Skip to main content
SpringerLink
Log in

Subspace Structures in Inner Product Spaces and von Neumann Algebras

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

We study subspaces of inner product spaces that are invariant with respect to a given von Neumann algebra. The interplay between order properties of the poset of affiliated subspaces and the structure of a von Neumann algebra is investigated. We extend results on nonexistence of measures on incomplete structures to invariant subspaces. Results on inner product spaces as well as on the structure of affiliated subspaces are reviewed.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Buhagiar, D., Chetcuti, E., Dvurečenskij, A.: Algebraic and measure-theoretic properties of classes of subspaces of an inner product space. In: Handbook of Quantum Logic and Quantum Structures: Quantum Structures, pp. 75–120. Elsevier, Amsterdam (2007)

    Chapter  Google Scholar 

  2. Dvurečenskij, A.: Gleason’s Theorem and its Applications. Kluwer, Boston (1993)

    MATH  Google Scholar 

  3. Hamhalter, J.: Quantum Measure Theory. Kluwer, Boston (2003)

    MATH  Google Scholar 

  4. Hamhalter, J.: Quantum structures and operator algebras. In: Handbook of Quantum Logic and Quantum Structures: Quantum Structures, pp. 285–333. Elsevier, Amsterdam (2007)

    Chapter  Google Scholar 

  5. Piron, C.: Axiomatique quantique. Helv. Phys. Acta 37, 439–468 (1964)

    MathSciNet  MATH  Google Scholar 

  6. Amemiya, I., Araki, H.: A remark on Piron’s paper. Publ. Res. Inst. Math. Sci., Ser. A 2, 423–427 (1967)

    Article  MathSciNet  Google Scholar 

  7. Hamhalter, J., Pták, P.: A completeness criterion for inner product spaces. Bull. Lond. Math. Soc. 19, 259–263 (1987)

    Article  MATH  Google Scholar 

  8. Chetcuti, E., Dvurečenskij, A.: The existence of finitely additive states on orthogonally closed subspaces of incomplete inner product spaces. Lett. Math. Phys. 67(1), 75–80 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Buhagiar, D., Chetcuci, E.: Only “free” measures are admissible on F(S) when S is incomplete. Proc. Am. Math. Soc. 136, 919–922 (2008)

    Article  MATH  Google Scholar 

  10. Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37(4), 823–824 (1936)

    Article  Google Scholar 

  11. Sherstnev, A.N., Turilova, E.A.: Classes of subspaces affiliated with a von Neumann algebra. Russ. J. Math. Phys. 6(4), 426–434 (1999)

    MathSciNet  MATH  Google Scholar 

  12. Turilova, E.A.: On certain classes of subspaces affiliated with a von Neumann algebra in the representation space of the algebra B(H) associated with a weight. Russ. Math. 50(9), 55–62 (2006)

    MathSciNet  Google Scholar 

  13. Turilova, E.A.: Measures on classes of subspaces affiliated with a von Neumann algebra. Int. J. Theor. Phys. 48(11), 3083–3091 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  15. Takesaki, M.: Theory of Operator Algebras II. Springer, New York (2003)

    MATH  Google Scholar 

  16. Pták, P., Weber, H.: Lattice properties of subspace families in inner product space. Proc. Am. Math. Soc. 129, 2111–2117 (2001)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Electrical Engineering, Department of Mathematics, Czech Technical University, Technicka 2, 166 27, Prague 6, Czech Republic

    Jan Hamhalter

  2. Faculty of Computational Mathematics and Cybernetics, Kazan State University, Kremlevskaya 18, Kazan, Russia

    Ekaterina Turilova

Authors
  1. Jan Hamhalter
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ekaterina Turilova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jan Hamhalter.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hamhalter, J., Turilova, E. Subspace Structures in Inner Product Spaces and von Neumann Algebras. Int J Theor Phys 50, 3812–3820 (2011). https://doi.org/10.1007/s10773-011-0665-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-011-0665-6

Keywords

Search

Navigation