= Boundary inclusion of spectra in a closed unital subalgebra
{title2=$\partial\sigma_B(x)\subseteq\partial\sigma_A(x)$}
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.
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