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Czechoslovak Mathematical Journal

, Volume 61, Issue 4, pp 873–880 | Cite as

Order bounded orthosymmetric bilinear operator

  • Elmiloud Chil
Article

Abstract

It is proved by an order theoretical and purely algebraic method that any order bounded orthosymmetric bilinear operator b: E×EF where E and F are Archimedean vector lattices is symmetric. This leads to a new and short proof of the commutativity of Archimedean almost f-algebras.

Keywords

vector lattice positive bilinear operator orthosymmetric bilinear operator lattice bimorphism 

MSC 2010

06F25 46A40 47A65 

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References

  1. [1]
    C.D. Aliprantis, O. Burkinshaw: Positive Operators. Springer, Berlin, 2006.MATHGoogle Scholar
  2. [2]
    M. Basly, A. Triki: FF-algèbres Archimédiennes réticulées. University of Tunis, Preprint, 1988.Google Scholar
  3. [3]
    S. J. Bernau, C.B. Huijsmans: Almost f-algebras and d-algebras. Math. Proc. Camb. Philos. Soc. 107 (1990), 287–308.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    A. Bigard, K. Keimel, S. Wolfenstein: Groupes et Anneaux Réticulés. Lecture Notes in Mathematics Vol. 608. Springer, Berlin-Heidelberg-New York, 1977.MATHGoogle Scholar
  5. [5]
    G. Birkhoff, R. S. Pierce: Lattice-ordered rings. Anais Acad. Brasil. Ci. 28 (1956), 41–69.MathSciNetGoogle Scholar
  6. [6]
    Q. Bu, G. Buskes, A.G. Kusraev: Bilinear Maps on Product of Vector Lattices: A Survey. Positivity. Trends in Mathematics. Birkhäuser, Basel, 2007, pp. 97–126.Google Scholar
  7. [7]
    G. Buskes, B. de Pagter, A. van Rooij: Functional calculus in Riesz spaces. Indag. Math. New Ser. 4 (1991), 423–436.CrossRefGoogle Scholar
  8. [8]
    G. Buskes, A.G. Kusraev: Representation and extension of orthoregular bilinear operators. Vladikavkaz. Math. Zh. 9 (2007), 16–29.MathSciNetGoogle Scholar
  9. [9]
    G. Buskes, A. van Rooij: Small Riesz spaces. Math. Proc. Camb. Philos. Soc. 105 (1989), 523–536.MATHCrossRefGoogle Scholar
  10. [10]
    G. Buskes, A. van Rooij: Almost f-algebras: Commutativity and the Cauchy-Schwarz inequality. Positivity 4 (2000), 227–231.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    G. Buskes, A. van Rooij: Squares of Riesz spaces. Rocky Mt. J. Math. 31 (2001), 45–56.MATHCrossRefGoogle Scholar
  12. [12]
    G. Buskes, A. van Rooij: Bounded variation and tensor products of Banach lattices. Positivity 7 (2003), 47–59.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    J. J. Grobler, C.C.A. Labuschagne: The tensor product of Archimedean ordered vector spaces. Math. Proc. Camb. Philos. Soc. 104 (1988), 331–345.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    C.B. Huijsmans, B. de Pagter: Subalgebras and Riesz subspaces of an f-algebra. Proc. Lond. Math. Soc. III. Ser. 48 (1984), 161–174.MATHCrossRefGoogle Scholar
  15. [15]
    W.A. J. Luxemburg, A.C. Zaanen: Riesz spaces I. North-HollandMathematical Library, Amsterdam-London, 1971.Google Scholar
  16. [16]
    H. Nakano: Product spaces of semi-ordered linear spaces. J. Fac. Sci., Hakkaidô Univ. Ser. I. 12 (1953), 163–210.MATHGoogle Scholar
  17. [17]
    A.C. Zaanen: Riesz spaces II. North-Holland Mathematical Library, Amsterdam-New York-Oxford, 1983.Google Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2011

Authors and Affiliations

  1. 1.Institut préparatoire aux études d’ingenieurs de TunisMontfleuryTunisia

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