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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.
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