Characterizations of (Jordan) derivation on Banach algebra with local actions

TL;DR

This paper investigates the structure of derivations and related mappings on unital Banach *-algebras, establishing conditions under which such mappings are Jordan derivations and providing characterizations of specific linear maps with zero products.

Contribution

It characterizes *-derivable and *-left derivable mappings at separating points as Jordan derivations and describes linear maps satisfying zero product conditions in Banach *-algebras.

Findings

*- ext{derivable mappings at W are Jordan derivations

Complete description of linear maps satisfying zero product conditions

Conditions for *-left derivable mappings to be Jordan left derivations

Abstract

Let $\mathcal{A}$ be a unital Banach $*$-algebra and $\mathcal{M}$ be a unital $*$-$\mathcal{A}$-bimodule. If $W$ is a left separating point of $\mathcal{M}$, we show that every $*$-derivable mapping at $W$ is a Jordan derivation, and every $*$-left derivable mapping at $W$ is a Jordan left derivation under the condition $W \mathcal{A}=\mathcal{A}W$. Moreover we give a complete description of linear mappings $\delta$ and $\tau$ from $\mathcal{A}$ into $\mathcal{M}$ satisfying $\delta(A)B^*+A\tau(B)^*=0$ for any $A, B\in \mathcal{A}$ with $AB^*=0$ or $\delta(A)\circ B^*+A\circ\tau(B)^*=0$ for any $A, B\in \mathcal{A}$ with $A\circ B^*=0$, where $A\circ B=AB+BA$ is the Jordan product.