Nice operators into \(L_1\)-preduals

Abstract

A Banach space X is said to be “nice” if every extreme operator from any Banach space into X is a nice operator (that is, its adjoint preserves extreme points). We prove that a nice \(L_1\)-predual space is isometrically isomorphic to \(c_0(I)\) for some set I. In fact, by using the structure topology, we get a more general result which allows us to conclude that there exist extreme non-nice operators into certain spaces of continuous affine functions which are not \(L_1\)-preduals.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Al-Halees, H., Fleming, R.J.: Extreme points methods and Banach–Stone theorems. J. Aust. Math. Soc. 75, 125–143 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Al-Halees, H., Fleming, R.J.: Extreme contractions on continuous vector-valued function spaces. Proc. Am. Math. Soc. 134, 2661–2666 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Alfsen, E.M.: Compact Convex Sets and Boundary Integrals. Springer, New York (1971)

    Google Scholar 

  4. 4.

    Alfsen, E.M., Effros, E.G.: Structure in real Banach spaces. II. Ann. Math. 96, 129–173 (1972)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Asimow, L., Ellis, A.J.: Convexity Theory and its Applications in Funtional Analysis. Academic Press, London (1980)

    Google Scholar 

  6. 6.

    Brilloüet, N.: Extreme and nice operators. Southeast Asian Bull. Math. 9, 23–30 (1985)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Cabrera-Serrano, A.M., Mena-Jurado, J.F.: Extreme operators whose adjoints preserve extreme points. J. Convex Anal. 22, 247–258 (2015)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Cabrera-Serrano, A.M., Mena-Jurado, J.F.: Facial topology and extreme operators. J. Math. Anal. Appl. 427, 899–904 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Cabrera-Serrano, A.M., Mena-Jurado, J. F.: Nice operators into \(G\)-spaces. Bull. Malays. Math. Sci. Soc. doi:10.1007/s40840-015-0155-8

  10. 10.

    Choy, S.T.L.: Extreme operators on function spaces. Ill. J. Math. 33, 301–309 (1989)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Effros, E.G.: Structure in simplexes. Acta Math. 117, 103–121 (1967)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Fabian, M., Habala, P., Hájek, P., Montesinos Santalucía, V., Pelant, J., Zizler, V.: Funtional Analysis and Infinite-Dimensional Geometry. Springer, New York (2001)

    Google Scholar 

  13. 13.

    Gleit, A.: On the construction of split-face topologies. Trans. Am. Math. Soc. 194, 291–299 (1974)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Grothendieck, A.: Une caractérisation vectorielle-métrique des espaces \(L_1\). Can. J. Math. 7, 552–561 (1955)

    Article  MATH  Google Scholar 

  15. 15.

    Harmand, P., Werner, D., Werner, W.: M-Ideals in Banach Spaces and Banach Algebras. Lecture Notes in Mathematics, vol 1547. Springer, Berlin (1993)

  16. 16.

    Khalil, R., Salih, A.: Extreme and nice operators on certain function spaces. Sci. Math. Japan 65, 423–430 (2007)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Labuschagne, L.E., Mascioni, V.: Linear maps between \(C^*\)-algebras whose adjoints preserve extreme points of the dual ball. Adv. Math. 138, 15–45 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Lacey, H.E.: The Isometric Theory of Classical Banach Spaces. Springer, Berlin (1974)

    Google Scholar 

  19. 19.

    Lazar, A.J., Lindenstrauss, J.: Banach spaces whose duals are \(L_1\) spaces and their representing matrices. Acta Math. 126, 165–194 (1971)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Lima, Å.: Intersection properties of balls and subspaces in Banach spaces. Trans. Am. Math. Soc. 227, 1–62 (1977)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Lindenstrauss, J.: Extensions of compact operators. Mem. Am. Math. Soc. 48 (1964)

  22. 22.

    Mena-Jurado, J.F., Montiel-Aguilera, F.: A note on nice operators. J. Math. Anal. Appl. 289, 30–34 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Mena-Jurado, J.F., Navarro-Pascual, J.C.: A note on extreme points in dual spaces. Acta Math. Sin. (Engl. Ser.) 29, 471–476 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Morris, P.D., Phelps, R.R.: Theorems of Krein–Milman type for certain convex sets of operators. Trans. Am. Math. Soc. 150, 183–200 (1970)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Navarro-Pascual, J.C., Navarro, M.A.: Unitary operators in real von Neumann algebras. J. Math. Anal. Appl. 386, 933–938 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Navarro-Pascual, J.C., Navarro, M.A.: Nice operators and surjective isometries. J. Math. Anal. Appl. 426, 1130–1142 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Navarro-Pascual, J.C., Navarro, M.A.: Differentiable functions and nice operators. Banach J. Math. Anal. 10, 96–107 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Rao, T.S.S.R.K.: Nice surjections on spaces of operators. Proc. Indian Acad. Sci. Math. Sci. 116, 401–409 (2006)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

Thanks are due to the referees for several useful suggestions and remarks that improved the final version of the paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Juan F. Mena-Jurado.

Additional information

Supported by Spanish MICINN and FEDER projects No. MTM2012-31755 and MTM2015-65020-P and by Junta de Andalucía and FEDER Grants FQM-185 and FQM-3737.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cabrera-Serrano, A.M., Mena-Jurado, J.F. Nice operators into \(L_1\)-preduals. Rev Mat Complut 30, 417–426 (2017). https://doi.org/10.1007/s13163-016-0219-9

Download citation

Keywords

  • Extreme operator
  • Nice operator
  • \(L_1\)-predual
  • Structure topology

Mathematics Subject Classification

  • 46B20
  • 46B04