Advertisement

Positivity

, Volume 20, Issue 1, pp 235–255 | Cite as

Tensor products of function systems revisited

  • Kyung Hoon HanEmail author
Article

Abstract

Based on the Archimedeanization developed by Paulsen and Tomforde, we give an explicit description for the positive cones of maximal tensor products of function systems. From this description, we obtain an approximation theorem for nuclear maps between function systems. As an application, we give elementary proofs on several characterizations of nuclear function systems that are already known.

Keywords

Function system Archimedean ordered space Archimedean ordered \(*\)-vector space Tensor product Nuclear 

Mathematics Subject Classification

46B40 46B28 

Notes

Acknowledgments

The author is grateful to the referee for careful reading and bringing his attention to Refs. [2, 4, 6, 7, 16].

References

  1. 1.
    Alfsen, E.M.: Compact Convex Sets and Boundary Integrals, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 57. Springer, New York (1971)CrossRefGoogle Scholar
  2. 2.
    Buskes, G.J.H.M., van Rooij, A.C.M.: The Archimedean l-group tensor product. Order 10(1), 93–102 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Effros, E.G.: Injectives and Tensor Products for Convex Sets and \(C^*\)-Algebras. University College of Swansea, NATO Advanced Study Institute, Swansea (1972)Google Scholar
  4. 4.
    Ellis, A.J.: Linear operators in partially ordered normed vector spaces. J. London Math. Soc. 41, 323–332 (1966)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Effros, E.G., Ozawa, N., Ruan, Z.-J.: On injectivity and nuclearity for operator spaces. Duke Math. J. 110, 489–521 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Fremlin, D.H.: Tensor products of Archimedean vector lattices. Am. J. Math. 94, 777–798 (1972)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Grobler, J.J., Labuschagne, C.C.A.: The tensor product of Archimedean ordered vector spaces. Math. Proc. Cambr. Philos. Soc. 104(2), 331–345 (1988)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Han, K.H., Paulsen, V.I.: An approximation theorem for nuclear operator systems. J. Funct. Anal. 261, 999–1009 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Kadison, R.V.: A representation theory for commutative topological algebra. Mem. Am. Math. Soc. 1951(7) (1951)Google Scholar
  10. 10.
    Kavruk, A., Paulsen, V.I., Todorov, I.G., Tomforde, M.: Tensor products of operator systems. J. Funct. Anal. 261(2), 267–299 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Kavruk, A., Paulsen, V.I., Todorov, I.G., Tomforde, M.: Quotients, exactness and WEP in the operator systems category. Adv. Math. 235, 321–360 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Lin, H.: An introduction to the classification of amenable \(C^*\)-algebras. World Scientific Publishing Co. Inc, River Edge (2001)CrossRefzbMATHGoogle Scholar
  13. 13.
    Namioka, I., Phelps, R.R.: Tensor products of compact convex sets. Pacific J. Math. 31, 469–480 (1969)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Paulsen, V.I., Tomforde, M.: Vector spaces with an order unit. Indiana Univ. Math. J. 58(3), 1319–1359 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Paulsen, V.I., Todorov, I.G., Tomforde, M.: Operator system structures on ordered spaces. Proc. London Math. Soc. 102(3), 25–49 (2011)Google Scholar
  16. 16.
    van Gaans, O., Kalauch, A.: Tensor products of Archimedean partially ordered vector spaces. Positivity 14(4), 705–714 (2010)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of MathematicsThe University of SuwonHwaseongKorea

Personalised recommendations