Rings of continuous functions vanishing at infinity

Aliabad, A. R. ; Azarpanah, F. ; Namdari, M.

Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 3, pp. 519-533

Résumé

We prove that a Hausdorff space $X$ is locally compact if and only if its topology coincides with the weak topology induced by $C_\infty (X)$. It is shown that for a Hausdorff space $X$, there exists a locally compact Hausdorff space $Y$ such that $C_\infty(X)\cong C_\infty(Y)$. It is also shown that for locally compact spaces $X$ and $Y$, $C_\infty(X)\cong C_\infty(Y)$ if and only if $X\cong Y$. Prime ideals in $C_\infty(X)$ are uniquely represented by a class of prime ideals in $C^*(X)$. $\infty$-compact spaces are introduced and it turns out that a locally compact space $X$ is $\infty$-compact if and only if every prime ideal in $C_\infty(X)$ is fixed. The existence of the smallest $\infty$-compact space in $\beta X$ containing a given space $X$ is proved. Finally some relations between topological properties of the space $X$ and algebraic properties of the ring $C_\infty(X)$ are investigated. For example we have shown that $C_\infty(X)$ is a regular ring if and only if $X$ is an $\infty$-compact $\operatorname{P}_\infty$-space.

Détail

Classification : 54C40, 54D45
Keywords: $\sigma $-compact; pseudocompact; $\infty $-compact; $\infty $-compactification; $\operatorname{P}_{\infty }$-space; P-point; regular ring; fixed and free ideals