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 need to be carefulwhen you say that "unital abelian C*-algebras are the 'same' as compact Hausdorff topological spaces". What Gelfand duality shows is that thecategoryof unital commutative C*-algebras and *-homomorphisms (important) is (anti-)equivalent to the category of compact topological spaces and continuous maps. With *-homomorphisms, you cannot "see" non-selfadjoint subalgebras... $\endgroup$7more comments