Jones Index Theory for C*-Algebras

  • Yasuo Watatani
Part of the Progress in Mathematics book series (PM, volume 84)


The notion of index [M : N] was introduced by Jones [13] as an invariant for subfactors N of a factor M of type II1. Subsequently Kosaki [18] defined an index E for a conditional expectation E of an arbitrary factor M onto a subfactor N using the spatial theory of Connes [6] and the theory of operator-valued weights of Haagerup [9]. We shall define an index E for a conditional expectation E on a C*-algebra. This index theory for C*-algebras is a mixture of the index theory by Jones and the theory of Morita equivalence by Rieffel [24], [25]. We establish the link between transfer in K -theory and a multiplication by Index E.




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  1. [1]
    J. Bion-Nadal, Von Neumann subalgebras of type II1 factors, correspondences and property T, preprint.Google Scholar
  2. [2]
    B. Blackadar, K-Theory for Operator Algebras, MSRI Publication 5, Springer-Verlag, Berlin, (1986).MATHCrossRefGoogle Scholar
  3. [3]
    M. Brillet, Y. Demizeau and F.F. Hauvet, Indice d’une espérance conditionnelle, Compositio Math., 66 (1988), 199–236.MathSciNetGoogle Scholar
  4. [4]
    M. Choda, Full II1-factors with non-integer index, preprint.Google Scholar
  5. [5]
    M. Choda, Index for factors generated by Jones two-sided sequence of projections, Pacific J. Math., 139 (1989), 1–16.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    A. Connes, Spatial theory of von Neumann algebras, J. Funct. Anal. 35 (1980), 153–164.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    E.G. Effros and C-L. Shen, Approximately finite dimensional C*-algebras and continued fractions, Indiana J. Math. 29 (1980), 191–204.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    F.M. Goodman, P. de la Harpe and V. Jones, Coxeter-Dynkin diagrams and towers of algebras, MSRI Publication 14, Springer-Verlag, Berlin (1989).Google Scholar
  9. [9]
    U. Haagerup, Operator valued weights in von Neumann algebras I, J. Funct. Anal. 32 (1979), 175–206; II, J. Func. Anal. 33 (1979), 339-361.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    T. Hamachi and H. Kosaki, Index and flow of weights of factors of type III, Proc. Japan Academy, Ser. A 64 (1988), 11–13.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    T. Hamachi and H. Kosaki, Inclusion of type III factors constructed from ergodic flows, preprint.Google Scholar
  12. [12]
    F. Hiai, Minimizing indices of conditional expectations onto a subfactor, Publ. RIMS, Kyoto Univ., 24 (1988), 673–678.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    V. Jones, Index for subfactors, Invent. Math. 72 (1983), 1–25.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    V. Jones, Index for subrings of rings, Contemp. Math. 43 (Am. Math. Soc. 1985), 181–190.CrossRefGoogle Scholar
  15. [15]
    V. Jones, Braid Groups, Hecke Algebras and Type II1 Factors in Geometric Methods in Operator Algebras, Pitman Research Notes in Mathematics Series 123, (1986), 242–273.Google Scholar
  16. [16]
    S. Kawakami and H. Yoshida, Finite group actions on finite von Neumann algebras and the relative entropy, J. Math. Soc. Japan 39 (1989), 609–626.MathSciNetCrossRefGoogle Scholar
  17. [17]
    S. Kawakami and H. Yoshida, The constituents of Jones’s index analyzed from the structure of the Galois group, Math. Japon. 33 (1988), 551–557.MathSciNetMATHGoogle Scholar
  18. [18]
    H. Kosaki, Extension of Jones’ theory on index to arbitrary factors, J. Funct. Anal. 66 (1986), 123–140.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    P.H. Loi, Sur la théorie de l’indice et les facteurs de type III, C.R. Acad. Sci. Paris, 305 (1987), 423–426.MathSciNetMATHGoogle Scholar
  20. [20]
    M. Nagisa and G. Song, Inheritance of the solvability of the similarity problem within a C*-algebra and its C*-subalgebras, Math. Japon. 34 (1989), 73–80.MathSciNetMATHGoogle Scholar
  21. [21]
    A. Ocneanu, Quantized groups, string algebras and Galois theory for algebras, London Math. Soc, Lecture Note 136 (1988), 119–172, Cambridge Univ. Press.MathSciNetGoogle Scholar
  22. [22]
    M. Pimsner and S. Popa, Entropy and index for subfactors, Ann. Sci. Ecole Norm. Sup. 19 (1986), 57–106.MathSciNetMATHGoogle Scholar
  23. [23]
    M.A. Rieffei, C*-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415–429.MathSciNetCrossRefGoogle Scholar
  24. [24]
    M.A. Rieffel, Morita equivalence for operator algebras, in Proc. Symposium Pure Math. 38, Part 1, 285–298.Google Scholar
  25. [25]
    M.A. Rieffel, Applications of strong Morita equivalence to transformation group C*-algebras, in Proc. Symposium Pure Math. 38, Part 1, 299–310.Google Scholar
  26. [26]
    S. Sakai, C*-Algebras and W*-Algebras, Ergebnisse der Math. vol. 60 (1971), Berlin-Heidelberg-New York.Google Scholar
  27. [27]
    M. Takesaki, Conditional expectation on von Neumann algebras, J. Func. Anal. 9 (1972), 306–321.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    J. Tomiyama, On the projection of norm one in W*-algebras, I, Proc. Japan Acad. 33 (1957), 608–612; II, Tohoku Math. J. 10 (1958), 204-209; III, ibid., 11 (1959), 125-129.MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    H. Umegaki, Conditional expectation in an operator algebra I, Tohoku Math. J. 6 (1954), 177–181; II, 8 (1956), 86-100, III, Kodai Math. Sem. Rep. 11 (1959), 51-74; IV, 14 (1962), 59-85.MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    Y. Watatani, L’indice d’une C*-sous-algè bre d’une C*-algè bre simple, C.R. Acad. Sci. Paris, 305, Serie 1, (1987), 23–26.MathSciNetMATHGoogle Scholar
  31. [31]
    H. Wenzl, Representations of Hecke algebras and subfactors, Thesis, Univ. of Pennsylvania, 1985.Google Scholar
  32. [32]
    H. Wenzl, On sequences of projections, C.R. Math. Rep. Acad. Sci. Canada 9 (1987), 5–9.MathSciNetMATHGoogle Scholar
  33. [33]
    H. Yoshida, On crossed products and relative entropy, preprintGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Yasuo Watatani
    • 1
  1. 1.Osaka Kyoiku UniversityTennoji, OsakaJapan

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