Ternary derivations of separable associative and Jordan algebras (Q1937759)

Let \(A\) be an algebra. A linear map \(D: A\to A\) is called a generalized derivation is there exist linear maps \(F,G: A\to A\) such that \(D(xy)=F(x)y+xG(y)\) for all \(x,y\in A\). In this case, the triple \((D,F,G)\) is called a ternary derivation. The first part of the paper considers generalized derivations of associative algebras. It is shown that every generalized derivation of a finite dimensional simple central associative algebra \(A\) is inner, i.e., of the form \(D(x)=ax + xb\) for some \(a,b\in A\). The same is true if \(A\) is a separable associative algebra. The second part is devoted to Jordan algebras. In particular, it is shown that a generalized derivation of a finite dimensional simple central Jordan algebra is the sum of a derivation and a scalar multiple of the identity.