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.