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 $f(x)=0$.