**Double commutant theorem:** For a unital $C^*$-subalgebra $M \subset B(H)$, one has
$$\smash{\overline M}^\text{SOT}=\smash{\overline M}^\text{WOT}=M''.$$

**My question:**
 For a  $C^*$-subalgebra $M \subset B(H)$ but don't assume $M$ contains identity operator  $1$, does
$$\smash{\overline M}^\text{SOT}=\smash{\overline M}^\text{WOT}=M''?$$