Jordan *-derivation pairs on a complex *-algebra
Let \(A\) be a ring with involution * and \(M\) be an \(A\)-bimodule. The paper considers pair of functions \(E,F\colon A\to M\) satisfying \(E(x^3)=E(X)x^{*^2}+xF(x)x^*+x^2E(x)\) and \(F(x^3)=F(X)x^{*^2}+xE(x)x^*+x^2F(x)\) for all \(x\in A\), and thereby extends the treatment of the so-called Jordan *-derivation pairs introduced by \textit{B. Zalar} [Aequationes Math. 54, No. 1-2, 31-43 (1997; Zbl 0881.39019)]. The main result asserts that if \(A\) is a complex algebra and \(M\) has the property that \(A^2f=0\) of \(fA^2=0\), where \(f\in M\), implies \(f=0\), then \(E\) and \(F\) are of the form \(E(x)=T_1(x^*)+S_2(x)\), \(F(x)=-(T_2(x^*)+S_1(x))\), where \((T_1,S_1)\) and \((T_2,S_2)\) are double centralizers of \(A\) into \(M\).
scientific article; zbMATH DE number 1053598