Sums of lattice homomorphisms

Bernau, S. J.; Huijsmans, C. B.; de Pagter, B.

doi:10.1090/S0002-9939-1992-1086322-8

Sums of lattice homomorphisms
HTML articles powered by AMS MathViewer

by S. J. Bernau, C. B. Huijsmans and B. de Pagter PDF

Proc. Amer. Math. Soc. 115 (1992), 151-156 Request permission

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.

References

Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Pure and Applied Mathematics, vol. 119, Academic Press, Inc., Orlando, FL, 1985. MR 809372
S. J. Bernau, Orthomorphisms of Archimedean vector lattices, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 1, 119–128. MR 591978, DOI 10.1017/S030500410005800X
David C. Carothers and William A. Feldman, Sums of homomorphisms on Banach lattices, J. Operator Theory 24 (1990), no. 2, 337–349. MR 1150624
Michel Duhoux and Mathieu Meyer, A new proof of the lattice structure of orthomorphisms, J. London Math. Soc. (2) 25 (1982), no. 2, 375–378. MR 653395, DOI 10.1112/jlms/s2-25.2.375
C. B. Huijsmans and B. de Pagter, Disjointness preserving and diffuse operators, Compositio Math. 79 (1991), no. 3, 351–374. MR 1121143

Riesz spaces

Mathieu Meyer, Le stabilisateur d’un espace vectoriel réticulé, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 5, Aii, A249–A250. MR 433191
Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039
A. C. Zaanen, Riesz spaces. II, North-Holland Mathematical Library, vol. 30, North-Holland Publishing Co., Amsterdam, 1983. MR 704021