Construct a banach algebra $\displaystyle \mathcal{B}$ and a unital subalgebra $\displaystyle \mathcal{A}$ such that $\displaystyle \sigma_{\mathcal{A}}(x)\neq\sigma_{\mathcal{B}}(x)$ for some $\displaystyle x\in\mathcal{A}$
Let $A = A(D)$, the disk algebra. That is $D = \{z \in C : |z|\le 1\}$ and $A$ be the algebra of all functions $f: D \to C$ which are continuous on $D$ and analytic on the interior of $D$. Let $\Gamma = \{z \in C : |z|= 1\}$ and $B = C(\Gamma)$, the algebra of all continuous functions on $\Gamma$. Then with the sup norm, both of them are Banach Algebras. Also the elements of $A(D)$ can be realized as elements of $C(\Gamma)$ by restricting them to $\Gamma$. Thus, in this way, we can say that $A(D)$ is a Banach subalgebra of $C(\Gamma)$. Now let $f : D \to C$ be defined by $f(z) = z$, then the spectrum of $f$ computed in $A$ is $D$ and computed in $B$ is $\Gamma$. I hope this answers your question.