A Duality Theorem for Real C* Algebras

  • M. Andrew Moshier
  • Daniela Petrişan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5728)

Abstract

The full subcategory of proximity lattices equipped with some additional structure (a certain form of negation) is equivalent to the category of compact Hausdorff spaces. Using the Stone-Gelfand-Naimark duality, we know that the category of proximity lattices with negation is dually equivalent to the category of real C* algebras. The aim of this paper is to give a new proof for this duality, avoiding the construction of spaces. We prove that the category of C* algebras is equivalent to the category of skew frames with negation, which appears in the work of Moshier and Jung on the bitopological nature of Stone duality.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramsky, S.: Domain theory in logical form. Ann. Pure Appl. Logic 51 (1991)Google Scholar
  2. 2.
    Banaschewski, B., Mulvey, C.J.: A globalisation of the Gelfand duality theorem. Ann. Pure Appl. Logic 137(1-3), 62–103 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Coquand, T., Spitters, B.: Constructive Gelfand duality for C*-algebras. In: Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press, Cambridge (2009)Google Scholar
  4. 4.
    Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Approximating labelled Markov processes. Inf. Comput. 184(1), 160–200 (2003)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Johnstone, P.: Stone Spaces. Cambridge University Press, Cambridge (1982)MATHGoogle Scholar
  6. 6.
    Jung, A., Moshier, M.A.: On the bitopological nature of Stone duality. Technical Report CSR-06-13, School of Computer Science, University of Birmingham (2006)Google Scholar
  7. 7.
    Jung, A., Sünderhauf, P.: On the duality of compact vs. open. In: Andima, S., Flagg, R.C., Itzkowitz, G., Misra, P., Kong, Y., Kopperman, R. (eds.) Papers on General Topology and Applications: Eleventh Summer Conference at the University of Southern Maine. Annals of the New York Academy of Sciences, vol. 806, pp. 214–230 (1996)Google Scholar
  8. 8.
    Dexter, K.: Semantics of probabilistic programs. In: SFCS 1979: Proceedings of the 20th Annual Symposium on Foundations of Computer Science (sfcs 1979), Washington, DC, USA, pp. 101–114. IEEE Computer Society Press, Los Alamitos (1979)Google Scholar
  9. 9.
    Mislove, M.W., Ouaknine, J., Pavlovic, D., Worrell, J.B.: Duality for labelled markov processes. In: Walukiewicz, I. (ed.) FOSSACS 2004. LNCS, vol. 2987, pp. 393–407. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  10. 10.
    Moshier, M.A.: On the relationship between compact regularity and Gentzen’s cut rule. Theoretical Comput. Sci. 316, 113–136 (2004)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Saheb-Djahromi, N.: Cpo’s of measures for nondeterminism. Theor. Comput. Sci. 12, 19–37 (1980)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Smyth, M.: Power domains and predicate transformers: a topological view. In: Díaz, J. (ed.) ICALP 1983. LNCS, vol. 154. Springer, Heidelberg (1983)Google Scholar
  13. 13.
    Smyth, M.: Topology. In: Handbook of Logic in Computer Science. OUP (1993)Google Scholar
  14. 14.
    van Breugel, F., Mislove, M.W., Ouaknine, J., Worrell, J.B.: An intrinsic characterization of approximate probabilistic bisimilarity. In: Gordon, A.D. (ed.) FOSSACS 2003. LNCS, vol. 2620, pp. 200–215. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  15. 15.
    Vickers, S.J.: Topology Via Logic. CUP (1989)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • M. Andrew Moshier
    • 1
  • Daniela Petrişan
    • 2
  1. 1.Department of Mathematics and Computer ScienceChapman UniversityUSA
  2. 2.Department of Computer ScienceUniversity of LeicesterUK

Personalised recommendations