Do subalgebras of C(X) admit a description in terms of the compact Hausdorff space X? 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.
 A: Closed subalgebras of $C(X)$ are called uniform algebras, and there is a huge literature on them.  You might start by reading Gamelin's book, Uniform Algebras, or simple by Googling "uniform algebras". 
A: This may be an interesting add-on for Martin's answer. In this paper Pavlov and Troitskii show that an inclusion of commutative $C^*$-algebras $C(X) \to C(Y)$ (with $X$ and $Y$ compact Hausdorff), which allows a positive unital conditional expectation $E \colon C(Y) \to C(X)$ that satisfies an index condition, corresponds via Gelfand duality to a branched covering $p \colon Y \to X$. The latter means that $p$ is a continuous closed and open surjection with boundedly many preimages $p^{-1}(x)$ at every $x \in X$,
A: A bit more exotic, a finitely generated subalgebra (no matter $*$-subalgebra or not) corresponds to a continuous map to an affine variety over $\mathbb C$ (continuous in the euclidean topology), such that image is Zariski dense. Sometimes this is a useful observation.
A: 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 $X$.
A: To add to the dictionary, some strange spaces called extremely disconnected correspond to rather natural abelian C* algebras, the Von Neumann. They are natural, since they correspond to C* algebras acting on $L^2(X,m)$ for not too bad measure spaces $(X,m)$; roughly, you look at the spectrum of $L^\infty(X,m)$ acting on $L^2$.
