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Duality for crossed products and the structure of von Neumann algebras of type III

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References

  1. Araki, H., A classifiction of factors, II.Publications of Research Institute for Math. Sciences, Kyoto University, Ser. A, 4 (1968), 585–593.

    MathSciNet  Google Scholar 

  2. Araki, H.,Structure of some von Neumann algebras with isolated discrete modular spectrum. To appear.

  3. Araki, H. &Woods, J., A classification of factors.Publications of Research Institute for Math. Sciences, Kyoto University, Ser. A, 4 (1968), 51–130.

    MathSciNet  MATH  Google Scholar 

  4. Arveson, W.,On groups of automorphisms of operator algebras. To appear.

  5. Blattner, R. J., On induced representations.Amer. J. Math., 83 (1961), 79–98.

    MATH  MathSciNet  Google Scholar 

  6. —, On a theorem of G. W. Mackey.Bull. Amer. Math. Soc., 68 (1962), 585–587.

    Article  MATH  MathSciNet  Google Scholar 

  7. Bourbaki, N.,Intégration, Chap. 1–4, Paris (1952).

  8. Bourbaki, N.,Intégration, Chap. 7–8, Paris (1963).

  9. Combes, F., Poids associés à une algèbre hilbertienne à gauche.Comp. Math., 23 (1971), 49–77.

    MATH  MathSciNet  Google Scholar 

  10. Connes, A.,Une classification des facteurs de type III. Thesis, to appear.

  11. Dixmier, J.,Les algèbres d'opérateurs dans l'espace hilbertien, 2nd edition, Paris, Gauthier-Villars, (1969).

    MATH  Google Scholar 

  12. Dotlicher, S., Kaspler, D. &Robinson, D., Covariance algebras in field theory and statistical mechanics.Comm. Math. Phys., 3 (1966), 1–28.

    Article  MathSciNet  Google Scholar 

  13. Haga, Y.,On subalgebras of a crossed product von Neumann algebra. To appear.

  14. Haga, Y. & Takeda, Z., Correspondence between subgroups and subalgebras in a cross product von Neumann algebra. To appear inTôhoku Math. J.

  15. Herman, R. &Takesaki, M., States and automorphism groups of operator algebras.Comm. Math. Phys., 19 (1940), 142–160.

    Article  MathSciNet  Google Scholar 

  16. Ionescu Tulcea, A. andIonescu Tulcea, C.,Topics in the theory of lifting. Springer-Verlag, New York, (1969).

    MATH  Google Scholar 

  17. Ionescu Tulcea, A. andIonescu Tulcea, C.,On the existence of a lifting commuting with the left translations of an arbitrary locally compact groups, Proc. Fifth Berkeley Symp. in Math. Stat. and Prob. Univ. of California Press, (1967), 63–97.

  18. Loomis, L. H., Note on a theorem of Mackay.Duke Math. J., 19 (1952), 641–645.

    Article  MATH  MathSciNet  Google Scholar 

  19. Mackey, G. W., A theorem of Stone and von Neumann.Duke Math. J., 16 (1949), 313–326.

    Article  MATH  MathSciNet  Google Scholar 

  20. —, Induced representations of locally compact groups, I.Ann. Math., 55 (1952), 101–139.

    Article  MATH  MathSciNet  Google Scholar 

  21. —, Unitary representations of group extensions.Acta Math., 99 (1958), 265–311.

    MATH  MathSciNet  Google Scholar 

  22. Maréchal, O., Champs measureables d'espaces hilbertiens.Bull. Sc. Math., 93 (1969), 113–143.

    MATH  Google Scholar 

  23. McDuff, D., A countable infinity of II1 factors.Ann. Math., 90 (1969), 361–371.

    Article  MATH  MathSciNet  Google Scholar 

  24. —, Uncountably many II1 factors.Ann. Math., 90 (1969), 372–377.

    Article  MATH  MathSciNet  Google Scholar 

  25. Murray, F. J. &von Neumann, J., On rings of operators.Ann. Math., 37 (1936), 116–229.

    Article  Google Scholar 

  26. —, On rings of operators IV.Ann. Math., 44 (1943), 716–808.

    Article  Google Scholar 

  27. Nakamura, M. &Takeda, Z., On some elementary properties of the crossed products of von Neumann algebras.Proc. Japan Acad., 34 (1958), 489–494.

    MathSciNet  MATH  Google Scholar 

  28. —, A Galois theory for finite factors.Proc. Japan Acad., 36 (1960), 258–260.

    MathSciNet  MATH  Google Scholar 

  29. —, On the fundamental theorem of the Galois theory for finite factors.Proc. Japan Acad., 36 (1960), 313–318.

