Non-maximal closed prime ideals in a unital commutative Banach algebra are accessible

Abstract

It is proved that in a commutative unital Banach algebra, every non-maximal closed prime ideal is accessible. Specifically, it can be represented as the intersection of all closed ideals of the algebra that properly contain it. Consequently, all derivations and epimorphisms on commutative unital semi-prime Banach algebras are continuous. Moreover, any separating ideal in a commutative unital Banach algebra is nilpotent and, therefore, a nil ideal.