Function algebras and the lattice of compactifications

We provide some conditions as to when $K(X) \cong K(Y)$ for two locally compact spaces $X$ and $Y$ (where $K(X)$ is the lattice of all Hausdorff compactifications of $X$). More specifically, we prove that $K(X) \cong K(Y)$ if and only if $C^*(X)/C_0(X) \cong C^*(Y)/C_0(Y)$. 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.