Skip to main content
Log in

Hilbert lattices: New results and unsolved problems

  • Part I. Invited Papers Dedicated To The Memory Of Charles H. Randall (1928–1987)
  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

The class of Hilbert lattices that derive from orthomodular spaces containing infinite orthonormal sets (normal Hilbert lattices) is investigated. Relevant open problems are listed. Comments on form-topological orthomodular spaces and results on arbitrary orthomodular spaces are appended.

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

References

  1. E. Artin,Geometric Algebra (Interscience, New York, 1957).

    Google Scholar 

  2. A. Dvurečenskij, “Converse of the Eilers-Horst theorem,”Int. J. Theor. Phys. 26(7), 609–612 (1987).

    Google Scholar 

  3. A. Dvurečenskij and L. Mišík, Jr., “Gleason's theorem and completeness of inner product spaces,”Int. J. Theor. Phys. 27(4), 417–426 (1988).

    Google Scholar 

  4. A. Dvurečenskij and Sylvia Pulmannová, “State on splitting subspaces and completeness of inner product spaces,”Int. J. Theor. Phys. 27(9), 1059–1067 (1988).

    Google Scholar 

  5. Angela Fässler-Ullmann, “Nonclassical Hilbert spaces,”Expo Math. 1, 275–277 (1983).

    Google Scholar 

  6. H. Gross, “Quadratic forms in infinite-dimensional vector spaces,”Progress in Mathematics, Vol. 1 (Birkhäuser, Boston, 1979).

    Google Scholar 

  7. H. Gross, “Quadratic forms and Hilbert lattices, Contributions to general algebra,” inProceedings, Vienna Conference, June 21–24, 1984 (Teubner-Verlag, Stuttgart, and Hölder-Pichler-Tempsky, Vienna, 1985), pp. 181–190.

    Google Scholar 

  8. H. Gross, “Orthogonal geometry in infinite-dimensional vector spaces,” expanded version of a lecture at the University of Kaiserslautern, June 7, 1986. Preprint, pp. 1–29.

  9. H. Gross, “Different orthomodular orthocomplementations on a lattice,”Order 4, 79–92 (1987).

    Google Scholar 

  10. H. Gross, “Hilbert lattices with the extension property,”Geometriae Dedicata 29, 153–161 (1989).

    Google Scholar 

  11. H. Gross, “On orthomodular spaces. Contributions to general algebra,” inProceedings, Krems Conference, August 21–27, 1988 (North-Holland/Elsevier, Amsterdam, 1989).

    Google Scholar 

  12. H. Gross and H. A. Keller, “On the Definition of Hilbert Space,”Manuscripta Math. 23, 67–90 (1977).

    Google Scholar 

  13. H. Gross and U.-M. Künzi, “On a class of orthomodular quadratic spaces,”L'Enseignement Mathématique 31, 187–212 (1985).

    Google Scholar 

  14. S. S. Holland, Jr., “*-Valuations and ordered *-fields,”Trans. Am. Math. Soc. 262, 219–243 (1980).

    Google Scholar 

  15. H. A. Keller, “Una nota sobre espacios de Hilbert,”Notas Matemáticas imuc, Universidad Católica de Chile, Santiago, No. 3, June 1974, pp. 23–32.

  16. H. A. Keller, “Ein nicht-klassischer Hilbertscher Raum,”Math. Z. 172, 41–49 (1980).

    Google Scholar 

  17. H. A. Keller, three letters to H. Gross (concerning the proof of Theorem I.2.1), dated January 5, 1988, January 10, 1988, and January 12, 1988, respectively.

  18. H. A. Keller, letter to H. Gross (proof of Theorem III.3.1), July 1988.

  19. U.-M. Künzi, “Orthomodulare Räume über bewerteten Körpern,” Ph.D. Thesis, University of Zürich (1984).

  20. F. Maeda and S. Maeda,Theory of Symmetric Lattices (Grundlehren173), (Springer, New York, 1970).

    Google Scholar 

  21. R. P. Morash, “Angle bisection and orthoautomorphisms in Hilbert Lattices,”Can. J. Math. 25, 261–272 (1973).

    Google Scholar 

  22. Herminia Ochsenius, “Los espacios ortomodulares y el teorema de Kakutani-Mackey,” Tesis para optar al grado académico de Doctor en Ciencias Exactas (Matemática), Pontificia Universidad Católica de Chile, Santiago, in preparation.

  23. P. Ribenboim,Théorie des valuations (Presses Université Montréal, Montréal, 1964).

    Google Scholar 

  24. P. Ribenboim, “Equivalent forms of Hensel's lemma,”Expo. Math. 3, 3–24 (1985).

    Google Scholar 

  25. W. J. Wilbur, “On characterizing the standard quantum logics,”Trans. Am. Math. Soc. 233, 265–282 (1980).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Deceased (October 29, 1989).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gross, H. Hilbert lattices: New results and unsolved problems. Found Phys 20, 529–559 (1990). https://doi.org/10.1007/BF01883238

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01883238

Keywords

Navigation