On left Jordan derivations of rings and Banach algebras

It is well known that there are no nonzero linear derivations on complex commutative semisimple Banach algebras. In this paper we prove the following extension of this result. Let ▫$A$▫ be a complex semisimple Banach algebra and let ▫$D: A to A$▫ be a linear mapping satisfying the relation ▫$D(x^2) = 2xD(x)$▫ for all ▫$x in R$▫. In this case ▫$D = 0$▫. Throughout, ▫$R$▫ will represent an associative ring with center Z(R). A ring ▫$R$▫ is ▫$n$▫-torsion free, where ▫$n > 1$▫ is an integer, if ▫$nx = 0$▫, ▫$x in R$▫ implies ▫$x = 0$▫. As usual the commutator ▫$xy - yx$▫ will be denoted by ▫$[x,y]$▫. We shall use the commutator identities ▫$[xy,z] = [x,z]y + x[y,z]$▫ and ▫$[x,yz] = [x,y]z + y[x,z]$▫ for all ▫$x,y,z in R$▫. Recall that a ring ▫$R$▫ is prime if for ▫$a,b in R$▫, ▫$aRb = (0)$▫ implies that either ▫$a=0$▫ or ▫$b=0$▫, and is semiprime in case ▫$aRa = (0)$▫ implies that ▫$a=0$▫. An additive mapping ▫$D$▫ is called a derivation if ▫$D(xy) = D(x)y + xD(y)$▫ holds for all pairs ▫$x,y in R$▫, and is called a Jordan derivation in case ▫$D(x^2) = D(x)x + xD(x)$▫ is fulfilled for all ▫$x in R▫$. Obviously, any derivation is a Jordan derivation. The converse is in general not true. Herstein has proved that any Jordan derivation on a 2-torsion free prime ring is a derivation. Cusack has generalized Herstein\u27s result to 2-torsion free semiprime rings. An additive mapping ▫$D: R to R$▫ is called a left derivation if ▫$D(xy) = yD(x) + xD(y)$▫ holds for all pairs ▫$x,y in R$▫ and is called a left Jordan derivation (or Jordan left derivation) in case ▫$D(x^2) = 2xD(x)$▫ is fulfilled for all ▫$x in R$▫. In this paper by a Banach algebra we mean a Banach algebra over the complex field