Jordan derivations on certain Banach algebras

TL;DR

This paper investigates Jordan derivations on Banach algebras with a right identity, showing conditions under which they are derivations and describing their ranges, thus deepening understanding of algebraic structure and derivation behavior.

Contribution

It establishes that Jordan derivations are derivations when $eA$ is commutative and semisimple, and characterizes the range of Jordan left derivations in Banach algebras.

Findings

Jordan derivations are derivations if $eA$ is commutative and semisimple.

Jordan triple left/right derivations are Jordan left/right derivations.

Jordan left derivations map $A$ into $eA$.

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 $eA$.