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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
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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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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