Characterizing linear mappings through zero products or zero Jordan products

Abstract

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