A Characterization of Riesz n-Morphisms and Applications

Abstract

Let X ₁, X ₂,…, X _n be realcompact spaces and Z be a topological space. Let π:C(X ₁) × C(X ₂) × ··· × C(X _n ) → C(Z) be a Riesz n-morphism. We show that there exist functions σ _i : Z → X _i (i = 1, 2,…, n) and w ∊ C(Z) such that

and σ₁, σ₂,…, σ _n are continuous on {z : w(z) ≠ 0}. This fact extends a result in Boulabiar (2002) and leads to one of the main results in Boulabiar (2004) with a more elementary proof.

Key Words:

• f-algebras
• Realcompact space
• Riesz n-morphism

2000 Mathematics Subject Classification:

• Primary 46E25, 06F25

ACKNOWLEDGMENTS

This work is presented in the Workshop/Conference on Postivity, CARTHAPO'06. The first author would like to thank Karim Boulabiar for his interest, comments, and his hospitality. Boulabiar has informed the first author that Ph.D. student N. Khleifia has an alternative proof of Theorem 4, which appears in his Ph.D. thesis.

We would like to thank the referee for many helpfull corrections.

Notes

Communicated by K. M. Rangaswamy.