Jordan, Jordan right and Jordan left derivations on convolution algebras
A Jordan derivation on a ring $R$ is an additive mapping $d$ that satisfies \[ d(x^2) = d(x) x + x d(x) \] for all $x \in R$; $d$ is said to be a Jordan left derivation if \[ d(x^2) = 2xd(x) \] for all $x \in R$. Jordan right derivations are defined similarly. Previous work, which is reviewed in the introduction of the paper, for Jordan derivations, Jordan left derivations and Jordan right derivations has been done for semiprime rings and semisimple Banach algebras. Now let $\mathcal{G}$ be a locally compact group and denote by $L_0^{\infty}(\mathcal{G})$ the subspace of $L^{\infty}({\mathcal{G}})$ consisting of all functions that vanish at infinity. With respect to the first Arens product, $L_0^{\infty}(\mathcal{G})^{\ast}$ is a noncommutative Banach algebra that is not semiprime and not semisimple. The goal of the paper under review is to see what results concerning Jordan derivations, Jordan left derivations and Jordan right derivations for semiprime rings or semisimple Banach algebras are true in the $L_0^{\infty}(\mathcal{G})^{\ast}$ setting. Results are given for Jordan derivations, Jordan left derivations and Jordan right derivations on $L_0^{\infty}(\mathcal{G})^{\ast}$ when $\mathcal{G}$ is a locally compact abelian group. An example of such a result is that when $\mathcal{G}$ is abelian, $d$ is a Jordan right derivation on $L_0^{\infty}(\mathcal{G})^{\ast}$ if and only if it is a right derivation on $L_0^{\infty}(\mathcal{G})^{\ast}$.
scientific article; zbMATH DE number 7040769
scientific article