Jordan, Jordan Right and Jordan Left Derivations on Convolution Algebras

Abstract

In this paper, we investigate Jordan derivations, Jordan right derivations and Jordan left derivations of $$L_0^\infty ({{\mathcal {G}}})^*$$ . We show that any Jordan (right) derivation on $$L_0^\infty ({{\mathcal {G}}})^*$$ is a (right) derivation on $$L_0^\infty ({{\mathcal {G}}})^*$$ and the zero map is the only Jordan left derivation on $$L_0^\infty ({{\mathcal {G}}})^*$$ . Then, we prove that the range of a Jordan (right) derivation on $$L_0^\infty ({{\mathcal {G}}})^*$$ is contained into $$\hbox {rad}(L_0^\infty ({{\mathcal {G}}})^*)$$ . Finally, we establish that the product of two Jordan (right) derivations of $$L_0^\infty ({{\mathcal {G}}})^*$$ is always a derivation on $$L_0^\infty ({{\mathcal {G}}})^*$$ and there is no nonzero centralizing Jordan (right) derivation on $$L_0^\infty ({{\mathcal {G}}})^*$$ .