Additivity of Jordan derivations on Jordan algebras with idempotents (Q2081675)

Additivity is one of the most active topics in the study of mappings on rings and operator algebras. The aim of this paper is to study the additivity of Jordan derivations on Jordan algebras. The following result is obtained. Let \(J\) be a Jordan algebra with a nontrivial idempotent \(e\) and let \(J=J_1\oplus J_{\frac{1}{2}}\oplus J_0\) be the Peirce decomposition of \(J\) with respect to \(e\). Then every multiplicative Jordan derivation on \(J\) is additive if \(J\) satisfies the following conditions: \begin{itemize} \item [(i)] Let \(a_i\in J_i (i=0, 1)\). If \(t_{\frac{1}{2}}a_i=0\) for all \(t_{\frac{1}{2}}\in J_{\frac{1}{2}}\), then \(a_i=0\). \item [(ii)] Let \(a_0\in J_0\). If \(t_0a_0=0\) for all \(t_0\in J_0\), then \(a_0=0\). \item [(iii)] Let \(a_{\frac{1}{2}}\in J_{\frac{1}{2}}\). If \(t_0a_{\frac{1}{2}}=0\) for all \(t_0\in J_0\), then \(a_{\frac{1}{2}}=0\). \end{itemize}