Afrika Matematika

, Volume 25, Issue 2, pp 379–396 | Cite as

The Dineen problem for homogeneous orthogonally-additive polynomials

Article
First Online:
Received:
Accepted:

Abstract

In this paper, we study the extensions to the Dedekind completion and the Aron-Berner extensions of orthogonally-additive polynomials and we show that this class is preserved by these extensions, which gives a positive answer to a problem posed by Dineen (LNM 332:77–111, 1973).

Keywords

Orthosymmetric multimorphisms Orthogonally additive polynomials 

Mathematics Subject Classification (2000)

Primary 06F25 46G25 47B65 Secondary 46G20 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Academic Press, Orlando (1985)MATHGoogle Scholar
  2. 2.
    Aron, R., Berner, P.: A Hahn–Banach extension theorem for analytic mappings. Bull. Soc. Math. France 106(1), 3–24 (1978)MATHMathSciNetGoogle Scholar
  3. 3.
    Benyamini, Y., Lassalle, S., Llavona, J.L.G.: Homogeneous orthogonally-additive polynomials on Banach lattices. Bull. London Math. Soc. 38(3), 459–469 (2006)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bernau, S.J., Huijsmans, C.B.: The order bidual of almost f-algebras and d-algebras. Trans. Am. Math. Soc. 347, 4259–4275 (1995)MATHMathSciNetGoogle Scholar
  5. 5.
    Buskes, G., van Rooij, A.: Almost f-algebras: commutativity and the Cauchy-Schwarz inequality. Positivity 4, 227–331 (2000)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Carando, D., Lassalle, S., Zalduendo, I.: Orthogonally additive polynomials over C(K) are measures—a short proof. Int. Equ. Oper. Theory 56(4), 597–602 (2006)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    De Pagter, B.: f-Algebras and Orthomorphisms. Thesis, Leiden (1981)Google Scholar
  8. 8.
    Dineen, S.: Holomorphically complete locally convex topological vector spaces. LNM 332, 77–111 (1973)MathSciNetGoogle Scholar
  9. 9.
    Fuchs, L.: Partially Ordered Algebraic Systems. Pergamon Press, Oxford (1963)MATHGoogle Scholar
  10. 10.
    Grobler, J.J., Labuschagne, C.C.A.: The tensor product of Archimedean ordered vector spaces. Math. Proc. Camb. Phil. Soc. 104, 331–345 (1988)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Huijsmans, C.B., De Pagter, B.: The order bidual of lattice ordered algebras. J. Funct. Anal. 59, 61–74 (1984)CrossRefGoogle Scholar
  12. 12.
    Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces I. North-Holland, Amsterdam (1971)MATHGoogle Scholar
  13. 13.
    Meyer-Neiberg, P.: Banach Lattices. Springer, Berlin (1974)Google Scholar
  14. 14.
    Pérez-Garcıa, D., Villanueva, I.: Orthogonally additive polynomials on spaces of continuous functions. J. Math. Anal. Appl. 306(1), 97–105 (2005)MATHMathSciNetGoogle Scholar
  15. 15.
    Schaefer, H.H.: Banach Lattices and Positive Operators. Springer, New York (1974)CrossRefMATHGoogle Scholar
  16. 16.
    Sundaresan, K.: Geometry of Spaces of Homogeneous Polynomials on Banach Lattices. Applied Geometry and Discrete Mathematics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science,vol. 4, pp. 571–586. American Mathematical Society, Providence (1991)Google Scholar
  17. 17.
    Toumi, M.A.: Extensions of orthosymmetric lattice bimorphisms. Proc. Am. Math. Soc. 134, 1615–1621 (2006)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Toumi, M.A.: Commutativity of almost f-algebras. Positivity 11(2), 357–368 (2007)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Toumi, M.A.: A decomposition theorem for orthogonally additive polynomials on archimedean vector lattices. Bull. Belgian Math. Soc. (2013, to appear)Google Scholar
  20. 20.
    Zaanen, A.C.: Riesz Spaces II. North-Holland, Amsterdam (1983)MATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Département de MathématiquesFaculté des Sciences de BizerteZarzouna, BizerteTunisia

Personalised recommendations