Skip to main content
Log in

Introduction to the theory of noncommutative integration

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

Abstract

We present a survey of the contemporary state of the theory of integration in von Neumann and JBW algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Literature cited

  1. R. Z. Abdullaev, “Lp spaces for Jordan algebras with a semifinite trace,” Izv. Akad. Nauk UzSSR, Ser. Fiz.-Mat. Nauk (to appear).

  2. R. Z. Abdullaev, “Nonassociative Lp spaces,” Izv. Akad. Nauk UzSSR, Ser. Fiz.-Mat. Nauk, No. 6, 3–5 (1983).

    Google Scholar 

  3. Sh. A. Ayupov, “Integration of Jordan algebras,” Izv. Akad. Nauk SSSR, Ser. Mat.,47, No. 1, 3–25 (1983).

    Google Scholar 

  4. Sh. A. Ayupov and R. Z. Abdullaev, “The Radon-Nikodym theorem and Lp spaces for weights on semifinite JBW-algebras,” Izv. Akad. Nauk UzSSR, Ser. Mat.-Fiz. Nauk (to appear).

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

    Google Scholar 

  6. V. Ya. Golodets, “Conditional expectations and modular automorphisms of von Neumann algebras,” Funkts. Anal. Prilozhen.,6, No. 3, 68–69 (1972).

    Google Scholar 

  7. A. A. Zolotarev, “Lp spaces with respect to a state on a von Neumann algebra and interpolation,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 8, 36–43 (1982).

    Google Scholar 

  8. A. A. Zolotarev, “On the interpolation theory of Lp spaces with respect to a state on a von Neumann algebra,” Kazan. Univ., Kazan (1983).

    Google Scholar 

  9. G. D. Lugovaya, “Bilinear forms defining measures on projections,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 2, 88 (1983).

    Google Scholar 

  10. G. D. Lugovaya, Unbounded Measures on the Projections of a von Neumann Algebra [in Russian], Thesis, Kazan (1983).

  11. G. D. Lugovaya, “On the structure of unbounded measures on the projections of a Hilbert space,” Issled. Prikl. Mat., No. 10, 202–205 (1984).

    Google Scholar 

  12. G. D. Lugovaya and A. N. Sherstnev, “On Gleason's theorem for unbounded measures,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 12, 30–32 (1980).

    Google Scholar 

  13. G. D. Lugovaya and A. N. Sherstnev, “On the integration of bounded operators with respect to measures on ideals of projections,” Konstruktivn. Teoriya Funktsii i Funkts. Analiz, No. 3, 44–50 (1981).

    Google Scholar 

  14. G. D. Lugovaya and A. N. Sherstnev, “Gleason's theorem for unbounded measures on the projections of a Hilbert space,” Tezisy Dokl. III Mezhd. Konf. po Teorii Veroyatn. i Mat. Statistike, Vol. 2, Vilnius (1981), pp. 13–14.

    Google Scholar 

  15. G. D. Lugovaya and A. N. Sherstnev, “On the realization of the L1 space with respect to an unbounded measure on projections,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 12, 35–42 (1984).

    Google Scholar 

  16. M. S. Matviechuk, “A theorem for states on quantum logic,” Teor. Mat. Fiz.,45, No. 2, 244–250 (1980).

    Google Scholar 

  17. M. S. Matveichuk, “A theorem for states on quantum logic. II,” Teor. Mat. Fiz.,48, No. 3, 261–265 (1981).

    Google Scholar 

  18. M. S. Matveichuk, “The description of finite measures on semifinite algebras,” Funkts. Anal. Prilozhen.,15, No. 3, 41–53 (1981).

    Google Scholar 

  19. M. S. Matveichuk, “The linearity of a state on a nonassociative logic,” Izv. Vyssh. Uchebn Zaved., Mat., No. 11, 119–120 (1983).

    Google Scholar 

  20. M. S. Matveichuk, “A theorem for states on quantum logic. States on Jordan algebras,” Teor. Mat. Fiz.,57, No. 3, 465–468 (1983).

    Google Scholar 

  21. M. S. Matveichuk and N. I. Nessonov, “A description of finite measures on W*-factors of type III,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 2, 13–16 (1984).

    Google Scholar 

  22. O. E. Tikhonov, “Integration with respect to a Hermitian functional on a von Neumann algebra,” Kazan Univ. (1981).

  23. O. E. Tikhonov, “Integrable bilinear forms and integral with respect to an operator-valued measure,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 3, 76–80 (1982).

    Google Scholar 

  24. O. E. Tikhonov, “Lp-type spaces with respect to a weight on a von Neumann algebra,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 8, 76–78 (1982).

    Google Scholar 

  25. N. V. Trunov, “Locally finite weights on von Neumann algebras,” Kazan Univ. (1978).

  26. N. V. Trunov, “On the noncommutative analog of an Lp space,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 11, 69–77 (1979).

