The compactifications to which an element of $C^{\ast } (X)$ extends

Abstract:

We first determine a necessary and sufficient condition for a function $f \in {C^\ast }(X)$, which extends to a compactification of $X$, to extend to a smaller compactification. We apply this result to show that when $|\beta X\backslash X| \leqq {\aleph _0}$ there is an $f \in {C^\ast }(X)$ which extends to no compactification other than $\beta X$. Two examples show that when ${\aleph _0} < |\beta X\backslash X| \leqq c$ no such definite result may be obtained.