Let $X = \mathbb{R}^n$, and consider a nondegenerate representation $\rho: C_0(X) \to B(H)$ where $B(H)$ is the algebra of bounded operators on a separable Hilbert space. The support of a vector $v \in H$ is defined to be the complement in $X$ of the union of all open sets $U$ such that $\rho(f)v = 0$ for every $f \in C_0(U)$.

Suppose $v$ has compact support. My intuition is that any function $g \in C_0(X)$ which restricts to the zero function on $supp(v)$ should satisfy $\rho(g)v = 0$, but I can't quite prove it. Here is what I have so far.

Let $\mathcal{F}$ denote the collection of open sets $U$ such that $\rho(f)v = 0$ for $f \in C_0(U)$. By definition $V = supp(v)^c$ is the union of the open sets in $\mathcal{F}$, and moreover a simple partition of unity argument shows that $\mathcal{F}$ is closed under finite unions. If we can show that $V \in \mathcal{F}$ then we are done because by the hypotheses $g \in C_0(V)$. I'm sure I'm just missing something simple; can anyone help?

This result should be true if $X$ is any separable metric space equipped with a proper coarse structure, so I suppose the proper setting for this question is metric geometry. That should explain the title of the question and the tags.