This question comes up after I read the chapter 7 : Banach algberas and spectrum theory from Conway's book. As we have known that if $X$ is compact space, then all the maximal ideals of $C(X)=\{ f : X\to \mathbb{C}\}$ are of the forms $m_x =\{f\in C(X) : f(x)=0\}$. Now assume that $X$ is just locally compact and let $X^{+}$ and $\overline{X}$ be the one-point compactification, and Stone-Cech compactification of $X$ respectively. Let $I_{0}(X)$, $I(X^+), I(\overline{X})$ be the set of all maximal ideals of $C_{0}(X), C(X^+), C(\overline{X})$ respectively. My question here is that: do we have any relation between these sets $I_{0}(X)$, $I(X^+), I(\overline{X})$ ? For example, if we know $I(X^+)$, how can we find $I_0(X)$ (because if we know the answer for this, it seems for me that one proof of the Stone-Cech theorem could be found from here) ?

PS. It seems for me that the forum does not let me write parentheses by TeX.

This question comes up after I read the chapter 7 : Banach algebras and spectrum theory from Conway's book. As we have known that if $X$ is compact space, then all the maximal ideals of $C(X)=\{ f : X\to \mathbb{C}\}$ are of the forms $m_x =\{f\in C(X) : f(x)=0\}$. Now assume that $X$ is just locally compact and let $X^{+}$ and $\overline{X}$ be the one-point compactification, and Stone-Čech compactification of $X$ respectively. Let $I_{0}(X)$, $I(X^+), I(\overline{X})$ be the set of all maximal ideals of $C_{0}(X), C(X^+), C(\overline{X})$ respectively. My question here is that: do we have any relation between these sets $I_{0}(X)$, $I(X^+), I(\overline{X})$ ? For example, if we know $I(X^+)$, how can we find $I_0(X)$ (because if we know the answer for this, it seems for me that one proof of the Stone-Čech theorem could be found from here) ?