Compactification of topological spaces

Question

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 $f(x)=0$.

Answer 1

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.

Answer 2

Note : In order to define the alexandrov compactification of $X$ you have to take the algebra of function converging at infinity (not just those converging to $0$)

If by a compactification of $X$ you mean a compact Hausdorff topological space $Y$ equiped a continuous map $ X \rightarrow Y$ having a dense image, then the answer is true :

If you consider instead the algebra of complex function then it is immediate from the gelfand duality :

A such space is just a compact Hausdorff topological space equipped with a surjective continuous map form $\overline{X}^{sc}$. But the gelfand duality assert that compact Hausdorff topological space corresponds to commutative $C^ * $ algebra. And it is easy to see, that a continuous map between compact space is surjective if and only if the induce map between the $C^ * $ algebra is injective (and hence isometric by classical result on $C^ * $ algebra).

So, compactification of $X$ corresponds to sub-$C^*$-algebra of $C^b(X)$. ie sub-algebra of $C^b(X)$ which are closed under complexe conjugaition, closed for the normt topology, and which contain $1$.

One can then easily see that such algebra also corresponds to sub-algebra of the algbera of real bounded function, which contain $1$ and are closed for the norm topology.

NB : local compactness is actually useless for all this to work.