    Google Scholar 

  27. N. V. Trunov, “On the noncommutative analog of an L2 space,” Konstruktivn. Teoriya Funktsii i Funkts. Analiz, No. 2, Kazan (1979), pp. 93–114.

    Google Scholar 

  28. N. V. Trunov, “Lp spaces associated with a weight on a semifinite von Neumann algebra,” Konstruktivn. Teoriya Funktsii i Funkts. Analiz, No. 3, Kazan (1981), pp. 88–92.

    Google Scholar 

  29. N. V. Trunov, “Integration in von Neumann algebras and regular weights,” Konstruktivn. Teoriya Funktsii i Funkts. Analiz, No. 3, Kazan (1981), pp. 73–87.

    Google Scholar 

  30. N. V. Trunov, “On the theory of normal weights on von Neumann algebras,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 8, 61–70 (1982).

    Google Scholar 

  31. N. V. Trunov, “On the theory of noncommutative L1 and L2 spaces,” Konstruktivn. Teoriya Funktsii i Funkts. Analiz, No. 4, Kazan (1983), pp. 96–105.

    Google Scholar 

  32. N. V. Trunov, “Integration with respect to a weight on Jordan algebras,” Kazan Univ., Kazan (1984).

    Google Scholar 

  33. M. V. Trunov and A. N. Sherstnev, “Conditional expectation in a setting of noncommutative probability theory,” Trans. 8th Prague Conf. Inform. Theory, Statist. Decis. Funct., Random Processes, Vol. B, Prague (1978), pp. 287–299.

    Google Scholar 

  34. N. V. Trunov and A. N. Sherstnev, “On the general theory of integration in operator algebras with respect to a weight. I,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 7, 79–88 (1978).

    Google Scholar 

  35. N. V. Trunov and A. N. Sherstnev, “On the general theory of integration in operator algebras with respect to a weight. II,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 12, 88–98 (1978).

    Google Scholar 

  36. A. S. Kholevo, “Investigations in the general theory of statistical decisions,” Trudy Mat. Inst. AN SSSR,124 (1976).

  37. A. N. Sherstnev, “On the general theory of states on von Neumann algebras,” Funkts. Anal. Prilozhen.,8, No. 3, 89–90 (1974).

    Google Scholar 

  38. A. N. Sherstnev, “Every smooth weight is anl-weight,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 8, 88–91 (1977).

    Google Scholar 

  39. E. Alfsen, F. Schultz, and E. Størmer, “A Gelfand-Neumark theorem for Jordan algebras,” Adv. Math.,28, No. 1, 11–56 (1978).

    Google Scholar 

  40. H. Araki and T. Masuda, “Positive cones and Lp-spaces for von Neumann algebras,” Publ. Res. Inst. Math. Sci.,18, No. 2, 339–411 (1982).

    Google Scholar 

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

    Google Scholar 

  42. L. Ciach, “Linear-topological spaces of operators affiliated with a von Neumann algebra,” Bull. Pol. Acad. Sci. Math.,31, No. 3–4, 161–166 (1983).

    Google Scholar 

  43. F. Combes, “Poids associe a une algbre hilbertienne a gauche,” Compositio Math.,23, No. 1, 49–77 (1971).

    Google Scholar 

  44. F. Combes, “Poids et esperances conditionelles dans les algebres de von Neumann,” Bull. Soc. Math. France,99, No. 4, 73–112 (1971).

    Google Scholar 

  45. A. Connes, “On the spatial theory of von Neumann algebras,” J. Funct. Anal.,35, No. 2, 153–164 (1980).

    Google Scholar 

  46. J. Dixmier, “Formes lineaires sur un anneau d'operateurs,” Bull. Soc. Math. France,81, 9–39 (1953).

    Google Scholar 

  47. J. Dixmier, Les algebres d'operateurs dans l'espace Hilbertien (algebres de von Neumann), Gauthier-Villars, Paris (1969).

    Google Scholar 

  48. H. Dye, “The Radon-Nikodym theorem for finite rings of operators,” Trans. Am. Math. Soc.,72, 243–280 (1952).

    Google Scholar 

  49. A. M. Gleason, “Measures on the closed subspaces of a Hilbert space,” J. Math. Mech.,6, No. 6, 885–893 (1957).

    Google Scholar 

  50. L. Gross, “Existence and uniqueness of physical ground states,” J. Funct. Anal.,10, No. 1, 52–109 (1972).

    Google Scholar 

  51. J. Gunson, “Physical states on quantum logics,” Ann. Inst. H. Poincare,17, No. 4, 295–311 (1972).

    Google Scholar 

  52. U. Haagerup, “Normal weights on W*-algebras,” J. Funct. Anal.19, No. 3, 302–317 (1975).

    Google Scholar 

  53. U. Haagerup, “Lp-spaces associated with an arbitrary von Neumann algebra,” Colloq. Int. CNRS, No. 274, 175–184 (1979).

