Let $A$ be a $C^*$ algebra of operators acting on some Hilbert space $H$, and $A_0$ is a norm dense $*$-subalgebra of $A$. Suppose there exists some unit vector $\xi \in H$, such that (i) $A_0 \xi$ is dense in $H$; (ii) $a\xi = 0$ if and only if $a=0$ for all $a \in A_0$, i.e. $\xi$ is separating for $A_0$.
Question: is $\xi$ also separating for $A$, i.e. is it true that $a \xi = 0$ implies $a=0$ for all $a \in A$?
It is clear that if the vector state $\omega_\xi$ of $\xi$ is a trace on $A$, then the answer is affirmative by (i). More generally, if $\omega_\xi$ is a KMS state (so that we have sufficient control of $\omega_\xi$ from being away from a trace using suitable automorphisms of $A$), then the answer is still affirmative. But in general, I don't know the answer, and suspect that there is a counter-example.
Best Answer
Here's a counter-example. Take $A_0:=\mathbb{C}[F(s,t)]\subset A:=\mathrm{C}^*_{\mathrm{r}}(F(s,t))$, where $F(s,t)$ is the free group on $\{s,t\}$, $E\colon A\to \mathrm{C}^*_{\mathrm{r}}(F(s))$ the canonical conditional expectation, and $\psi\colon \mathrm{C}^*_{\mathrm{r}}(F(s)) \cong C(\mathbb{R}/\mathbb{Z})\ni f\mapsto 2\int_0^{1/2} f(r)\,dr$. Note that the conditional expectation $E$ is faithful and the state $\psi$ is faithful on the algebra $\mathbb{C}[F(s)]$ of trigonometric polynomials. Put $\varphi=\psi\circ E$ and let $(\pi,H,\xi)$ denote the GNS-triplet. Since $\varphi$ is faithful on $A_0$, the vector $\xi$ is separating on $\pi(A_0)$. It is also cyclic because it is a GNS vector. However $\xi$ is not separating for $\pi(A)$, since $\psi$ is supported on $[0,1/2]$ and $\pi$ is faithful.