Abstract: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 $P_\infty $-space.
Keywords: $\sigma $-compact, pseudocompact, $\infty $-compact, $\infty $-compactification, $P_{\infty }$-space, P-point, regular ring, fixed and free ideals
AMS Subject Classification: 54C40