*(English)*Zbl 0888.46024

The paper is devoted to some classes of Banach algebras in which Jordan and Lie derivations are reduced to (associative) derivations.

A Banach algebra is symmetrically amenable if it has an approximate diagonal consisting of symmetric tensors. A Jordan derivation from a Banach algebra $U$ into a Banach $U$-bimodule $X$ is a linear map $D$ with $D\left({a}^{2}\right)=aD\left(a\right)+D\left(a\right)a$, $a\in U$.

A Lie derivation $D:U\to X$ is a linear map which satisfies $D(ab-ba)=aD\left(b\right)-D\left(b\right)a+D\left(a\right)b-bD\left(a\right)$, $a,b\in U$.

It is clear that if $D$ is a (ordinary) derivation (i.e. $D\left(ab\right)=aD\left(b\right)+D\left(a\right)b)$ then it is a Jordan and Lie derivation as well.

The author proves that if $U$ is symmetrically amenable then every continuous Jordan derivation into a $U$-bimodule is a derivation. This result can be extended to other algebras, for example all ${C}^{*}$-algebras. If the identity of $U$ is contained in a subalgebra isomorphism to the full matrix algebra ${M}_{n}$ $(n\ge 2)$ then every Jordan derivation from $U$ is a derivation.

Similar results are developed for Lie derivation. In similar situations every continuous Lie derivation is the sum of an ordinary derivation and a map ${\Delta}$ from the algebra $U$ into the $U$-bimodule $X$ with ${\Delta}(ab-ba)=0$ and $a{\Delta}\left(b\right)={\Delta}\left(b\right)a$ for all $a,b\in U$.

##### MSC:

46H25 | Normed modules and Banach modules, topological modules |

46L57 | Derivations, dissipations and positive semigroups in ${C}^{*}$-algebras |

46H70 | Nonassociative topological algebras |