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Zbl 0703.16020
Brešar, M.; Vukman, J.
On left derivations and related mappings.
(English)
[J] Proc. Am. Math. Soc. 110, No. 1, 7-16 (1990). ISSN 0002-9939; ISSN 1088-6826/e

If $X$ is a left $R$ module, then an additive map $D: R\to X$ is a left derivation if $D(ab)=aD(b)+bD(a)$ for all $a,b\in R$, and is a Jordan left derivation if $D(a\sp 2)=2aD(a)$ 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 $L(A)$ of continuous operators on $A$, a Hausdorff locally convex vector space, into either $A$ or $L(A)$. When $D$ is a left derivation, the torsion assumption in the main theorem can be removed, and also, if $D: R\to R$, then $D(R)$ is central when $R$ is a semi-prime ring. \par 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 $D(A)\subset rad(A)$, and if $D$ is a continuous linear Jordan derivation with $D(x)x- xD(x)\in rad(A)$ for all $x\in A$, then again $D(A)\subset rad(A)$. A final application is to functional equations. Let $X$ be a Banach space, $B(X)$ the algebra of bounded linear operators on $X$, and $f$ and $g$ additive maps of $B(X)$ into either $X$ or $B(X)$. If $f(U)=U\sp 2g(U\sp{-1})$ for all invertible $U\in B(X)$, then $f=g$ and $f(T)=Tf(I)$ for all $T\in B(X)$.
[C.Lanski]
MSC 2000:
*16W25 Derivations, actions of Lie algebras (assoc. rings and algebras)
16N60 Prime and semiprime assoc. rings
16U70 Commutativity theorems for assoc. rings
39B42 Matrix and operator functional equations
46H99 Topological algebras, etc.

Keywords: commutativity theorems; Jordan left derivations; semi-prime rings; Banach algebras; continuous linear left derivations; functional equations

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