Let $A\subseteq B(H) $ be a $C^*$ algebra. I know that if $A$ is unital, then by von Neumann density theorem SOT closure$(A)=A''$ and $A''$ is the von Neumann algebra generated by $A$ in $B(H)$.
Is it true that if $A$ is not unital, still SOT closure $(A)=A ''$?