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On left derivations and related mappings. (English) Zbl 0703.16020

If $X$ is a left $R$ module, then an additive map $D:R\to X$ is a left derivation if $D\left(ab\right)=aD\left(b\right)+bD\left(a\right)$ for all $a,b\in R$, and is a Jordan left derivation if $D\left({a}^{2}\right)=2aD\left(a\right)$ for all $a\in R$. The main theorem of the paper shows that if $X$ is 6-torsion free and no nonzero submodule has an annihilator in $R$, then the existence of a nonzero Jordan left derivation forces $R$ to be commutative. One corollary of this result shows that there are no nonzero Jordan left derivations of the algebra $L\left(A\right)$ of continuous operators on $A$, a Hausdorff locally convex vector space, into either $A$ or $L\left(A\right)$. When $D$ is a left derivation, the torsion assumption in the main theorem can be removed, and also, if $D:R\to R$, then $D\left(R\right)$ is central when $R$ is a semi-prime ring.

The authors apply their results to a Banach algebra $A$ by showing that if $D:A\to A$ is a continuous linear left derivation, then $D\left(A\right)\subset rad\left(A\right)$, and if $D$ is a continuous linear Jordan derivation with $D\left(x\right)x-xD\left(x\right)\in rad\left(A\right)$ for all $x\in A$, then again $D\left(A\right)\subset rad\left(A\right)$. A final application is to functional equations. Let $X$ be a Banach space, $B\left(X\right)$ the algebra of bounded linear operators on $X$, and $f$ and $g$ additive maps of $B\left(X\right)$ into either $X$ or $B\left(X\right)$. If $f\left(U\right)={U}^{2}g\left({U}^{-1}\right)$ for all invertible $U\in B\left(X\right)$, then $f=g$ and $f\left(T\right)=Tf\left(I\right)$ for all $T\in B\left(X\right)$.

Reviewer: C.Lanski

MSC:
 16W25 Derivations, actions of Lie algebras (associative rings and algebras) 16N60 Prime and semiprime associative rings 16U70 Center, normalizer (invariant elements) for associative rings 39B42 Matrix and operator functional equations 46H99 Topological algebras, normed rings and algebras, Banach algebras