= Connected-component permanence of subalgebra invertibility
{title2=$G(A)\text{ is clopen in }A\cap G(B)$}
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 $A$.
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