On left Jordan derivations of rings and Banach algebras (Q933704)

Let \(A\) be a semisimple Banach algebra. The main result of the present paper states that there are no nonzero linear maps \(D:A\to A\) satisfying \(D(x^2)=2xD(x)\) for all \(x\in A\). This result can be viewed as a noncommutative extension of the well-known fact that there are no nonzero derivations on commutative semisimple Banach algebras. The proof is based on some purely algebraic results on maps of prime and semiprime rings.