Function Algebras and the Lattice of Compactifications

We provide some conditions as to when K(X) = K(Y) for two locally compact spaces X and Y (where K(X) is the lattice of all Hausdorr compactiications of X). More speciically, we prove that K(X) = K(Y) if and only if C (X)=C 0 (X) = C (Y)=C 0 (Y). Using this result, we prove several extensions to the case where K(X) is embedded as a sub-lattice of K(Y) and to where X and Y are not locally compact. One major contribution is in the use of function algebra techniques. The use of these techniques makes the extensions simple and clean and brings new tools to the subject. 1. Introduction The study of the Hausdorr compactiication of a Tychonoo space X is a well-established branch of topology (the books PW] and Ch] are nice references in this area). There are many results examining the types of compactiications that X may have or the structure of the collection of all the compactiications of X. In fact, if we denote by K(X) the collection of all Hausdorr compactiications of X, we may put an order structure on K(X) which makes it a complete upper semi-lattice (which is a lattice if and only if X is locally compact). In this paper, we examine when K(X) is isomorphic to K(Y) for two spaces X and Y. We also examine when K(X) can be embedded in K(Y). Under what conditions are K(X) and K(Y) isomorphic? One simple suucient