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Sums of Lattice Homomorphisms

S. J. Bernau, C. B. Huijsmans and B. De Pagter
Proceedings of the American Mathematical Society
Vol. 115, No. 1 (May, 1992), pp. 151-156
DOI: 10.2307/2159578
Stable URL: http://www.jstor.org/stable/2159578
Page Count: 6
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Sums of Lattice Homomorphisms
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Abstract

Let E and F be Riesz spaces and T1, T2,..., Tn be linear lattice homomorphisms (henceforth called lattice homomorphisms) from E to F. If T = ∑n i = 1Ti, then it is easy to check that T is positive and that if x0, x1,... xn ∈ E and $x_i \wedge x_j = 0$ for all i ≠ j, then $\bigwedge^n_{i = 0}Tx_i = 0$. 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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