Characterizing linear mappings through zero products or zero Jordan products

Let A be a ∗ -algebra and M be a ∗ - A -bimodule. We study the local properties of ∗ -derivations and ∗ -Jordan derivations from A into M under the following orthogonality conditions on elements in A : a b ∗ = 0 , a b ∗ + b ∗ a = 0 and a b ∗ = 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.