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Sums of lattice homomorphisms


Authors: S. J. Bernau, C. B. Huijsmans and B. de Pagter
Journal: Proc. Amer. Math. Soc. 115 (1992), 151-156
MSC: Primary 46A40; Secondary 06F20, 47B60, 47B65
DOI: https://doi.org/10.1090/S0002-9939-1992-1086322-8
MathSciNet review: 1086322
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Abstract: Let $ E$ and $ F$ be Riesz spaces and $ {T_1},{T_2}, \ldots ,{T_n}$ be linear lattice homomorphisms (henceforth called lattice homomorphisms) from $ E$ to $ F$. If $ T = \sum\nolimits_{i = 1}^n {{T_i}} $, then it is easy to check that $ T$ is positive and that if $ {x_0},{x_1}, \ldots {x_n} \in E$ and $ {x_i} \wedge {x_j} = 0$ for all $ i \ne j$, then $ \wedge _{i = 0}^nT{x_i} = 0$. The purpose of this note is to show that if $ F$ is Dedekind complete, the above necessary condition for $ T$ to be be the sum of $ n$ lattice homomorphisms is also sufficient. The result extends to sums of disjointness preserving operators, thereby leading to a characterization of the ideal of order bounded operators generated by the lattice homomorphisms.


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  • [1] C. D. Aliprantis and O. Burkinshaw, Positive operators, Academic Press, Orlando, 1985. MR 809372 (87h:47086)
  • [2] S. J. Bernau, Orthomorphisms of archimedean vector lattices, Math. Proc. Cambridge Philos. Soc. 89 (1981), 119-128. MR 591978 (82b:47043)
  • [3] David C. Carothers and William A. Feldman, Sums of homomorphisms on Banach lattices, J. Operator Theory (to appear). MR 1150624 (93a:47039)
  • [4] Michel Duhoux and Mathieu Meyer, A new proof of the lattice structure of orthomorphisms, J. London Math. Soc. (2) 25 (1981), 375-378. MR 653395 (83d:46006)
  • [5] C. B. Huijsmans and B. de Pagter, Disjointness preserving and diffuse operators, Compositio Math. 79 (1991), 351-374. MR 1121143 (92k:47071)
  • [6] W. A. J. Luxemburg and A. C. Zaanen, Riesz spaces. I, North-Holland, Amsterdam, 1971.
  • [7] M. Meyer, Le stabilisateur d'un espace vectoriel réticulé, C. R. Acad Sci. Paris Sér. A 283 (1976), 249-250. MR 0433191 (55:6170)
  • [8] H. H. Schaefer. Banach lattices and positive operators. Grundlehren Math. Wiss., vol. 215, Springer, Berlin, 1974. MR 0423039 (54:11023)
  • [9] A. C. Zaanen, Riesz spaces. II, North-Holland, Amsterdam, 1983. MR 704021 (86b:46001)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1086322-8
Keywords: Lattice homomorphism, disjointness preserving operator, $ n$disjoint operator, Dedekind complete Riesz space, band projection, minimal positive extension
Article copyright: © Copyright 1992 American Mathematical Society

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