Relation between spectrums of algebra and subalgebra

Question

I want to ask this. Suppose that $\mathcal{A}$ is a Banach algebra and $\mathcal{B}$ a subalgebra of $\mathcal{A}$. I want to show that if $σ_{\mathcal{A}}(x)$ does not seperate $\mathbb{C}$ then $σ_{\mathcal{A}}(x) = σ_{\mathcal{B}}(x)$.

Any help/hint would be grateful. Thanks.