Derivations on Banach algebras (Q1811953)

For a closed left ideal \(J\) in a Banach algebra \(A\) and a derivation \(D\) on \(A\) with separating space \(S(D)\), the ideal \(S(D)\cap J\) is shown to be nilpotent if and only if the restriction of \(D^2\) to \(\overline{\bigcap^\infty_{n=1} (S(D)\cap J)^n}\) is continuous. Furthermore, \(S(D)\) is nilpotent if and only if \(M(D):= \{x\in S(D)\cap R: D(x)\in R\}\) is a nil ideal (where \(R\) denotes the Jacobson radical) or, equivalently, if \(\bigcap^\infty_{n=1} M(D)^n= \{0\}\). The continuity of \(D\) is characterized by \(M(D)= \{0\}\).