    Google Scholar 

  54. U. Haagerup, “Operator-valued weights in von Neumann algebras. I,” J. Funt. Anal.,32, No. 2, 175–206 (1979).

    Google Scholar 

  55. U. Haagerup, “Operator-valued weights in von Neumann algebras. II,” J. Funct. Anal.,33, No. 3, 339–361 (1979).

    Google Scholar 

  56. U. Haagerup and H. Hanche-Olsen, “Tomita-Takesaki theory for Jordan algebras,” Odense Univ., Preprint No. 4, 1–35 (1982).

    Google Scholar 

  57. H. Hanche-Olsen, “A Tomita-Takesaki theory for JBW-algebras,” Operator Algebras and Appl., Proc. Symp. Pure Math. Am. Math. Soc., Kingston, 1980, Part 2, Am. Math Soc., Providence, R.I. (1982), pp. 301–303.

    Google Scholar 

  58. M. Hilsum, “Les espaces Lp d'une algbre de von Neumann definies par la derivee spatiale,” J. Funct. Anal.,40, No. 2, 151–169 (1981).

    Google Scholar 

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

    Google Scholar 

  60. H. Kosaki, “Noncommutative Lorentz spaces associated with a von Neumann algebra and applications,” Proc. Jpn. Acad., Ser. A,57, No. 6, 303–306 (1981).

    Google Scholar 

  61. H. Kosaki, “Applications of the complex interpolation method to a von Neumann algebra: noncommutative LP-spaces,” J. Funct. Anal.,56, No. 1, 25–79 (1984).

    Google Scholar 

  62. R. Kunze, “Fourier transforms on locally compact unimodular groups,” Trans. Am. Math. Soc.,89, No. 2, 519–540 (1958).

    Google Scholar 

  63. T. Masuda, “Lp-spaces for von Neumann algebra with reference to a faithful normal semifinite weight,” Publ. Res. Inst. Math. Sci.,19, No. 2, 673–727 (1983).

    Google Scholar 

  64. F. Murray and J. von Neumann, “On rings of operators. II,” Trans. Am. Math. Soc.,41, 208–248 (1937).

    Google Scholar 

  65. E. Nelson, “Notes on noncommutative integration,” J. Funct. Anal.,15, No. 2, 103–116 (1974).

    Google Scholar 

  66. T. Ogasawara and K. Yoshinaga, “A noncommutative theory of integration for operators,” J. Sci. Hiroshima Univ.,18, No. 3, 311–347 (1955).

    Google Scholar 

  67. G. Pedersen and M. Takesaki, “The Radon-Nikodym theorem for von Neumann algebras,” Acta Math.,130, No. 1–2, 53–87 (1973).

    Google Scholar 

  68. D. Petz, “A dual in von Neumann algebras with weights,” Math. Inst. Hung. Acad. Sci., Preprint No. 9, Budapest (1983), pp. 1–14.

    Google Scholar 

  69. S. Sankaran, “The*-algebra of unbounded operators,” J. London Math. Soc.,34, No. 3, 337–344 (1959).

    Google Scholar 

  70. I. Segal, “A noncommutative extension of abstract integration,” Ann. Math.,57, 401–457 (1953).

    Google Scholar 

  71. F. Schultz, “On normed Jordan algebras which are Banach dual spaces,” J. Funct. Anal.,31, 360–379 (1979).

    Google Scholar 

  72. M. Takesaki, “Tomita's theory of modular Hilbert algebras and its applications,” Lect. Notes Math.,128, Springer-Verlag, Berlin-Heidelberg-New York (1970).

    Google Scholar 

  73. M. Takesaki, “Conditional expectations in von Neumann algebras,” J. Funct. Anal.,9, No. 3, 306–321 (1972).

    Google Scholar 

  74. M. Takesaki, “Duality for crossed products and the structure of von Neumann algebras of type III,” Acta Math.,131, 249–308 (1973).

    Google Scholar 

  75. M. Terp, “Interpolation spaces between a von Neumann algebra and its predual,” J. Operator Theory,8, No. 2, 327–360 (1982).

    Google Scholar 

  76. F. Yeadon, “Noncommutative LP-spaces,” Math. Proc. Cambridge Philos. Soc.,77, No. 1, 91–102 (1975).

    Google Scholar 

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

    Google Scholar 

  78. F. Yeadon, “Finitely additive measures on projections in finite W*-algebras,” Bull. London Math. Soc.,16, No. 2, 145–150 (1984).

    Google Scholar 

Download references

Authors

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki (Noveishie Dostizheniya), Vol. 27, pp. 167–190, 1985.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Trunov, N.V., Sherstnev, A.N. Introduction to the theory of noncommutative integration. J Math Sci 37, 1504–1523 (1987). https://doi.org/10.1007/BF01103856

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01103856

Keywords

Navigation