Characterisation of $C_{c}(X)$ as a dense subalgebra of $C_{0}(X)$.

Question

Consider a non-unital commutative $C^{*}$-algebra $A$ and a dense subalgebra $B\subset A$. We know that $A\cong C_{0}(X)$, for some locally compact Hausdorff space $X$. I would like to know if we are able to impose properties on $B$ which assure that $B\cong C_{c}(X)$ under the isomorphism from $A$ to $C_{0}(X)$. Is this possible?

Answer 1

An element $x$ of $B$ is characterized by the property: the set of characters not vanishing at $x$ has compact closure in the weak-$\ast$ topology. Equivalently, the set of characters not vanishing at $x$ is contained in a set of the form $\{\gamma \mid \lvert\gamma(a)\rvert\geq\epsilon\}$ for some $a\in A$ and positive $\epsilon$.