Jordan derivations on certain Banach algebras

Abstract

In this paper, we study the types of Jordan derivations of a Banach algebra $A$ with a right identity $e$. We show that if $eA$ is commutative and semisimple, then every Jordan derivation of $A$ is a derivation. In this case, Jordan derivations map $A$ into the radical of $A$. We also prove that every Jordan triple left (right) derivation of $A$ is a Jordan left (right) derivation. Finally, we investigate the range of Jordan left derivations and establish that every Jordan left derivation of $A$ maps $A$ into $eA$.