Resolvent components in a closed unital subalgebra (${\rho }_B(x)\subseteq {\rho }_A(x)$)

For a closed unital subalgebra $B$ of a unital Banach algebra $A$ and $x\in B$, the smaller-algebra algebra resolvent set is relatively open and closed in the larger one. Openness follows from the Neumann series; closedness follows from continuity of inversion in $A$ and norm closedness of $B$. Every larger-algebra resolvent component is therefore either retained whole or absorbed into the smaller-algebra Banach algebra spectrum. The unbounded component is always retained, since inverses at large modulus are norm-convergent series in $x$.

For a closed subalgebra of a Banach algebra $B\subseteq A$ with common identity, the algebra resolvent set ${\rho }_B(x)$ is clopen in ${\rho }_A(x)$. A point in ${\rho }_A(x)$ has a small connected disc on which membership of ${\rho }_B(x)$ is constant, so it cannot lie in $\partial {\sigma }_B(x)$. Also ${\sigma }_A(x)\subseteq {\sigma }_B(x)$; a point in the interior of the former is in the interior of the latter. Together these observations prove the boundary inclusion, without assuming a spectral-boundary theorem.