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

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

Thanks so much for your time and your answers.