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 $$\smash{\overline M}^\text{SOT}=\smash{\overline M}^\text{WOT}=M''?$$