Skip to main content
SpringerLink
Log in

Jordan Invariants of Von Neumann Algebras Given by Abelian Subalgebras and Choquet Order on State Spaces

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

Abstract

New complete invariants for Jordan parts of von Neumann algebras are presented. We shall prove that the poset of all finite dimensional abelian von Neumann subalgebras ordered by set theoretic inclusion is a complete Jordan invariant for von Neumann algebras. On the other hand, we exhibit an example showing that not any order isomorphism on this structure is derived from a Jordan isomorphism. We apply our results to the Choquet order of orthogonal measures on state spaces of von Neumann algebras. Among others we show that the poset of decompositions of a fixed faithful normal state on a von Neumann algebra endowed with the Choquet order is a complete Jordan invariant for σ-finite von Neumann algebras.

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. Brateli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, vol. 1. Springer, Berlin (1997)

    Book  Google Scholar 

  2. Bunce, L.J., Wright, J.D.M.: The Mackey-Gleason problem for vector measures on projections in von Neumann algebras. J. London Math. Soc. 49(2), 133–149 (1994)

    Article  MathSciNet  Google Scholar 

  3. Döring, A., Harding, J.: Abelian subalgebras and the Jordan structure of von Neumann algebras, arXiv:1009.4945

  4. Dye, H. A.: On the geometry of projections in certain operator algebras. Ann. Math. 61(1), 73–89 (1955)

    Article  MathSciNet  Google Scholar 

  5. Halvorson, H. (ed.): Deep beauty: understanding the quantum world through mathematical innovation. Cambridge University Press, Cambridge (2011)

  6. Hamhalter, J.: Isomorphisms of ordered structures of abelian C ∗-subalgebras of C∗-algebras. J. Math. Anal. Appl. 383, 391–399 (2011)

    Article  MathSciNet  Google Scholar 

  7. Hamhalter, J., Turilova, E.: Structure of associative subalgebras of Jordan operator algebras. Q. J. Math., Oxford Press 64(2), 397–408 (2013)

    Article  MathSciNet  Google Scholar 

  8. Hamhalter, J., Turilova, E.: Automorphisms of ordered structures of abelian parts of operator algebras and their role in quantum theory. Int. J. Theor. Phys. 53(10), 3333–3345 (2014)

    Article  Google Scholar 

  9. Hamhalter, J., Turilova, E.: Orthogonal measures on state spaces and context structures of quantum theory. Int. J. Theor. Phys. 55, 3353–3365 (2016). https://doi.org/10.1007/s10773-016-2964-4

    Article  MathSciNet  MATH  Google Scholar 

  10. Hamhalter, J., Turilova, E.: Choquet order and Jordan maps. Lobachevskii J. Math. 39(3), 340–347 (2018)

    Article  MathSciNet  Google Scholar 

  11. Harding, J., Navara, M.: Subalgebras of orthomodular lattices. Order 28, 549–563 (2011)

    Article  MathSciNet  Google Scholar 

  12. Heunen, C., Landsman, N.P., Spitters, B.: Bohrification of operator algebras and quantum logic. Synthese, 3 186, 719–752 (2012)

    Article  MathSciNet  Google Scholar 

  13. Kadison, R.V., Ringrose, J.R.: Theory of Operator Alegebras I, II. Academic Press (1986)

  14. Landsman, K.: Foundations of quantum theory, from classical concepts to operator algebras, Springer Open, Fundamental Theories of Physics 188 (2017)

  15. Lindenhovius, B.: C(A). PhD thesis, Radbound University, Nijmegen (2016)

  16. Lukeš, J., Malý, J., Netuka, I., Spurný, J.: Integral Representation Theory, Applications to Convexity, Banach Spaces, and Potential Theory. de Gruyter, Berlin, New York (2010)

  17. Phelps, R.: Lectures on Choquet’s Theorem. D van Nostrand Company, Princeton, New Jersey (1966)

    MATH  Google Scholar 

  18. Takesaki, M.: Theory of Operator Algebras I, II, III. Springer, Berlin (2001)

    Google Scholar 

Download references

Acknowledgments

The work of the first author was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities (no. 1.7629.2017/8.9).

The work of the second author was supported by the “Grant Agency of the Czech Republic” grant number 17-00941S, “Topological and geometrical properties of Banach spaces and operator algebras II” and by the project OP VVV Center for Advanced Applied Science CZ.02.1.01/0.0/0.0/16_019/000077.

Author information

Authors and Affiliations

  1. Institute of Mathematics and Mechanics, Kazan Federal University, 18, Kremlevskaya, Kazan, 420008, Russia

    Ekaterina Turilova

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

    Jan Hamhalter

Authors
  1. Ekaterina Turilova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jan Hamhalter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ekaterina Turilova.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Turilova, E., Hamhalter, J. Jordan Invariants of Von Neumann Algebras Given by Abelian Subalgebras and Choquet Order on State Spaces. Int J Theor Phys 60, 597–607 (2021). https://doi.org/10.1007/s10773-019-04157-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-019-04157-w

Keywords

Search

Navigation