Derivations of noncommutative Banach algebras (Q1203572)

The purpose of this paper is to present some conditions which imply that a continuous derivation of a Banach algebra maps the algebra into its radical. Let \([u,v]\) denote the commutator \(uv-vu\). The authors show that, if \(D\) and \(G\) are continuous derivations of a Banach algebra \(A\) such that \([D^ 2 (x)+ G(x),x]\in \text{rad}(A)\) for all \(x\in A\), then both \(D\) and \(G\) map \(A\) into \(\text{rad}(A)\).