Weak-local triple derivations on \(\mathrm{C}^{\ast}\)-algebras and \(\mathrm{JB}^{\ast}\)-triples
A triple derivation on a \(\mathrm{JB}^\ast\)-triple \((E,\{\cdot,\cdot,\cdot\})\) is a linear mapping \(\delta: E\to E\) satisfying the Leibniz rule \(\delta\{a,b,c\} = \{\delta(a), b,c\} +\{a,\delta(b),c\}+\{a,b,\delta(c)\}\) for every \(a,b,c\in E\). A \(\mathrm{JB}^\ast\)-triple \(E\) is a weak-local triple derivation if, for each \(\phi \in E^\ast\) and each \(a\in E\), there exists a triple derivation \(\delta_{a,\phi} : E\to E\), depending on \(\phi\) and \(a\), such that \(\phi T(a) = \phi \delta_{a,\phi} (a)\). In the paper under review, the authors prove that every weak-local triple derivation on a \(\mathrm{JB}^\ast\)-triple \(E\) is a (continuous) triple derivation. In addition, they give some sufficient and necessary conditions for a linear map \(T\) on a \(\mathrm{JB}^\ast\)-triple \(E\) to be a triple derivation.
scientific article; zbMATH DE number 6617607
scientific article