Abstract
In this paper, we first show that $z_{{\kappa}X}:E_{cc}({\kappa}X){\rightarrow}{\kappa}X$ is $z^{\sharp}$-irreducible and that if $\mathcal{G}(E_{cc}({\beta}X))$ is a base for closed sets in ${\beta}X$, then $E_{cc}({\kappa}X)$ is $C^*$-embedded in $E_{cc}({\beta}X)$, where ${\kappa}X$ is the extension of X such that $vX{\subseteq}{\kappa}X{\subseteq}{\beta}X$ and ${\kappa}X$ is weakly Lindel$\ddot{o}$f. Using these, we will show that if $\mathcal{G}({\beta}X)$ is a base for closed sets in ${\beta}X$ and for any weakly Lindel$\ddot{o}$f space Y with $X{\subseteq}Y{\subseteq}{\kappa}X$, ${\kappa}X=Y$, then $kE_{cc}(X)=E_{cc}({\kappa}X)$ if and only if ${\beta}E_{cc}(X)=E_{cc}({\beta}X)$.
