Wallman-type compaerifications and function lattices

Caterino, Alessandro ; Vipera, Maria Cristina

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 82 (1988) no. 4, pp. 679-683

Résumé

Let $F \subset C^{\ast} (X)$ be a vector sublattice over $\mathbb{R}$ which separates points from closed sets of $X$. The compactification $e_{F}X$ obtained by embedding $X$ in a real cube via the diagonal map, is different, in general, from the Wallman compactification $\omega (Z(F))$. In this paper, it is shown that there exists a lattice $F_{z}$ containing $F$ such that $\omega (Z(F)) = \omega (Z(F_{z})) = e_{F}X$. In particular this implies that $\omega (Z(F)) \ge e_{F}X$. Conditions in order to be $\omega (Z(F)) = e_{F}X$ are given. Finally we prove that, if $\alpha X$ is a compactification of $X$ such that $Cl_{\alpha X} (\alpha X \setminus X)$ is $0$-dimensional, then there is an algebra $A \subset C^{ast} (X)$ such that $\omega (Z(A)) = e_{A} X = \alpha X$.