Characterizing linear mappings through zero products or zero Jordan products

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.