Let $X = [0, 1)$ be a ray and $C(X)$ the algebra of bounded continuous real functions. The spectrum of $C(X)$ is the Stone-Cech compactification $\beta [0,1) $ of the ray. It's easy to see the compactification is separable. It's somewhat harder to see the remainder $\beta [0,1) - [0,1)$ is not separable.
Suppose we consider instead the algebra $C(X) $ of functions on $X$ that tend to some limit. The spectrum of this algebra is just the arc $[0,1]$ and the remainder is a singleton which is separable.
More generally suppose $A \subset C(X)$ is an arbitrary closed unital subalgebra. For convenience replace $A$ with $A' = \{f \in C(X):$ there exists $x \in [0,1)$ and $g \in A$ such that $f$ and $g$ coincide on $[x,1)\}$. This ensures $A'$ separates points of [0,1) and the spectrum contains a copy of the ray. Again we have a remainder and by Gelfand duality it is a continuous image of the Stone-Cech remainder.
Is there a nice property of the algebras $A$ or $A'$ that is equivalent to the associated remainder being separable? By nice I mean it does not come down to expressing properties of the remainder in terms of maximal ideals and hull-kernels?
