Positivity

, Volume 22, Issue 2, pp 629–652 | Cite as

Polynomials in operator space theory: matrix ordering and algebraic aspects

Article
  • 65 Downloads

Abstract

We extend the \(\lambda \)-theory of operator spaces given in Defant and Wiesner (J. Funct. Anal. 266(9): 5493–5525, 2014), that generalizes the notion of the projective, Haagerup and Schur tensor norm for operator spaces to matrix ordered spaces and Banach \(*\)-algebras. Given matrix regular operator spaces and operator systems, we introduce cones related to \(\lambda \) for the algebraic operator space tensor product that respect the matricial structure of matrix regular operator spaces and operator systems, respectively. The ideal structure of \(\lambda \)-tensor product of \(C^*\)-algebras has also been discussed.

Keywords

Matrix regular operator spaces Operator systems Tensor products Ideals 

Mathematics Subject Classification

Primary 46L06 46L07 Secondary 46L05 47L25 

Notes

Acknowledgements

The authors would like to thank Andreas Defant for providing a copy of [19]. The authors would also like to thank the referee for useful comments and suggestions.

References

  1. 1.
    Allen, S.D., Sinclair, A.M., Smith, R.R.: The ideal structure of the Haagerup tensor product of \({C}^*\)-algebras. J. Reine Angew. Math. 442, 111–148 (1993)MathSciNetMATHGoogle Scholar
  2. 2.
    Blecher, D.P.: Geometry of the tensor product of C*-algebras. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 104, pp. 119–127. Cambridge University Press (1988)Google Scholar
  3. 3.
    Blecher, D.P., Paulsen, V.I.: Tensor products of operator spaces. J. Funct. Anal. 99(2), 262–292 (1991)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Defant, A., Wiesner, D.: Polynomials in operator space theory. J. Funct. Anal. 266(9), 5493–5525 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Effros, E.G., Ruan, Z.J.: Operator Spaces. Clarendon Press, Oxford (2000)MATHGoogle Scholar
  6. 6.
    Haagerup, U.: The Grothendieck inequality for bilinear forms on C*-algebras. Adv. Math. 56(2), 93–116 (1985)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Haagerup, U., Musat, M.: The Effros–Ruan conjecture for bilinear forms on C*-algebras. Inventiones Mathematicae 174(1), 139–163 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Han, K.H.: The predual of the space of decomposable maps from a \({C}^*\)-algebra into a von Neumann algebra. J. Math. Anal. Appl. 402(2), 463–476 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Itoh, T.: Completely positive decompositions from duals of \({C}^*\)-algebras to von Neumann algebras. Math. Japon. 51, 89–98 (2000)MathSciNetMATHGoogle Scholar
  10. 10.
    Jain, R., Kumar, A.: Operator space tensor products of C*-algebras. Mathematische Zeitschrift 260(4), 805–811 (2008)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras II, vol. 2. American Mathematical Society, Providence (1996)MATHGoogle Scholar
  12. 12.
    Kaniuth, E.: A Course in Commutative Banach Algebras, vol. 246. Springer, Berlin (2008)MATHGoogle Scholar
  13. 13.
    Kavruk, A.S., Paulsen, V.I., Todorov, I.G., Tomforde, M.: Tensor products of operator systems. J. Funct. Anal. 261(2), 267–299 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kumar, A.: Operator space projective tensor product of \( {C}^* \)-algebras. Math. Z. 237(2), 211–217 (2001)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ng, W.H.: Two characterizations of the maximal tensor product of operator systems. arXiv preprint arXiv:1503.07097 (2015)
  16. 16.
    Paulsen, V.I., Tomforde, M.: Vector spaces with an order unit. Indiana Univ. Math. J. 58, 1319–1359 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Rajpal, V., Kumar, A., Itoh, T.: Schur tensor product of operator spaces. Forum Math. 27, 3635–3655 (2015)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Schreiner, W.J.: Matrix regular operator spaces. J. Funct. Anal. 152(1), 136–175 (1998)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Wiesner, D.: Polynomials in operator space theory. Ph.D. Thesis, Der Andere Verlag, Tönning, Lübeck, Marburg (2009)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DelhiDelhiIndia
  2. 2.Department of Mathematics, Shivaji CollegeUniversity of DelhiDelhiIndia

Personalised recommendations