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Positivity

, Volume 17, Issue 3, pp 841–861 | Cite as

Quantum systems and representation theorem

  • Anar Dosi
Article
  • 131 Downloads

Abstract

In this paper we investigate quantum systems which are locally convex versions of abstract operator systems. Our approach is based on the duality theory for unital quantum cones. We prove the unital bipolar theorem and provide a representation theorem for a quantum system being represented as a quantum \(L^{\infty }\)-system.

Keywords

Quantum cone Multinormed \(W^{*}\)-algebra Quantum system  Quantum order 

Mathematics Subject Classification (1991)

Primary 46K10 Secondary 47L25 47L60 

References

  1. 1.
    Bourbaki, N.: Elements of Mathematics. General Topology, chap. 1–4. Springer, Berlin (1989)Google Scholar
  2. 2.
    Choi, M.D., Effros, E.G.: Injectivity and operator spaces. J. Funct. Anal. 24, 156–209 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Dosiev, A.A.: Local operator spaces, unbounded operators and multinormed \(C^{\ast }\)-algebras. J. Funct. Anal. 255, 1724–1760 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Dosiev, A.A.: Quantized moment problem. Comptes Rendus Math. 344(1), 627–630 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Dosi, A.A.: Local operator algebras, fractional positivity and quantum moment problem. Trans. AMS 363(2), 801–856 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Dosi, A.A.: Quantum duality, unbounded operators and inductive limits. J. Math. Phys. 51(6), 1–43 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dosi, A.A.: Noncommutative Mackey theorem. Int. J. Math. 22(3), 535–544 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dosi, A.A.: Bipolar theorem for quantum cones. Funct. Anal. Appl. 46(3), 228–231 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dosi, A.A.: Quantum cones and their duality. PreprintGoogle Scholar
  10. 10.
    Effros, E.G., Ruan, Z.-J.: Operator Spaces. Clarendon Press, Oxford (2000)zbMATHGoogle Scholar
  11. 11.
    Effros, E.G., Webster, C.: Operator Analogues of Locally Convex Spaces. Operator Algebras and Applications (Samos 1996) (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 495). Kluwer, Dordrecht (1997)Google Scholar
  12. 12.
    Effros, E.G., Winkler, S.: Matrix convexity: operator analogues of the bipolar and Hanh–Banach theorems. J. Funct. Anal. 144, 117–152 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Helemskii, A.Ya.: Quantum Functional Analysis. MCCME, Moscow (2009)Google Scholar
  14. 14.
    Kutateladze, S.S.: Fundamentals of Functional Analysis, vol. 12. Kluwer Academic Publishers, Dordrecht (1995)Google Scholar
  15. 15.
    Paulsen, V.: Completely bounded maps and operator algebras. In: Cambridge Studies in Advanced Mathematics, vol. 78. Cambridge University Press, Cambridge (2002)Google Scholar
  16. 16.
    Pisier, G.: Introduction to Operator Space Theory. Cambridge University Press, London (2003)zbMATHGoogle Scholar
  17. 17.
    Schaefer, H.: Topological Vector Spaces. Springer, Berlin (1970)Google Scholar
  18. 18.
    Webster, C.: Local operator spaces and applications. Ph.D. University of California, Los Angeles (1997)Google Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Middle East Technical University Northern Cyprus CampusMersin 10Turkey

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