# Characterizing linear mappings through zero products or zero Jordan products

## TL;DR

This paper investigates the structure of linear maps like derivations and Jordan derivations on *-algebras, characterizing them under orthogonality conditions and applying results to various operator algebras.

## Contribution

It provides new characterizations of *-derivations and *-Jordan derivations based on zero product and zero Jordan product conditions, with applications to several classes of operator algebras.

## Key findings

- Characterization of mappings on zero product determined algebras
- Characterization of mappings on zero Jordan product determined algebras
- Applications to C*-algebras, group algebras, and von Neumann algebras

## Abstract

Let $\mathcal{A}$ be a $*$-algebra and $\mathcal{M}$ be a $*$-$\mathcal A$-bimodule, we study the local properties of $*$-derivations and $*$-Jordan derivations from $\mathcal{A}$ into $\mathcal{M}$ under the following orthogonality conditions on elements in $\mathcal A$: $ab^*=0$, $ab^*+b^*a=0$ and $ab^*=b^*a=0$. We characterize the mappings on zero product determined algebras and zero Jordan product determined algebras. Moreover, we give some applications on $C^*$-algebras, group algebra, matrix algebras, algebras of locally measurable operators and von Neumann algebras.