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A characterization of Riesz \(n\)-morphisms and applications. (English) Zbl 1147.46023

Summary: Let \(X_1,X_2,\dots,X_n\) be realcompact spaces and \(Z\) be a topological space. Let \(\pi: C(X_1)\times C(X_2)\times \cdots\times C(X_n)\to C(Z)\) be a Riesz \(n\)-morphism. We show that there exist functions \(\sigma_i :Z\to X_i\) \((i = 1, 2,\dots, n)\) and \(w\in C(Z)\) such that
\[ \pi(f_1,f_2,\dots,f_n)= wf_1\circ\sigma_1f_2\circ \sigma_2\cdots f_n\circ\sigma_n \]
and \(\sigma_1,\sigma_2,\dots,\sigma_n\) are continuous on \(\{z : w(z)\neq 0\}\). This fact extends a result of K. Boulabiar [Indag. Math., New Ser. 13, No. 4, 419–432 (2002; Zbl 1035.47020)] and leads to one of the main results of K. Boulabiar [Commun. Algebra 32, No. 10, 3955–3967 (2004; Zbl 1060.06023)] with a more elementary proof.

MSC:

46E05 Lattices of continuous, differentiable or analytic functions
46A40 Ordered topological linear spaces, vector lattices
06F25 Ordered rings, algebras, modules
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References:

[1] Abramovich Y. A., An Invitation to Operator Theory 50 (2002) · Zbl 1022.47001
[2] Aliprantis C. D., Positive Operators (1985)
[3] Benamor F., J. Math. Anal. Appl. 322 (2) pp 599– (2006) · Zbl 1106.47063 · doi:10.1016/j.jmaa.2005.09.038
[4] Boulabiar K., Indag. Math. 13 (4) pp 419– (2002) · Zbl 1035.47020 · doi:10.1016/S0019-3577(02)80023-4
[5] Boulabiar K., Comm. Algebra 32 (10) pp 3955– (2004) · Zbl 1060.06023 · doi:10.1081/AGB-200027793
[6] Boulabiar K., Contemporary Mathematics 328 pp 99– (2003) · doi:10.1090/conm/328/05773
[7] Ercan Z., Proc. Amer. Math. Soc. 134 (3) (2006) · Zbl 1090.54025 · doi:10.1090/S0002-9939-05-08012-3
[8] Ercan Z., Proc. Amer. Math. Soc. 133 (12) pp 3609– (2005) · Zbl 1087.46038 · doi:10.1090/S0002-9939-05-07930-X
[9] Galindo A. M., Studia Mathematica 166 (1) pp 83– (2005) · Zbl 1129.46018 · doi:10.4064/sm166-1-6
[10] Gillman L., Rings of Continuous Functions (1960) · Zbl 0093.30001 · doi:10.1007/978-1-4615-7819-2
[11] Shirota T., Osoka Math. J. 4 pp 23– (1952)
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