In light of the well-known theorem of Gelfand that, bluntly put, ends up saying that unital abelian C\*-algebras are the 'same' as compact Hausdorff topological spaces, I tried to compile a dictionary of concepts between these two objects. More specifically, given a compact Hausdorff space $X$, I ask in what manner are topological properties of $X$ encoded into $C(X) := C(X, \mathbb{C})$? And, conversely, in what way do algebraic properties of the latter manifest topologically in the former? Here is the elementary list I was able to gather:


$\cdot$ $C(X)$ has $2^n$ idempotent elements $\Leftrightarrow$ $X$ has $n$ connected components 


$\cdot$ $C(X)$ separable $\Leftrightarrow$ $X$ metrizable


$\cdot$ $C(X)$ isomorphic to $C(Y)$ $\Leftrightarrow$ $X$ homeomorphic to $Y$


$\cdot$ continuous functions from $g:X \to Y$ induce \*-homomorphisms $\hat g: C(X) \to C(Y)$ and vice-versa


$\cdot$ there is a bijective correspondence between ideals of $C(X)$ and open
sets of $X$


What do subalgebras of $C(X)$ correspond to? If this is not a well-posed question please tell me why. Subalgebras are a very natural substructure to consider and yet I am at a loss as to how it translates over.


If you have any additions (or corrections) to the above dictionary, please share them.



Do subalgebras of C(X) admit a description in terms of the compact Hausdorff space X?
Gelfand duality asserts that $C(-)$ is an anti-equivalence from the category of compact hausdorff spaces to the category of commutative unital $C^{\ast}$-algebras. Now, for a continuous map $f : X \to Y$ it is not hard to show that


$f$ is surjective $\Longleftrightarrow$ $C(f) : C(Y) \to C(X)$ is injective


Sketch of proof: $\Rightarrow$ is trivial, and $\Leftarrow$ follows from Tietze extension theorem $\square$. By the way, we also have:


$f$ is injective $\Longleftrightarrow$ $C(f) : C(Y) \to C(X)$ is surjective


Therefore, $C^{\ast}$-subalgebras of $C(X)$ (i.e. closed unital $\ast$-subalgebras) correspond to surjective maps $X \to Y$, where $Y$ is compact hausdorff. It is well-known that these maps are quotient maps. The partial order of $C^{\ast}$-subalgebras of $C(X)$ is anti-isomorphic to the partial order of quotients of $X$.