    MathSciNet  MATH  Google Scholar 

  30. Nakamura, M. &Umegaki, H., Heisenberg's commutation relation and the Plancherel theorem.Proc. Japan Acad., 37 (1961), 239–242.

    Article  MathSciNet  MATH  Google Scholar 

  31. von Neumann, J., Die Eindeutigkeit der Schrödingerschen Operatoren.Math. Ann., 104 (1931), 570–578.

    Article  MATH  MathSciNet  Google Scholar 

  32. Nielsen, O. A.,The Mackey-Blattner theorem and Takesaki's generalized commutation relation for locally compact groups. To appear.

  33. Pedersen, G. K. &Takesaki, M., The Radon-Nikodym theorem for von Neumann algebras.Acta Math., 130 (1973), 53–87.

    MathSciNet  MATH  Google Scholar 

  34. Powers, R., Representations of uniformly hyperfinite algebras and their associated von Neumann rings.Ann. Math., 86 (1967), 138–171.

    Article  MATH  MathSciNet  Google Scholar 

  35. Rudin, W.,Fourier analysis on groups. Intersciences, New York, (1962).

    MATH  Google Scholar 

  36. Sakai, S., An uncountable number of II1 and II∞ factors.J. Functional Analysis, 5 (1970), 236–246.

    Article  MATH  MathSciNet  Google Scholar 

  37. Størmer, E.,Spectra of states, and asymptotically abelian C *-algebras. To appear.

  38. Suzuki, N., Cross products of rings of operators.Tôhoku Math. J., 11 (1959), 113–124.

    MATH  Google Scholar 

  39. Takesaki, M., Covariant representations ofC *-algebras and their locally compact automorphism groups.Acta Math., 119 (1967), 273–303.

    Article  MATH  MathSciNet  Google Scholar 

  40. —, A liminal crossed product of a uniformly hyperfiniteC *-algebra by a compact abelian automorphism group.J. Functional Analysis, 7 (1971), 140–146.

    Article  MATH  MathSciNet  Google Scholar 

  41. —, A generalized commutation relation for the regular representation.Bull. Soc. Math., France, 97 (1969), 289–297.

    MATH  MathSciNet  Google Scholar 

  42. Takesaki, M.,Tomita's theory of modular Hilbert algebras and its application. Lecture Notes in Mathematics, 128, Springer-Verlag, (1970).

  43. Takesaki, M.,States and automorphisms of operator algebras—Standard representations and the Kubo-Martin-Schwinger boundary condition. Summer Rencontres in Mathematics and Physics, Battelle Seattle, (1971).

  44. Takesaki, M., Periodic and homogeneous states on a von Neumann algebras, I, II, III. To appear inBull. Amer. Math. Soc.

  45. —, The structure of a von Neumann algebra with a homogeneous periodic state.Acta Math., 131 (1973), 79–121.

    MATH  MathSciNet  Google Scholar 

  46. —, Dualité dans les produits croisés d'algebres de von Neumann.C. R. Acad. Sc. Paris, Ser. A, 276 (1973), 41–43.

    MATH  MathSciNet  Google Scholar 

  47. Takesaki, M., Algèbres de von Neumann proprement infinits et produits croisés.C. R. Acad. Sc. Paris, Ser. A, 276 (1973).

  48. Takesaki, M., Duality in crossed products and von Neumann algebras of type III. To appearBull. Amer. Math. Soc.

  49. Takesaki, M. &Tatsuuma, N., Duality and subgroups.Ann. Math., 93 (1971), 344–364.

    Article  MathSciNet  Google Scholar 

  50. Tomita, M., Standard forms of von Neumann algebras.The 5th Functional Analysis Symposium of the Math. Soc. of Japan, Sendai, (1967).

  51. Turumaru, T., Crossed product of operator algebras.Tôhoka Math. J., 10 (1958), 355–365.

    MATH  MathSciNet  Google Scholar 

  52. Vesterstrøm, J. &Wils, W., Direct integrals of Hilbert spaces, II.Math. Scand., 26 (1970), 89–102.

    MathSciNet  Google Scholar 

  53. Zeller-Meier, G., Produits croisés d'uneC *-algebre par un groupe d'automorphismes.J. Math. Pures et appl., 47 (1968), 101–239.

    MATH  MathSciNet  Google Scholar 

  54. A. Connes & Van Daele, The group property of the invariant S. To appear inMath. Scand.

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Authors and Affiliations

  1. University of California, Los Angeles, Calif., USA

    Masamichi Takesaki

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  1. Masamichi Takesaki
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The preparation of this paper was supported in part by NSF Grant GP-33696X MOS Number 46L10.

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Takesaki, M. Duality for crossed products and the structure of von Neumann algebras of type III. Acta Math. 131, 249–310 (1973). https://doi.org/10.1007/BF02392041

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