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WHEN LATTICE HOMOMORPHISMS OF ARCHIMEDEAN VECTOR LATTICES ARE RIESZ HOMOMORPHISMS

  • MOHAMED ALI TOUMI (a1)

Abstract

Let A, B be Archimedean vector lattices and let (ui)iI, (vi)iI be maximal orthogonal systems of A and B, respectively. In this paper, we prove that if T is a lattice homomorphism from A into B such that for each λ∈ℝ+ and iI, then T is linear. This generalizes earlier results of Ercan and Wickstead (Math. Nachr279 (9–10) (2006), 1024–1027), Lochan and Strauss (J. London Math. Soc. (2) 25 (1982), 379–384), Mena and Roth (Proc. Amer. Math. Soc.71 (1978), 11–12) and Thanh (Ann. Univ. Sci. Budapest. Eotvos Sect. Math.34 (1992), 167–171).

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References

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[1]Aliprantis, C. D. and Burkinshaw, O., Positive Operators (Academic Press, Orlando, FL, 1985).
[2]De Pagter, B., f-Algebras and orthomorphisms. Thesis, Leiden, 1981.
[3]Ercan, Z. and Wickstead, A. W., ‘When a lattice homomorphism is a Riesz homomorphism’, Math. Nachr. 279(9–10) (2006), 10241027.
[4]Lochan, R. and Strauss, D., ‘Lattice homomorphisms of spaces of continuous functions’, J. London Math. Soc. (2) 25 (1982), 379384.
[5]Luxemburg, W. A. J. and Zaanen, A. C., Riesz Spaces I (North-Holland, Amsterdam, 1971).
[6]Mena, R. and Roth, B., ‘Homomorphisms of lattices on continuous functions’, Proc. Amer. Math. Soc. 71 (1978), 1112.
[7]Schaefer, H. H., Banach Lattices and Positive Operators, Grundlehren Math. Wiss., 215 (Springer, Berlin, 1974).
[8]Thanh, D. T., ‘A generalization of a theorem of R. Mena and B. Roth’, Ann. Univ. Sci. Budapest. Eotvos Sect. Math. 34 (1992), 167171.
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WHEN LATTICE HOMOMORPHISMS OF ARCHIMEDEAN VECTOR LATTICES ARE RIESZ HOMOMORPHISMS

  • MOHAMED ALI TOUMI (a1)

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