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.

For a closed unital subalgebra $A\subseteq B$ with common identity, its group of invertible elements of a Banach algebra is relatively open in $A\cap G(B)$ by the Neumann series. It is relatively closed because inversion in $B$ is continuous and $A$ is closed. Consequently each connected component of $A\cap G(B)$ either consists entirely of elements invertible in $A$, or entirely of elements whose inverses do not belong to $A$.