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.



Can one detect a cyclic and separating vector for a concrete $C^*$-algebra using a dense subalgebra?
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.


