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Jordan operator algebras

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Abstract

A survey of recent results in classification of JW-algebras (weakly closed Jordan algebras of self-adjoint operators in the Hilbert space) is given along with connections of JW-algebras with their enveloping W*-algebras. It is shown how these results are applied in the proofs of analogs of many important results in the theory of W*-algebras.

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Literature cited

  1. Sh. A. Ayupov, “Statistical ergodic theorems in Jordan algebras,” Usp. Mat. Nauk,36, No. 6, 201–202 (1981).

    Google Scholar 

  2. Sh. A. Ayupov, “On construction of Jordan algebras of self-adjoint operators,” Dokl. AN SSSR,267, No. 3, 521–524 (1982).

    Google Scholar 

  3. Sh. A. Ayupov, “Modular Jordan algebras of self-adjoint operators,” Teor. Mat. Fiz.,53, No. 1, 77–82 (1982).

    Google Scholar 

  4. Sh. A. Ayupov, “Integration on Jordan algebras,” Izv. AN SSSR, Ser. Mat.,47, No. 1, 3–25 (1983).

    Google Scholar 

  5. Sh. A. Ayupov, “Types of Jordan algebras of self-adjoint operators and their enveloping von Neumann algebras,” Funkts. Anal. Prilozhen.17, No. 1, 65–66 (1983).

    Google Scholar 

  6. Sh. A. Ayupov, “Classification, representations, and probabilistic aspects of ordered Jordan algebras,” Author's Abstract of Doctoral Dissertation, Tashkent (1983).

  7. Sh. A. Ayupov, “Locally measurable operators for JW-algebras and representation of ordered Jordan algebras,” Izv. AN SSSR, Ser. Mat.,48, No. 2, 211–236 (1984).

    Google Scholar 

  8. Sh. A. Ayupov, “Classification of injective JW-factors,” Funkts. Anal. Prilozhen.,18, No. 3, 67–68 (1984).

    Google Scholar 

  9. Sh. A. Ayupov, “On the existence of Jordan algebras of self-adjoint operators of a given type,” Sib. Mat. Zh.,25, No. 5, 3–8 (1984).

    Google Scholar 

  10. Sh. A. Ayupov, “JW-factors and antiautomorphisms of von Neumann; algebras,” Izv. AN SSSR, Ser. Mat.,49, No. 1, 211–220 (1984).

    Google Scholar 

  11. M. A. Berdikulov, “The spaces L1 and L2 for semifinite JBW-algebras,” Dokl. AN UzSSR, No. 6, 3–4 (1982).

    Google Scholar 

  12. A. A. Lodkin, “Each measure on projectors of a W*-algebra can be extended to a state,” Funkts. Anal. Prilozhen.,8, No. 4, 54–58 (1974).

    Google Scholar 

  13. M. S. Matveichuk, “Description of finite measures in semifinite algebras,” Funkts. Anal. Prilozhen.,15, No. 3, 41–53 (1981).

    Google Scholar 

  14. M. S. Matveichuk, “One theorem on states on quantum logics. States in Jordan algebras,” Teor. Mat. Fiz.,57, No. 3, 465–468 (1983).

    Google Scholar 

  15. M. S. Matveichuk and N. I. Nessonov, “A description of measures in Neumann factors of type III,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 2, 68–71 (1984).

    Google Scholar 

  16. T. A. Sarymsakov and Sh. A. Ayupov, “Partially ordered Jordan algebras,” Dokl. AN SSSR,249, No. 4, 789–792 (1979).

    Google Scholar 

  17. T. A. Sarymsakov, Sh. A. Ayupov, Dzh. Khadzhiev and V. I. Chilin, Ordered Algebras [in Russian], Fan, Tashkent (1983).

    Google Scholar 

  18. J. von Neumann, “A generalization of the mathematical apparatus of quantum mechanics by methods of abstract algebra, Part 1,” Mat. Sb.,1, No. 4, 415–484 (1936).

    Google Scholar 

  19. J. F. Aarnes, “Physical states on a C*-algebra,” Acta Math.,122, Nos. 3–4, 161–172 (1969).

    Google Scholar 

  20. J. F. Aarnes, “Quasistates on C*-algebras,” Trans. Am. Math. Soc.,149, No. 2, 601–625 (1970).

    Google Scholar 

  21. Sh. A. Ajupov, “Extension of traces and type criterions for Jordan algebras of selfadjoint operators,” Math. Z.,181, 253–268 (1982).

    Google Scholar 

  22. E. M. Alfsen, H. Hanche-Olsen and F. W. Shultz, “State spaces of C*-algebras,” Acta Math.,144, Nos. 3–4, 267–305 (1980).

