Equations arising from Jordan *-derivation pairs

Summary.

We study certain functional equations derived from the definition of a Jordan *-derivation pair.

More precisely, if A is a complex *-algebra and M is a bimodule over A, having the structure of a complex vector space compatible with the structure of A, such that $Am = 0\, (m \in M)$ implies m = 0 and $mA = 0\, (m \in M)$ implies m = 0 and if $E, F : A \to M$ are unknown additive mappings satisfying

$E(aba) = E(a)b^{*}a^{*} + aF(b)a^{*} + abE(a) \qquad (a,b \in A),$

then E and F can be represented by double centralizers. The obtained result implies that one of the equations in the definition of a Jordan *-derivation pair is redundant.

Furthermore, a remark on the extension of this result to unknown additive mappings $E, F : A \to M$ such that

$E(a^{3}) = E(a){a^*}^{2} + aF(a)a^{*} + a^{2}E(a) \qquad (a \in A)$

is given in a special case.