Compactnes of the space of minimal prime ideals of C(X) and C(Y) vs. that of C(XxY)
by
Melvin Henriksen
Harvey Mudd College, Claremont CA 91711

All spaces considered are Tychonoff (i.e., subspaces of a compact (Hausdoff) space.) For a space X, C(X) denotes its ring of real-valued continuous functions and MinC(X) the space of minimal prime ideals in the hull-kernel topology. MinC(X)is always 0-dimensional, countably compact, and is known to be compact iff the classical ring of fractions of C(X)is a von Neumann regular ring iff for each f in C(X), there is a g in C(X)such that fg = 0 and |f| + |g| is not a zero divisor. (Thus coz(f) and coz(g)have empty intersection and dense union.) Such spaces are called cozero complemented. In a recent preprint, R.Levy and J.Shapiro posed some questions about the relationship between X and Y being cozero complemented vs. XxY being cozero complemented. Some samples of partial answers are: Theorem 1 If XxY is cozero complemented and either (i) X and Y are compact, or (ii) both X and Y have an isolated point, then X and Y are cozero complemented.(It is not known if either (i) or (ii) is needed.) Theorem 2 Because every space in which every family of pairwise disjoint open sets is countable, an arbitrary product of separable cozero complemented spaces is cozero complemented. Let D(m) denote the discrete space of infinite cardinality m, and betaD(m) its Stone-Cech compactification. Theorem 3 betaD(m)xbetaD(m) is cozero complemented iff m is countable. More complicated product theorems are given and questions posed. This is part of joint research with R.G.Woods

Date received: December 6, 2002

Copyright © 2002 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 # cakg-05.