Closed subalgebra of a Banach algebra
= Closed subalgebra of a Banach algebra
= Banach subalgebra
{c}
{synonym}
A norm-closed <subalgebra> of a <Banach algebra> is itself a Banach algebra with the inherited norm: completeness follows from closedness, and the submultiplicative norm bound restricts to it. A unital inclusion uses the same identity as the ambient algebra. Inverse membership may still differ between the two algebras; <spectrum in a closed unital subalgebra> describes that difference.