Nice Operators into G-Spaces



G-spaces are a class of \(L_1\)-preduals introduced by Grothendieck. We prove that if every extreme operator from any Banach space into a G-space, X, is a nice operator (that is, its adjoint preserves extreme points), then X is isometrically isomorphic to \(c_0(I)\) for some set I. One of the main points in the proof is a characterization of spaces of type \(c_0(I)\) by means of the structure topology on the extreme points of the dual space.


Banach space Extreme operator Nice operator G-space  Structure topology 

Mathematics Subject Classification

46B20 46B04 



This study was supported by Spanish MICINN and FEDER Project No. MTM2012-31755 and by Junta de Andalucía and FEDER Grants FQM-185 and FQM-3737.


  1. 1.
    Al-Halees, H., Fleming, R.J.: Extreme points methods and Banach–Stone theorems. J. Aust. Math. Soc. 75, 125–143 (2003)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Blumenthal, R.M., Lindenstrauss, J., Phelps, R.R.: Extreme operators into \(C(K)\). Pac. J. Math. 15, 747–756 (1965)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Brilloüet, N.: Extreme and nice operators. Southeast Asian Bull. Math. 9, 23–30 (1985)MathSciNetMATHGoogle Scholar
  5. 5.
    Cabrera-Serrano, A.M., Mena-Jurado, J.F.: On extreme operators whose adjoints preserve extreme points. J. Convex Anal. 22, 247–258 (2015)MathSciNetMATHGoogle Scholar
  6. 6.
    Cabrera-Serrano, A.M., Mena-Jurado, J.F.: Facial topology and extreme operators. J. Math. Anal. Appl. 427, 899–904 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Choy, S.T.L.: Extreme operators on function spaces. Ill. J. Math. 33, 301–309 (1989)MathSciNetMATHGoogle Scholar
  8. 8.
    Effros, E.G.: On a class of real Banach spaces. Isr. J. Math. 9, 430–458 (1971)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fakhoury, H.: Préduaux de \(L\)-espace: Notion de Centre. J. Funct. Anal. 9, 189–207 (1972)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gendler, A.: Extreme operators in the unit ball of \({\cal L}({\cal C}(X),{\cal C}(Y))\) over the complex field. Proc. Am. Math. Soc. 57, 85–88 (1976)MathSciNetMATHGoogle Scholar
  11. 11.
    Grothendieck, A.: Une caractérisation vectorielle-métrique des espaces \(L_1\). Can. J. Math. 7, 552–561 (1955)CrossRefMATHGoogle Scholar
  12. 12.
    Harmand, P., Werner, D., Werner, W.: M-Ideals in Banach Spaces and Banach Algebras. Lecture Notes in Mathematics, vol. 1547. Springer, Berlin (1993)Google Scholar
  13. 13.
    Jerison, M.: Characterizations of certain spaces of continuous functions. Trans. Am. Math. Soc. 70, 103–113 (1951)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Khalil, R., Salih, A.: Extreme and nice operators on certain function spaces. Sci. Math. Jpn. 65, 423–430 (2007)MathSciNetMATHGoogle Scholar
  15. 15.
    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)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Lacey, H.E.: The Isometric Theory of Classical Banach Spaces. Springer, Berlin-Heidelberg-New York (1974)CrossRefMATHGoogle Scholar
  17. 17.
    Lazar, A.J., Lindenstrauss, J.: Banach spaces whose duals are \(L_1\) spaces and their representing matrices. Acta Math. 126, 165–194 (1971)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lima, Å.: Intersection properties of balls and subspaces in Banach spaces. Trans. Am. Math. Soc. 227, 1–62 (1977)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Lima, Å., Olsen, G., Uttersrud, U.: Intersections of \(M\)-ideals and \(G\)-spaces. Pac. J. Math. 104, 175–177 (1983)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Lindenstrauss, J.: Extensions of Compact Operators. Memoirs of the American Mathematical Society, Number 48, Providence (1964)MATHGoogle Scholar
  21. 21.
    Mena-Jurado, J.F., Montiel-Aguilera, F.: A note on nice operators. J. Math. Anal. Appl. 289, 30–34 (2004)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    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)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    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)MathSciNetMATHGoogle Scholar
  24. 24.
    Navarro-Pascual, J.C., Navarro, M.A.: Unitary operators in real von Neumann algebras. J. Math. Anal. Appl. 386, 933–938 (2012)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Navarro-Pascual, J.C., Navarro, M.A.: Nice operators and surjective isometries. J. Math. Anal. Appl. 426, 1130–1142 (2015)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Rao, T.S.S.R.K.: Characterizations of some classes of \(L^1\)-preduals by the Alfsen–Effros structure topology. Isr. J. Math. 42, 20–32 (1982)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Rao, T.S.S.R.K.: Nice surjections on spaces of operators. Proc. Indian Acad. Sci. Math. Sci. 116, 401–409 (2006)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Roy, N.M.: An \(M\)-ideal characterization of \(G\)-spaces. Pac. J. Math. 92, 151–160 (1981)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Sharir, M.: Characterizations and properties of extreme operators into \({\cal C}(Y)\). Isr. J. Math. 12, 174–183 (1972)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Sharir, M.: Extremal structure in operator spaces. Trans. Am. Math. Soc. 186, 91–111 (1973)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Sharir, M.: A counterexample on extreme operators. Isr. J. Math. 24, 320–337 (1976)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Sharir, M.: A non-nice extreme operator. Isr. J. Math. 26, 306–312 (1977)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Uttersrud, U.: On M-ideals and the Alfsen–Effros structure topology. Math. Scand. 43, 369–381 (1978)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Universiti Sains Malaysia 2015

Authors and Affiliations

  1. 1.Departamento de Análisis Matemático, Facultad de CienciasUniversidad de GranadaGranadaSpain

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