, Volume 19, Issue 3, pp 539–546 | Cite as

Probability measure on conjugation logics

  • Marjan Matvejchuk


In the paper we describe probability measures on the conjugation logics of projections and on the logic of all projections.


Hilbert space Conjugation operator Projection  Measure 

Mathematics Subject Classification

28A60 46C20 46C50 47A63 47B50 81P10 


  1. 1.
    Birhoff, G.: Lattice Theory. Providence Rhode, Island (1967). Russian transl: Moscow. Nauka (1984)Google Scholar
  2. 2.
    Mackey, G.W.: The mathematical foundations of quantum mechanics. The mathematical physics monograph series. W.A. Benjamin Inc., New York (1963)Google Scholar
  3. 3.
    Nagy, K.: State Vector Spaces with Indefinite Metric in Quantum Field Theory. Akad. Kiado, Budapest (1966)MATHGoogle Scholar
  4. 4.
    Gleason, A.: Measures on the closed subspaces of a Hilbert space. J. Math. Mech. 6, 885–983 (1957)MathSciNetMATHGoogle Scholar
  5. 5.
    Dixmier, J.: Les algebras d’operateurs dans l’espace Hilbertin (algebras de von Neumann), Deuxieme Edition. Gauthier-Villiars, Paris (1969)Google Scholar
  6. 6.
    Mel’tser, M.: On a classification of von Neumann \(J\)-algebra. Funct. Anal. Appl. 13(4), 83–84 (1979) [in Russian]. English transl. 13(4), 305–307 (1979)Google Scholar
  7. 7.
    Azizov, T., Iokhvidov, I.: Linear Operators in Space with an Indefinite Metric. Nauka, Moscow [in Russian]. Wiley, New York (1986)Google Scholar
  8. 8.
    Matveichuk, M.: Measures on the quantum logic of subspaces of a J-space. Sibirskii Matemat. Zhurnal. 32(2), 104–112 (1991) [in Russian]. English transl. Siberian Mathem. J. 32(2) 265–272 (1991)Google Scholar
  9. 9.
    Matvejchuk, M.: A description of indefinite measures in \(W^{*}J\)-factors. Dokl. Akad. Nauk SSSR. 319, 558–561 (1991)[in Russian]. English transl. Soviet Math. Dokl. 44, 161–165 (1992)Google Scholar
  10. 10.
    Matvejchuk, M.: Idempotents in a space with conjugation. Linear Algebra Appl. 438(1), 71–79 (2013)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Matvejchuk, M.: Idempotens as J-projections:II. Lobachevskii J. Math. 33(2), 139–143 (2012)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Mushtari D.: Projection logics in Banash spaces, Russian Mathem. Izvestiya VUZov. Matematica, No.8, 44–49 (1989) [in Russian]. English transl: Sov. Math. 33(8), 59–70 (1989)Google Scholar
  13. 13.
    Matvejchuk, M.: Any regular measure on conjugation logic is complex measure. Intern. J. Theoret. Phys. 51(1), 259–262 (2012)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Matvejchuk, M.: Hermitian measure in \(W^{*}J\)-algebras in Hilbert spaces with conjugation. Intern. J. Theoret. Phys. 39(3), 777–791 (2000)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Kazan Federal UniversityKazanRussia

Personalised recommendations