 Spectrum in a closed unital subalgebra

If a closed unital subalgebra $A\subseteq B$ contains $x$, then ${\sigma }_A(x)$ is obtained from ${\sigma }_B(x)$ by adjoining some bounded connected components of its complement. Membership of $(x-\lambda 1{)}^{-1}$ in $A$ is constant on every connected component of the resolvent set, and it always holds on the unbounded component.