Closed ideals in C(X) and related algebraic structures

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Stokke, Ross Thomas
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Given a topological space X, the ring C(X) of continuous real-valued functions on X is endowed with what is called the 'uniform metric'. The closed ideals of C(X) in this metric are of much interest, and a new, purely algebraic characterization of these ideals is provided. The result is applied to describe the real maximal ideals of C(X), and to characterize several types of topological spaces. A $\Phi$-algebra is an archimedian lattice-ordered algebra closely related to C(X). z-ideals in $\Phi$-algebras are examined, and as an application to this study, several conditions equivalent to regularity in a $\Phi$-algebra are obtained. A uniform metric may also be placed upon a $\Phi$-algebra, and in this metric the closed ideals of a $\Phi$-algebra have received a fair amount of research attention. We give necessary and sufficient conditions to ensure that an ideal of a $\Phi$-algebra is closed, and for two broad classes of $\Phi$-algebras show that these conditions are equivalent, thus generalizing our characterization from the C(X) case.