Advertisement

Positivity

, Volume 20, Issue 2, pp 337–342 | Cite as

A relationship between the space of orthomorphisms and the centre of a vector lattice revisited

  • Mohamed Ali ToumiEmail author
Article
  • 147 Downloads

Abstract

In (Chil et al. Positivity, 2014), the authors claim to give a counterexample to the main result, about Wickstead’s question, in a recent paper of Toumi (see Theorem 3, When orthomorphisms are in the ideal center, Positivity 18(3):579–583, 2014). In this note we show that their example is consistent with the main result of Toumi and not a counterexample.

Keywords

Center of a vector lattice Orthomorphism 

Mathematics Subject Classification

Primary 06F25 46A40 

Notes

Acknowledgments

The author is grateful to Professor A. E. Gutman for helping in characterizing maximal ideals of \(C^{\infty }\left( X\right) \) and for communicating a second method of the proof of Theorem 3. The author is much indebted to the Referee for his careful reading of the manuscript and for his valuable suggestions.

References

  1. 1.
    Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Academic Press, London (1985)zbMATHGoogle Scholar
  2. 2.
    Bigard, A., Keimel, K.: Sur les endomorphismes conservant les polaires d’un groupe réticulé archimédien. Bull. Soc. Math. France 97, 381–398 (1969)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Buck, R.C.: Multiplication operators. Pacific J. Maths. 11, 95–104 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chil, E., Bourokba, H., Mokaddem, M.: A relationship between the space of orthomorphisms and the centre of a vector lattice. Positivity (2014) (in press) Google Scholar
  5. 5.
    Gutman, A.E.: A private communication (2015)Google Scholar
  6. 6.
    Kouki, N., Toumi, M.A.: Derivations on universally complete f-algebras. Indag. Math. 26(1), 1–18 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces I. North-Holland, Amsterdam (1971)zbMATHGoogle Scholar
  8. 8.
    De Pagter, B.: \(f\)-Algebras and Orthomorphisms. Ph. D. Thesis, Leiden (1981)Google Scholar
  9. 9.
    Schaefer, H.H.: Banach Lattice and Positive Operators. Springer, Berlin (1974)CrossRefzbMATHGoogle Scholar
  10. 10.
    Toumi, M.A.: When orthomorphisms are in the ideal center. Positivity 18(3), 579–583 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Wickstead, A.W.: Representation and duality of multiplication operators on Archimedean Riesz spaces. Compositio Math. 35(3), 225–238 (1977)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Wójtowicz, M., Wiśniewska, H.: The problem of central orthomorphisms in a class of F-lattices. Indag. Math. New Ser. 26(2), 393–403 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Département de Mathématiques,Faculté des Sciences de BizerteUniversité de CarthageBizerteTunisia

Personalised recommendations