Using the duality between locally compact Hausdorff spaces and commutative $C^*$-algebras one can write down a vocabulary list translating topological notions regarding a locally compact Hausdorff space $X$ into algebraic notions ragarding its ring of functions $C_0(X)$ (see Wegge-Olsen's book, for instance). For example, we have the following correspondences: \begin{align*} \text{open subset of $X$}\quad &\longleftrightarrow\quad\text{ideal in $C_0(X)$}\newline \text{dense open subset of $X$}\quad &\longleftrightarrow\quad\text{essential ideal in $C_0(X)$}\newline \text{closed subset of $X$}\quad &\longleftrightarrow\quad\text{quotient of $C_0(X)$}\newline \text{locally closed subset of $X$}\quad &\longleftrightarrow\quad\text{subquotient of $C_0(X)$}\newline \text{???}\quad &\longleftrightarrow\quad\text{$C^*$-subalgebra in $C_0(X)$} \end{align*} By ideal I always mean a two-sided closed (and hence self-adjoint) ideal.
Well, I can't quite see how to reconvert a $C^*$-subalgebra in $C_0(X)$ into something topological involving only the space $X$. Can you come up with something handy?
Example: A simple example of a subalgebra of a commutative $C^*$-algebra not being an ideal is $$ \mathbb C\cdot(1,1)\subset \mathbb C\oplus\mathbb C. $$
(Alternatively, we could think about this question within the duality of affine algebraic varieties and finitely generated commutative reduced algebras or even within the duality between affine schemes and commutative rings.)
Edit: Since I was not completely satisfied by the response I got here, I reposted this question on MO.