Communications in Mathematical Physics

, Volume 319, Issue 3, pp 595–648 | Cite as

A Planar Calculus for Infinite Index Subfactors

Article

Abstract

We develop an analog of Jones’ planar calculus for II1-factor bimodules with arbitrary left and right von Neumann dimension. We generalize to bimodules Burns’ results on rotations and extremality for infinite index subfactors. These results are obtained without Jones’ basic construction and the resulting Jones projections.

References

  1. Bis97.
    Bisch, D.: Bimodules, higher relative commutants and the fusion algebra associated to a subfactor. In: Operator Algebras and Their Applications (Waterloo, ON, 1994/1995), Fields Inst. Commun., 13, Providence, RI: Amer. Math. Soc., 1997, pp. 13-63Google Scholar
  2. BMPS09.
    Bigelow, S., Morrison S., Peters E., Snyder N., Constructing the extended Haagerup planar algebra. Acta. Math. 209, 29–82 (2012)Google Scholar
  3. Bur03.
    Burns, M.: Subfactors, planar algebras, and rotations, Ph.D. thesis at the University of California, Berkeley, 2003, http://arxiv.org/abs/1111.1362v1 [math,OA], 2011
  4. Con74.
    Connes, A.: Caractérisation des espaces vectoriels ordonnés sous-jacents aux algèbres de von Neumann. Ann. Inst. Fourier (Grenoble) 24 , no. 4, x, 121–155 (1974)Google Scholar
  5. Con80.
    Connes A.: On the spatial theory of von Neumann algebras. J. Funct. Anal. 35(2), 153–164 (1980)MathSciNetMATHCrossRefGoogle Scholar
  6. EN96.
    Enock M., Nest R.: Irreducible inclusions of factors, multiplicative unitaries, and Kac algebras. J. Funct. Anal. 137(2), 466–543 (1996)MathSciNetMATHCrossRefGoogle Scholar
  7. EV00.
    Enock M., Vallin J.-M.: Inclusions of von Neumann algebras, and quantum groupoids. J. Funct. Anal. 172(2), 249–300 (2000)MathSciNetMATHCrossRefGoogle Scholar
  8. GdlHJ89.
    Goodman, F. M., de la Harpe, P., Jones, V.F.R.: Coxeter graphs and towers of algebras. Mathematical Sciences Research Institute Publications, 14. New York: Springer-Verlag, 1989Google Scholar
  9. GJS10.
    Guionnet, A., Jones, V.F.R., Shlyakhtenko, D.: Random matrices, free probability, planar algebras and subfactors. In: Quanta of maths, Clay Math. Proc., Vol. 11, Providence, RI: Amer. Math. Soc., 2010, pp. 201–239Google Scholar
  10. Haa79.
    Haagerup U.: Operator-valued weights in von Neumann algebras. I. J. Funct. Anal. 32(2), 175–206 (1979)MathSciNetMATHCrossRefGoogle Scholar
  11. HO89.
    Herman R.H., Ocneanu A.: Index theory and Galois theory for infinite index inclusions of factors. C. R. Acad. Sci. Paris Sér. I Math. 309(17), 923–927 (1989)MathSciNetMATHGoogle Scholar
  12. IJMS11.
    Izumi, M., Jones, V. F. R., Morrison, S., Snyder, N.: Subfactors of index less than 5, part 3: quadruple points. Commun. Math. Phys. 316, 531–554 (2012)Google Scholar
  13. ILP98.
    Izumi M., Longo R., Popa S.: A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras. J. Funct. Anal. 155(1), 25–63 (1998)MathSciNetMATHCrossRefGoogle Scholar
  14. Izu91.
    Izumi M.: Application of fusion rules to classification of subfactors. Publ. Res. Inst. Math. Sci. 27(6), 953–994 (1991)MathSciNetMATHCrossRefGoogle Scholar
  15. Jon83.
    Jones V.F.R.: Index for subfactors. Invent. Math. 72(1), 1–25 (1983)MathSciNetADSMATHCrossRefGoogle Scholar
  16. Jon99.
    Jones, V.F.R.:Planar algebras I. http://arxiv.org/abs/math/9909027v1 [math.QA], 1999
  17. JP11.
    Jones V.F.R., Penneys D.: The embedding theorem for finite depth subfactor planar algebras. Quantum Topol. 2(3), 301–337 (2011)MathSciNetMATHCrossRefGoogle Scholar
  18. Lie72.
    Lieberman A.: The structure of certain unitary representations of infinite symmetric groups. Trans. Amer. Math. Soc. 164, 189–198 (1972)MathSciNetCrossRefGoogle Scholar
  19. MPPS12.
    Morrison S., Penneys D., Peters E., Snyder N.: Subfactors of index less than 5, part 2: triple points. Internat. J. Math. 23(3), 1250016 (2012)MathSciNetCrossRefGoogle Scholar
  20. MS11.
    Morrison, S., Snyder, N.: Subfactors of index less than 5, part 1: the principal graph odometer. Commun. Math. Phys. 312, 1–35 (2012)Google Scholar
  21. Ocn88.
    Ocneanu, A.: Quantized groups, string algebras and Galois theory for algebras. In: Operator algebras and applications, Vol. 2, London Math. Soc. Lecture Note Ser., Vol. 136, Cambridge: Cambridge Univ. Press, 1988, pp. 119–172Google Scholar
  22. Pen12.
    Penneys D.: A cyclic approach to the annular Temperley-Lieb category. J. Knot Theory and its Ramifications 21(6), 1250049 (2012)MathSciNetCrossRefGoogle Scholar
  23. Pet10.
    Peters E.: A planar algebra construction of the Haagerup subfactor. Int’l J. Math. 21(8), 987–1045 (2010)MATHCrossRefGoogle Scholar
  24. Pop86.
    Popa, S.: Correspondences. INCREST Preprint, 1986.Google Scholar
  25. Pop93.
    Popa S.: Markov traces on universal Jones algebras and subfactors of finite index. Invent. Math. 111(2), 375–405 (1993)MathSciNetADSMATHCrossRefGoogle Scholar
  26. Pop94.
    Popa S.: Classification of amenable subfactors of type II. Acta Math. 172(2), 163–255 (1994)MathSciNetMATHCrossRefGoogle Scholar
  27. Pop95.
    Popa S.: An axiomatization of the lattice of higher relative commutants of a subfactor. Invent. Math. 120(3), 427–445 (1995)MathSciNetADSMATHCrossRefGoogle Scholar
  28. PT12.
    Penneys D., Tener J.: Subfactors of index less than 5, part 4: vines. Internat. J. Math. 23(3), 1250017 (2012)MathSciNetCrossRefGoogle Scholar
  29. Sau83.
    Sauvageot J.-L.: Sur le produit tensoriel relatif d’espaces de Hilbert. J. Operator Theory 9(2), 237–252 (1983)MathSciNetMATHGoogle Scholar
  30. Sau85.
    Sauvageot, J.-L.: Produits tensoriels de Z-modules et applications. In: Operator algebras and their connections with topology and ergodic theory (1983), Lecture Notes in Math., Vol. 1132, Berlin: Springer, 1985, pp. 468–485Google Scholar
  31. Tak03.
    Takesaki M., Theory of operator algebras. II. In: Encyclopaedia of Mathematical Sciences, Vol. 125, Berlin: Springer-Verlag, 2003Google Scholar
  32. Yam94.
    Yamagami S.: Modular theory for bimodule. J. Funct. Anal. 125(2), 327–357 (1994)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, BerkeleyBerkeleyUSA
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada

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