Consider the real line $\mathbb R$ and the Banach algebra of bounded real functions $C_b(X, \mathbb R)$.
By Gelfand duality the space of maximal ideals of $C_b(X, \mathbb R)$ becomes a compact Hausdorff space when equipped with the hull-kernel topology. This is one of many equivalent definitions of the Stone-Cech compactification $\beta \mathbb R$. Although $\beta \mathbb R$ separable, it is well-known that the remainder $\mathbb R^* = \beta \mathbb R - \mathbb R$ is not separable.
Now consider a closed unital subalgebra $A \subset C_b(X, \mathbb R)$. We can again equip the space $I(A)$ of maximal ideals of $A$ with the hull-kernel topology to get a compact Hausdorff space. It turns out the inclusion map $A \to C_b(X, \mathbb R) $ induces a continuous map $\beta \mathbb R \to I(A)$.
For example letting $A$ be the subalgebra of all functions that tend to a limit it's easy to see the only maximal ideals are $\{f \in A: f(x)=0\}$ for each $x \in \mathbb R$ and $\{f \in A: \lim f =0 \}$. The corresponding space is the one-point compactification of $\mathbb R$. The remainder is a singleton and clearly a separable space.
Is there a nice condition on the subalgebra $A$ that ensures the associated 'remainder' is separable? By nice I mean it does not come down to restating separability in terms of maximal ideals and hull-kernels.