    Google Scholar 

  23. E. M. Alfsen and F. W. Shultz, “State spaces of Jordan algebras,” Acta Math.,140, Nos. 3–4, 155–190 (1978).

    Google Scholar 

  24. E. M. Alfsen and F. W. Shultz, “On noncommutative spectral theory and Jordan algebras,” Proc. London Math. Soc.,38, No. 3, 497–516 (1979).

    Google Scholar 

  25. E. M. Alfsen, F. W. Shultz and E. Stormier, “A Gelfand-Neumark theorem for Jordan algebras,” Adv. Math.,28, No. 1, 11–55 (1978).

    Google Scholar 

  26. H. Araki, “Characterization of the state space of quantum mechanics,” Commun. Math. Phys.,75, No. 3, 1–25 (1980).

    Google Scholar 

  27. J. Bellisard and B. Iochum, “Homogeneous selfdual cones versus Jordan algebras. The theory revisited,” Ann. Inst. Fourier,28, No. 1, 27–67 (1978).

    Google Scholar 

  28. E. Christensen, “Measures on projections and physical states,” Commun. Math. Phys.,86, No. 4, 529–538 (1982).

    Google Scholar 

  29. A. Connes, “Une classification des facteurs de type III,” Ann. Scient. Ecole Norm. Super.,6, No. 2, 133–252 (1973).

    Google Scholar 

  30. A. Connes, “Classification of injective factors,” Ann. Math.,104, No. 1, 73–115 (1976).

    Google Scholar 

  31. A. Connes and M. Takesaki, “The flow of weights on factors of type III,” Tohoku Math. J., 2 Ser.,29, No. 4, 473–575 (1977).

    Google Scholar 

  32. T. Drisch, “Generalization of Gleason's theorem,” Int. J. Theor. Phys.,18, No. 4, 239–243 (1979).

    Google Scholar 

  33. C. M. Edwards, “On the centres of hereditary JBW-subalgebras of JBW-algebras,” Math. Proc. Cambridge Phil. Soc.,85, No. 2, 317–324 (1975).

    Google Scholar 

  34. E. Effros and E. Stormier, “Jordan algebras of self-adjoint operators,” Trans. Am. Math. Soc.,127, No. 2, 313–316 (1967).

    Google Scholar 

  35. E. Effros and E. Stormer, “Positive projections and Jordan structure in operator algebras,” Math. Scand.,45, No. 1, 127–138 (1979).

    Google Scholar 

  36. G. G. Emch and W. P. C. King, “Faithful normal states on JBW-algebras,” in: Oper. Algebras and Appl. Proc. Symp. Pure Math. Amer. Math. Soc., Kingstone, 1980, Vol. 38, Part 2 (1982), pp. 305–307.

    Google Scholar 

  37. T., Giordano, “Antiautomorphismes involutifs des facteurs injectifs,” C. R. Acad. Sci. Paris, AB291, Nos. 10–17, A583-A585 (1980).

    Google Scholar 

  38. T. Giordano, “Antiautomorphismes involutifs des facteurs de von Neumann injectifs,” These, Univ. de Neuchatel, 1–106 (1981).

  39. T. Giordano, “Antiautomorphismes involutifs des facteurs de von Neumann injectifs,” J. Oper. Theory,10, No. 2, 252–287 (1983).

    Google Scholar 

  40. T. Giordano, “Antiautomorphismes involutifs des facteurs de von Neumann injectifs. II,” J. Funct. Anal.,51, No. 3, 326–360 (1983).

    Google Scholar 

  41. T. Giordano and V. Jones, “Antiautomorphismes involutifs du facteur hyperfini de type II,” C. R. Acad. Sci. Paris, AB290, No. 1–7, A29-A31 (1980).

    Google Scholar 

  42. U. Haagerup and H. Hanche-Olsen, “A Tomita-Takesaki theory for JBW-algebras,” Preprint Odense Universitet, 1–35 (1982).

  43. B. Iochum, Cones Autopolaires et Algebres de Jordan, Lecture Notes Math., No. 1049 (1984).

  44. B. Iochum and F. W. Shultz, “Normal state spaces of Jordan and von Neumann algebras,” J. Funct. Anal.,50, No. 3, 317–328 (1983).

    Google Scholar 

  45. N. Jacobson, Structure and Representations of Jordan Algebras, Amer. Math. Soc. Colloq. Publ., Vol. 39, Providence (1968).

