Closed Ideals in C(X) and Related Algebraic Structures
by
Ross Stokke
University of Manitoba

Given a topological space X, the ring C(X) is endowed with the `uniform metric'. The closed ideals of C(X) in this metric of 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 a real archimedian lattice-ordered algebra with an identity element that is a weak order unit. 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 we give necessary and sufficient conditions for an ideal in a \Phi-algebra to be (uniformly) closed. Moreover, we show that for two relatively broad classes of \Phi-algebras, these conditions are equivalent, thus generalizing our characterization from the C(X) case.

Date received: June 25, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caao-39.