Abstract
When are so happy in a vector lattice that all band preserving linear operators turn out to be in the ideal centre? This question was raised by Wickstead (Representation and duality in Representation and duality Riesz spaces. Compos Math 35(3):225–238, 1977) (though not explicitly stated) and by the first author and Chil and Meyer (On the centre of a vector lattice. Indag Math 23:167–183, 2012). The answer depends on the vector lattice in which the operator in question acts. However, in this article we focus our attention on this question. First, we give a complete description of those vector lattices \(E\) with the property that every orthomorphism on \(E\) is a central operator. Secondly, we provide 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, in Positivity (2014, in press). (Published online: 11 Dec 2013)].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Academic Press, Orlando (1985)
Bigard, A., Keimel, K., Wolfenstein, S.: Groupes et anneaux ré ticulés. In: Lecture Notes in Mathematics, vol. 608. Springer, Berlin (1977)
Birkhoff, G.: Lattice theory. In: American Mathematical Society Colloquium Publications, vol. 25. American Mathematical Society, Providence (1967)
Buck, R.C.: Multiplication operators. Pac. J. Math. 11, 95–104 (1961)
Conrad, P.F., Diem, J.E.: The ring of polar preserving endomorphisms of an abelian lattice ordered group. Ill. J. Math. 15, 224–240 (1971)
Chil, E.: Order Bounded orthosymmetric bilinear operator. Czechoslov. Math. J. 61, 873–880 (2011)
Chil, E., Meyer, M.: On the centre of a vector lattice. Indag. Math. 23, 167–183 (2012)
Goullet de Rugy, A.: La structure idéal des M-espaces. J. Math. Pures Appl. 51, 331–373 (1972)
Hart, D.R.: Disjointness Preserving Operators, Thesis. California Institute of Technology, Pasadena (1983)
Luxembourg, W.A.J., Zaanen, A.C.: Riesz spaces I. North-Holland, Amsterdam (1971)
Meyer, M.: Richesses du centre d’un espace réticulé. Math. Ann. 236, 147–169 (1978)
Meyer, M.: Quelques propriétés des homomorphismes d’espace vectoriel réticulé. Equipe d’Analyse, E. R. A. 294, Paris (1979)
Nakano, H.: Modern Spectral Theory. Maruzen, Tokyo (1950)
Quinn, J.: Intermediate Riesz spaces. Pac. J. Math. 56, 225–263 (1975)
Schaefer, H.H.: Banach Lattices and Positive Operators. Springer, Berlin (1974)
Toumi, M.A.: When orthomorphisms are in the ideal center. Positivity (2014, in press). (Published online: 11 Dec 2013)
Triki, A.: On algebra homomorphisms in complex almost \(f\)-algebras. Comment. Math. Univ. Carol. 43, 23–31 (2002)
Wickstead, A.W.: The ideal centre of a Banach lattice. In: Proceedings of the Royal Irish Academy. Section A, vol. 76, no. 4, pp. 15–23 (1976). MR0420214 (54 #8228)
Wickstead, A.W.: Representation and duality of multiplication operators on Archimedean Riesz spaces. Compos. Math. 35(3), 225–238 (1977)
Wickstead, A.W.: Banach lattices with trivial centre. In: Proceedings of the Royal Irish Academy. Section A, vol. 88, no. 1, pp. 71–83 (1988). MR974286
Zaanen, A.C.: Examples of orthomorphisms. J. Approx. Theory 13, 192–204 (1975)
Acknowledgments
The authors wish to express their thanks to the referee for stimulating comments and suggestions concerning the first version of this paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chil, E., Hassen, B. & Mohamed, M. A relationship between the space of orthomorphisms and the centre of a vector lattice. Positivity 19, 457–465 (2015). https://doi.org/10.1007/s11117-014-0308-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-014-0308-2