Let $A$ be a non-unital $C^*$-algebra. Let $S\subseteq A^*$ be a set of continuous functionals that separates the points of $A$. Every element $\omega \in A^*$ extends uniquely to a strictly continuous functional $\omega \in M(A)^*$, so we can ask: does $S$ also separate the points of $M(A)$? Concretely, if $m \in M(A)$ and $\omega(m)=0$ for all $\omega \in S$, do we have $m=0$?
In some cases, this is automatically true. For instance, if $AS \subseteq S$ or $SA\subseteq S$ this is easily seen to be true. However, it is not clear to me if this is true in general. My intuition tells me that the answer is "no" but I have trouble finding a concrete counterexample. It's not even clear to me what happens if $A= C_0(X)$ or $A=B_0(H)$.