  46. P. Jordan, J. von Neumann and E. Wigner, “On an algebraic generalization of the quantum mechanical formalism,” Ann. Math.,35, 29–64 (1934).

    Google Scholar 

  47. W. P. C. King, “Semifinite traces on JBW-algebras,” Math. Proc. Cambridge Phil. Soc.,93, No. 3, 503–509 (1983).

    Google Scholar 

  48. S. Kurepa, “A cosine functional equation in Hilbert space,” Can. J. Math.,12, No. 1, 45–50 (1960).

    Google Scholar 

  49. G. K. Pedersen and E. Stormer, “Traces on Jordan algebras,” Can. J. Math.,34, No. 2, 370–373 (1982).

    Google Scholar 

  50. G. K. Pedersen and M. Takesaki, “The Radon-Nikodym theorem for von Neumann algebras,” Acta Math.,130, Nos. 1–2, 53–87 (1973).

    Google Scholar 

  51. A. G. Robertson, “Automorphisms of spin factors and the decomposition of positive maps,” Quart. J. Math.,34, No. 133, 87–96 (1983).

    Google Scholar 

  52. A. G. Robertson and M. A. Youngson, “Positive projection with contractive complements on Jordan algebras,” J. London Math. Soc.,25, No. 2, 365–374 (1982).

    Google Scholar 

  53. S. Sakai, C*-Algebras and W*-Algebras, Springer, Berlin (1971).

    Google Scholar 

  54. S. Sakai, “On automorphism groups of II1-factors,” Tohoku Math. J.,26, No. 3, 423–430 (1974).

    Google Scholar 

  55. F. W. Shultz, “On normed Jordan algebras which are Banach dual spaces,” J. Funct. Anal.,31, No. 3, 360–376 (1979).

    Google Scholar 

  56. P. J. Stacey, “The structure of type I JBW-algebras,” Math. Proc. Cambridge Philos. Soc.,90, No. 3, 477–482 (1981).

    Google Scholar 

  57. P. J. Stacey, “Type I2 JBW-algebras,” Quart. J. Math.,33, No. 129, 115–127 (1982).

    Google Scholar 

  58. P. J. Stacey, “Real structure in σ-finite factors of type IIIλ where 0 < λ < 1,” Proc. London Math. Soc.,47, No. 2, 275–284 (1983).

    Google Scholar 

  59. E. Stormier, “On the Jordan structure of C*-algebras,” Trans. Am. Math. Soc.,120, No. 12, 438–447 (1965).

    Google Scholar 

  60. E. Stormer, “Jordan algebras of type I,” Acta Math.,115, Nos. 3–4, 165–184 (1966).

    Google Scholar 

  61. E. Stormer, “On anti-automorphisms of von Neumann algebras,” Pac. J. Math.,21, No. 2, 349–370 (1967).

    Google Scholar 

  62. E. Stormer, “Irreducible Jordan algebras of self-adjoint operators,” Trans. Am. Math. Soc.,130, No. 1, 153–166 (1968).

    Google Scholar 

  63. E. Stormer, “Real structure in the hyperfinite factor,” Duke Math. J.,47, No. 1, 145–153 (1980).

    Google Scholar 

  64. S. Stratila, Modular Theory in Operator Algebras, Ed. Academiei, Bucuresti; Abacus Press, Tunbridge Wells (1981).

    Google Scholar 

  65. S. Stratila and L. Zsido, Lectures on von Neumann Algebras, Ed, Academiei, Bucuresti; Abacus Press, Tunbridge Wells (1979).

    Google Scholar 

  66. M. Takesaki, Theory of Operator Algebras I, Springer, New York (1979).

    Google Scholar 

  67. J. Tisher, “Gleason's theorem for type I von Neumann algebras,” Pac. J. Math.,100, No. 2, 473–488 (1982).

    Google Scholar 

  68. D. Topping, “Jordan algebras of self-adjoint operators,” Mem. Am. Math. Soc., No. 53, 1–48 (1965).

    Google Scholar 

  69. D. Topping, “An isomorphism invariant for spin factors,” J. Math. Mech.,15, No. 6, 1055–1064 (1966).

    Google Scholar 

  70. F. J. Yeadon, “Measures on projections in W*-algebras of type II1,” Bull. London Math. Soc.,15, No. 2, 139–145 (1983).

    Google Scholar 

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  1. Sh. A. Ayupov
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki (Noveishie Dostizheniya), Vol. 27, pp. 67–98, 1985.

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Ayupov, S.A. Jordan operator algebras. J Math Sci 37, 1422–1448 (1987). https://doi.org/10.1007/BF01103853

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