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Symmetric amenability and the nonexistence of Lie and Jordan derivations. (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 $D\left(ab-ba\right)=aD\left(b\right)-D\left(b\right)a+D\left(a\right)b-bD\left(a\right)$, $a,b\in U$.

It is clear that if $D$ is a (ordinary) derivation (i.e. $D\left(ab\right)=aD\left(b\right)+D\left(a\right)b\right)$ 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 ${M}_{n}$ $\left(n\ge 2\right)$ 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 ${\Delta }\left(ab-ba\right)=0$ 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