International Journal of Theoretical Physics

, Volume 27, Issue 9, pp 1059–1067

State on splitting subspaces and completeness of inner product spaces

  • Anatolij Dvurečenskij
  • Sylvia Pulmannová


We show that an inner product spaceV is complete iff the system of all splitting subspaces, i.e., of all subspacesM for whichM + M=V, possesses at least one completely additive state.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Amemiya, I., and Araki, H. (1966). A remark on Piron's paper,Publication of the Research Institute for Mathematical Sciences, Series A,2, 423–427.Google Scholar
  2. Cattaneo, G., and Marino, G. (1986). Completeness of inner product spaces with respect to splitting subspaces.Letters in Mathematical Physics,11, 15–20.Google Scholar
  3. Dvurečenskij, A. (1988a). Note on a construction of unbounded measures on a nonseparable Hubert space quantum logic.Annales de l'Institut Henri Poincaré-Physique Theorique,48, 297–310.Google Scholar
  4. Dvurečenskij, A. (1988b). Completeness of inner product spaces and quantum logic of splitting subspaces.Letters in Mathematical Physics,15, 231–235.Google Scholar
  5. Dvurečenskij, A. (1989). A state criterion of the completeness for inner product spaces,Demonstratio Mathematique, in press.Google Scholar
  6. Dvurečenskij, A., and Misik, Jr., L. (1988). Gleason's theorem and completeness of inner product spaces,International Journal of Theoretical Physics,27, 417–426.Google Scholar
  7. Gross, H., and Keller, H. A. (1977). On the definition of Hubert space.Manuscripta Mathematica,23, 67–90.Google Scholar
  8. Gudder, S. P. (1974). Inner product spaces,American Mathematical Monthly,81, 29–36.Google Scholar
  9. Hamhalter, J., and Pták, P. (1987). A completeness criterion for inner product spaces,19, 259–263.Google Scholar
  10. Maeda, S. (1980).Lattice Theory and Quantum Logic, Mahishoten, Tokyo (in Japanese).Google Scholar
  11. Varadarajan, V. S. (1962). Probability in physics and a theorm on simultaneous observability,Communications in Pure and Applied Mathematics,15, 186–217 [Errata,18 (1965)].Google Scholar
  12. Varadarajan, V. S. (1968).Geometry of Quantum Theory, Vol. 1, Van Nostrand, Princeton, New Jersey.Google Scholar

Copyright information

© Plenum Publishing Corporation 1988

Authors and Affiliations

  • Anatolij Dvurečenskij
    • 1
  • Sylvia Pulmannová
    • 1
  1. 1.Mathematical InstituteSlovak Academy of SciencesBratislavaCzechoslovakia

Personalised recommendations