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 into a Banach -bimodule is a linear map with , .
A Lie derivation is a linear map which satisfies , .
It is clear that if is a (ordinary) derivation (i.e. then it is a Jordan and Lie derivation as well.
The author proves that if is symmetrically amenable then every continuous Jordan derivation into a -bimodule is a derivation. This result can be extended to other algebras, for example all -algebras. If the identity of is contained in a subalgebra isomorphism to the full matrix algebra then every Jordan derivation from 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 from the algebra into the -bimodule with and for all .