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 algebras, matrix algebras, algebras of locally measurable operators and von Neumann algebras.