Equations arising from Jordan *-derivation pairs (Q1881596)

Let \(A\) be a complex \(\ast\)-algebra and \(M\) a bimodule over \(A\) equipped with a complex vector space structure which is compatible with the structure of \(A.\) Furthermore, it is assumed that for any \(m\in M\), the conditions \(Am = 0\) or \(mA = 0\) imply \(m=0.\) Let \(E, F: A\to M\) be additive maps such that \(E(aba) = E(a)b^{\ast}a^{\ast} + aF(b)a^{\ast} + abE(a)\) for any \(a, b \in A.\) The main result of the paper under review asserts the existence of unique additive maps \(T_1 , S_1 , T_2 , S_2 : A\to M\) such that \(aT_{\sigma }(b) = S_{\sigma }(a)b\) for any \(a, b\in A\) and \(\sigma = 1, 2\) and being \(E(a) = T_1 (a^{\star }) + S_2 (a)\) and \(F(a) = -T_2 (a^{\ast}) - S_1 (a).\) This implies the redundance of the conditions in the definition of Jordan \(\ast\)-derivation in the considered case.