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Standard and split inclusions of von Neumann algebras

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References

  1. Albeverio, S., Høegh-Krohn, R.: Ergodic actions by compact groups onC *-algebras. Math. Z.174, 1–18 (1980)

    Google Scholar 

  2. Araki, H.: Von Neumann algebras of local observables for free scalar field. J. Math. Phys.5, 1–13 (1964)

    Google Scholar 

  3. Araki, H.: Some properties of modular conjugation operator of a vonNeumann algebra and a non-commutative Radon-Nikodym theorem with a chain rule. Pac. J. Math.50, 309–354 (1974)

    Google Scholar 

  4. Benfatto, G., Nicolo', F.: The local von Neumann algebras for the massless scalar free field and the free electromagnetic field. J. Math. Phys.19, 653–660 (1978)

    Google Scholar 

  5. Bisognano, J.J., Wichmann, E.H.: On the duality condition for a hermitian scalar field. J. Math. Phys.16, 985–1007 (1975)

    Google Scholar 

  6. Borchers, H.J.: A remark on a theorem of B. Misra. Commun. Math. Phys.4, 315–327 (1967)

    Google Scholar 

  7. Borchers, H.J.: On the converse of Reeh-Schlieder theorem. Commun. Math. Phys.10, 269–273 (1968)

    Google Scholar 

  8. Buchholz, D.: Product states for local algebras. Commun. Math. Phys.36, 287–304 (1974)

    Google Scholar 

  9. Connes, A.: Caractérisation des espaces vectoriels ordonnées soujacents aux algèbres de von Neumann. Ann. Inst. Fourier24, 121–155 (1974)

    Google Scholar 

  10. Connes, A.: Une classification des facteurs de type III. Ann. Ec. Norm. Sup.6, 133–252 (1973)

    Google Scholar 

  11. Connes, A., Takesaki, M.: The flow of weight on a factor of type III. tohoku Math. J.29, 473–575 (1977)

    Google Scholar 

  12. D'Antoni, C., Longo, R.: Interpolation by type I factors and the flip automorphism. J. Funct. Anal.51, 361–371 (1983)

    Google Scholar 

  13. Dixmier, J.:C *-algebras and their representations. North-Holland (1980)

  14. Dixmier, J., Marechal, O.: Vecteurs totalisateurs d'une algèbre de von Neumann. Commun. Math. Phys.22, 44–50 (1971)

    Google Scholar 

  15. Doplicher, S., Haag, R., Roberts, J.E.: Fields, observables and gauge transformations. I. Commun. Math. Phys.13, 1–23 (1969). II. Commun. Math. Phys.15, 173–200 (1969)

    Google Scholar 

  16. Doplicher, S., Kastler, D., Størmer, E.: Invariant states and asymptotic abelianess. J. Functional Anal.3, 419–432 (1969)

    Google Scholar 

  17. Doplicher, S.: Local aspects of superselection rules. Commun. Math. Phys.85, 73–86 (1982)

    Google Scholar 

  18. Doplicher, S., Longo, R.: Local aspects of superselection rules, II. Commun. Math. Phys.88, 399–409 (1983)

    Google Scholar 

  19. Eckmann, J.P., Fröhlich, J.: Unitary equivalence of local algebras in the quasi-free representation. Ann. Inst. H. Poincaré A20, 201–209 (1974)

    Google Scholar 

  20. Fröhlich, J.: New superselection “soliton state” in two dimensional bose quantum field models. Commun. Math. Phys.47, 269–310 (1974)

    Google Scholar 

  21. Fuglede, G., Kadison, R.V.: On a conjecture of Murray and von Neumann. Proc. Nat. Acad. Sci. USA37, 420–425 (1951)

    Google Scholar 

  22. Garber, W.D.: The connexion of duality and causal properties for generalized free fields. Commun. Math. Phys.42, 195–208 (1975)

    Google Scholar 

  23. Haagerup, U.: The standard form of a von Neumann algebra. Math. Scand.37, 271–283 (1975)

    Google Scholar 

  24. Haagerup, U.: Seminar talks at the University of Pennsylvania and University of Rome and Private Communication

  25. Hislop, P.D., Longo, R.: Modular structure of the local algebras associated with the free massless scalar field theory. Commun. Math. Phys.84, 71–85 (1982)

    Google Scholar 

  26. Hugenoltz, N.M., Wieringa, J.D.: On locally normal states in quantum statistical mechanics. Commun. Math. Phys.11, 183–197 (1969)

