Advertisement

Positivity

, Volume 23, Issue 1, pp 35–53 | Cite as

Quantization in \(*\)-algebras and an algebraic analog of Arveson’s extension theorem

  • G. H. EsslamzadehEmail author
  • F. Taleghani
Article
  • 16 Downloads

Abstract

In this paper our main goal is showing that many of quantization results in functional analysis are rather algebraic. Following Esslamzadeh and Taleghani (Linear Algebra Appl 438:1372–1392, 2013), we call every subspace [resp. self-adjoint unital subspace] of a unital \(*\)-algebra, a quasi operator space [resp. quasi operator system]. Local operator systems can be realized as quasi operator spaces. Arveson’s extension theorem asserts that \(\mathcal {B}(\mathcal {H})\) is an injective object in the category of operator systems. We show that Arveson’s theorem remains valid in the much larger category of quasi operator systems. This shows that Arveson’s theorem as a non commutative extension of Hahn–Banach theorem, is of purely algebraic nature. Moreover we prove an algebraic extension of Ruan’s theorem which gives a charcterization of bounded quasi operator spaces. Then we identify the largest and the smallest of such quasi quantizations of a seminormed space \(\mathcal {X}\), which we call \(QMAX(\mathcal {X})\) and \(QMIN(\mathcal {X})\).

Keywords

Completely positive map Quasi operator system Quasi operator space Matricially seminormed space Minimal and maximal quantization 

Mathematics Subject Classification

46L07 46B40 

Notes

Acknowledgements

The authors would like to express their sincere thanks to the anonymous referee, Professors Lyudmila Turowska, Ivan G. Todorov, Kyung Hoon Han and Matin Mathieu for their useful comments. Special thanks to Professor Turowska, who patiently wrote her comments on different versions of manuscript. Section 3 of this paper was written when the first author was visiting Department of Mathematical Sciences of Chalmers University of Technology and the University of Gothenburg. The first author wishes to express his sincere thanks to the Department and Professors L. Turowska and M. Asadzadeh for their hospitality during this visit.

References

  1. 1.
    Alizadeh, R., Esslamzadeh, G.H.: On the existence of unitarily invariant norm under some conditions. Linear Multilinear Algebra 58(3), 367–375 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arveson, W.: Subalgebras of \(C^*\)-algebras. Acta Math. 123, 141–224 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arveson, W.: Subalgebras of \(C^*\)-algebras II. Acta Math. 128, 271–308 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Berberian, S.K.: Baer \(*\)-Rings. Springer, Berlin (1972)CrossRefzbMATHGoogle Scholar
  5. 5.
    Choi, M.D., Effros, E.G.: Injectivity and operator spaces. J. Funct. Anal. 24, 156–209 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dosiev, A.: Local operator spaces, unbounded operators and multinormed C*-algebras. J. Funct. Anal. 255, 1724–1760 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Effros, E.G., Ruan, Z.J.: Operator Spaces. Clarendon Press, Oxford (2000)zbMATHGoogle Scholar
  8. 8.
    Esslamzadeh, G.H., Taleghani, F.: Structure of quasi operator systems. Linear Algebra Appl. 438, 1372–1392 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Esslamzadeh, G.H., Moazzami, M., Taleghani, F.: Structure of quasi ordered \(*\)-vector spaces. Iran. J. Sci. Technol. 38, 445–453 (2014)MathSciNetGoogle Scholar
  10. 10.
    Esslamzadeh, G. H., Turowska, L.: Arveson’s extension theorem in \(*\)-algebras. Preprint, arXiv:1311.5065 (2013)
  11. 11.
    Karn, A.K.: Adjoining an order unit to a matrix ordered space. Positivity 9, 207–223 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Karn, A.K.: A p-theory of ordered normed spaces. Positivity 14, 441–458 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Luthra, P., Kumar, A., Rajpal, V.: Polynomials in operator space theory, matrix ordering and algebraic aspects. (To appear in Positivity)Google Scholar
  14. 14.
    Palmer, T.W.: Banach Algebras and the General Theory of \(*\)-Algebras II. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
  15. 15.
    Paulsen, V.I.: Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  16. 16.
    Paulsen, V.I., Tomforde, M.: Vector spaces with an order unit. Indiana Univ. Math. J. 58, 1319–1359 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Paulsen, V.I., Todorov, I.G., Tomforde, M.: Operator system structures on ordered spaces. Proc. Lond. Math. Soc. 102, 25–49 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Phillips, N.C.: Inverse limits of \(C^*\)-algebras. J. Operat. Theory 19, 159–195 (1989)MathSciNetGoogle Scholar
  19. 19.
    Ruan, Z.J.: Subspaces of \(C^*\)-algebras. J. Funct. Anal. 76, 217–230 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ruan, Z.J.: Operator amenability of \(A(G)\). Am. J. Math. 117, 1449–1474 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Schreiner, W.J.: Matrix regular operator spaces. J. Funct. Anal. 152, 136–175 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Stinespring, W.F.: Positive functions on \(C^*\)-algebras. Proc. Am. Math. Soc. 6, 211–216 (1955)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Webster, C. J.: Local operator spaces and applications. UCLA Ph.D. thesis (1997)Google Scholar
  24. 24.
    Werner, W.: Subspaces of \(L(H)\) that are \(*\)-invariant. J. Funct. Anal. 193, 207–223 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of SciencesShiraz UniversityShirazIran
  2. 2.Department of MathematicsIslamic Azad University, Lahijan BranchLahijanIran

Personalised recommendations