Derivations on prime rings and Banach algebras (Q2770336)

For an associative Banach algebra \(A\), a Jordan derivation is a linear map \(D\colon A\to A\) such that \(D(x\circ y)=(Dx)\circ y+x\circ(Dy)\) for all \(x,y\in A\), where we denote as usual by \(x\circ y=(xy+yx)/2\) the Jordan product. The main result of the paper under review is that, if \(D\) and \(G\) are bounded linear Jordan derivations of the Banach algebra \(A\) such that \([D(x),x]x-x[G(x),x]\) belongs to the Jacobson radical of \(A\) for all \(x\in A\), then both \(D\) and \(G\) have the ranges contained in the Jacobson radical of \(A\) (Theorem~2.12). The proof of this fact relies on a purely algebraic statement (Theorem~2.8) concerning Jordan derivations of prime rings.