  MATH  MathSciNet  Google Scholar 

  • [Po6] —, Sous-facteurs, actions des groupes et cohomologie.C. R. Acad. Sci. Paris Sér I Math., 309 (1989), 771–776.

    MATH  Google Scholar 

  • [Po7] —, Sur la classification des sous-facteurs d'indice fini du facteur hyperfini.C. R. Acad. Sci. Paris Sér I Math., 311 (1990), 95–100.

    MATH  Google Scholar 

  • [Po8] —, Relative dimension, towers of projections and commuting squares.Pacific J. Math., 137 (1989), 181–207.

    MATH  MathSciNet  Google Scholar 

  • [Po9]Popa, S., Classification of actions of discrete amenable groups on subfactors. I.H.E.S. preprint, 1992.

  • [Po10]Popa, S., On amenable factors of type II1, inOperator Algebras and Applications. London Math. Soc. Lecture Note Ser., 136, pp. 173–184. Cambridge Univ. Press, 1988.

  • [Po11]Popa, S., Hyperfinite subalgebras normalized by a given automorphism and related topics, inLecture Notes in Math., 1132, pp. 421–433. Springer-Verlag, 1984.

  • [Po12]Popa, S., Some equivalent characterizations of amenability for inclusions of factors. To appear.

  • [PoWa]Popa, S. &Wassermann, A., Actions of compact Lie groups on von Neumann algebras.C. R. Acad. Sci. Paris Sér I Math., 315 (1992), 421–426.

    MathSciNet  MATH  Google Scholar 

  • [PS]Powers, R. &Størmer, E., Free states of the canonical anticommutation relations.Comm. Math. Phys., 16 (1970), 1–33.

    Article  MathSciNet  MATH  Google Scholar 

  • [Sk]Skau, C., Finite subalgebras of a von Neumann algebra.J. Funct. Anal., 25 (1977), 211–235.

    Article  MATH  MathSciNet  Google Scholar 

  • [StZ]Stratila, S. &Zsido, L.,Lectures on von Neumann Algebras, Abacus Press-Editura Academiei, London-Bucharest, 1979.

    MATH  Google Scholar 

  • [SuVi]Sunder, S. & Vijayavagan, A., On the non-occurence of the Coxeter graphsE 7,D 2n+1 as principal graphs of an inclusion of II1 factor. Preprint, 1991.

  • [T1]Takesaki, M., Duality for crossed products and the structure of von Neumann algebras of type III.Acta Math., 131 (1973), 249–310.

    MATH  MathSciNet  Google Scholar 

  • [T2] —, Conditional expectations in von Neumann algebras.J. Funct. Anal., 9 (1972), 306–321.

    Article  MATH  MathSciNet  Google Scholar 

  • [Wa1]Wassermann, A., Ergodic actions of compact groups on operator algebras III.Invent. Math., 9, (1988), 309–355.

    Article  MathSciNet  Google Scholar 

  • [Wa2]Wassermann, A., Coactions and Yang-Baxter equations for ergodic actions and subfactors, inOperator Algebras and Applications. London Math. Soc. Lecture Note Ser., 136, pp. 202–236. Cambridge Univ. Press, 1988.

  • [We1]Wenzl, H., Hecke algebras of typeA n and subfactors.Invent. Math., 92 (1988), 349–383.

    Article  MATH  MathSciNet  Google Scholar 

  • [We2] —, Quantum groups and subfactors of typeB, C, andD.Comm. Math. Phys., 133 (1990), 383–432.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of California (UCLA), 90024-1555, Los Angeles, CA, U.S.A.

    Sorin Popa

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This work was supported in part by an NSF Grant DMS-8908281.

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Popa, S. Classification of amenable subfactors of type II. Acta Math. 172, 163–255 (1994). https://doi.org/10.1007/BF02392646

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