TITLE: Compactification of topological spaces
QUESTION [5 upvotes]: Hello,
If we take a localy compact space $X$ and we put $A=C_{b}(X)$ the $\mathbb{R}$-algebra of bounded continous functions on $X$, we have an embeding of topological space 
$$\psi:X\longrightarrow \text{Spec}_{\max}(A) $$
defined by $\psi(x)=I_{x}$, where $I_{x}=\{ f\in A / ~ f(x)=0 \}$, and where $\text{Spec}_{\max}(A)$ is the set of maximal ideal of the ring $A$ with the Zariski topology. Then we define the Stone-Čech compactification of $X$ by $\displaystyle \overline{X}^{sc}=\overline{\psi(X)}$.
We can also define the Alexandroff compactification of $X$ with the same method, we just must to take $A=C_{0}(X)$ the algebra of continous functions with zero limit at infinity.
My question is: Can we define all compactification of $X$ with the same method, in other hand if $\widetilde{X}$ is a compactification of $X$ ($X$ locally compact), then there is a sub-algebra $A$ of the algebra $C_{b}(X)$ such that $\widetilde{X}$ is homeomorphic with $\psi(X)$, where 
$$\psi:X\longrightarrow \text{Spec}_{\max}(A) $$
is defined by $\psi(x)=I_{x}$ the maximal ideal of elements $f\in A$ such that $f(x)=0$.

REPLY [6 votes]: Yes. This is called Constantinescu-Cornea theorem,
Ideale Ränder Riemannscher Flächen, Springer 1963. They deal with 2-dimensional
manifolds only, but the result is true for any locally compact space.
It is stated and proved in this generality in the book
of Marcel Brelot, On topologies and boundaries in potential theory, Springer 1971.