    Google Scholar 

  27. Kadison, R.V.: Remarks on the type of von Neumann algebras of local observables in quantum field theory. J. Math. Phys.4, 1511–1516 (1963)

    Google Scholar 

  28. Kovacs, I., Szücs, J.: Ergodic type theorems in von Neumann algebras. Acta Sci. Math.27, 233–246 (1966)

    Google Scholar 

  29. Landau, L.J.: On local function of fields. Commun. Math. Phys.39, 49–62 (1974)

    Google Scholar 

  30. Landau, L.J., Wichmann, E.H.: On the translation invariance of local internal symmetries. J. Math. Phys.11, 306–311 (1970)

    Google Scholar 

  31. Longo, R.: A simple proof of the existence of the modular automorphisms in approximately finite dimensional von Neumann algebras. Pacific J. Math.75, 199–205 (1978)

    Google Scholar 

  32. Longo, R.: Algebraic and modular structure of von Neumann algebras of physics. Proceedings of Symposia in Pure Mathematics38, Part 2, 551–566 (1982)

    Google Scholar 

  33. Summers, S.J.: Normal product states for fermions and twisted duality for CCR- and CAR-type algebras with application to the Yukawa2 quantum field model. Commun. Math. Phys.86, 111–141 (1982)

    Google Scholar 

  34. Takesaki, M.: On the cross norm on the direct product ofC *-algebras. Tohoku Math. J.16, 111–122 (1964)

    Google Scholar 

  35. Takesaki, M.: Tomita's theory of modular Hilbert algebras and its applications. Lecture Notes in Math. Vol. 128. Berlin-Heidelberg-New York: Springer 1970

    Google Scholar 

  36. Takesaki, M.: Conditional expectation in von Neumann algebra. J. Functional Anal.9, 306–321 (1972)

    Google Scholar 

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

    Google Scholar 

  38. Takesaki, M.: Theory of operator algebras I. Berlin-Heidelberg-New York: Springer 1979

    Google Scholar 

  39. Von Neumann, J.: On infinite direct products. Compositio Math.6, 1–77 (1938)

    Google Scholar 

  40. Wichmann, E.H.: Private Communication

  41. Haag, R., Kastler, D.: An algebraic approach to quantum field theory. J. Math. Phys.5, 848–861 (1974)

    Google Scholar 

  42. Buchholz, D., Wichmann, E.H.: In preparation

  43. Segal, I.M.: Tensor algebras over Hilbert spaces, I. Trans. Amer. Math. Soc.81, 106–134 (1956)

    Google Scholar 

  44. Testard, D.: In Algebres d'Operateurs et leurs Applications en Physique Mathematique, Paris, Ed. du CNRS (1979)

    Google Scholar 

  45. Buchholz, D.: On the structure of local quantum fields with non-trivial interaction. In: Proc. of the Int. Conf. on “Operator Algebras Ideals and their Applications in Theoretical Physics” Baumgärtel, Lassner, Pietsch, Uhlmann (eds). Leipzig: Teubner 1978

    Google Scholar 

  46. D'Antoni, C., Fredenhagen, K., Longo, R., Doplicher, S.: in preparation

  47. Popa, S.: On a problem of R.V. Kadison on maximal abelian subalgebras. Invent. Math.65, 269–281 (1981)

    Google Scholar 

  48. Jones, V., Popa, S.: Some properties of MASA'S in factors. In “Invariant Subspaces and other Topics”, Birkhäuser 89-102 (1982)

  49. Bogolubov, N.N., Logunov, A.A., Todorov, I.T.: Introduction to axiomatic quantum field theory. Benjamin 1975

  50. Eckmann, J.P., Osterwalder, K.: An application of Tomita's theory of modular Hilbert algebras: Duality for free bose fields. J. Functional Anal.13, 1–12 (1973); Leyland, P., Roberts, J.E., Testard, D.: Duality for quantum free fields. Marseille preprint 1978

    Google Scholar 

  51. Longo, R.: Solution of the factorial Stone-Weierstrass conjecture. An application of the theory of standard splitW *-inclusions. Invent. Math. in press (1984)

  52. Longo, R., Peligrad, C.: Non-commutative topological dynamics and compact actions onC *-algebras. Preprint

  53. Buchholz, D., Doplicher, S., Longo, R.: In preparation

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Research supported by Ministero della Pubblica Istruzione and CNR-Gnafa

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Doplicher, S., Longo, R. Standard and split inclusions of von Neumann algebras. Invent Math 75, 493–536 (1984). https://doi.org/10.1007/BF01388641

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