$A$ is a C$^*\! $-algebra and $(x_n)_{n\in \mathbb{N}} \subseteq A $. If $\ $ $yx_n\to 0 $ for all $y\in A$, Is it true that $x_n$ is weakly convergent to $0$ ?

For unitals this is trivial. For characters like $w\in \Omega (A)$ we have $w(x_n)\to 0$ but if for all functionals, I don't know.