Let U be a unital *-algebra and d : U --> U be a linear map behaving like a derivation or an anti-derivation at the following orthogonality conditions on elements of U: xy = 0, xy* = 0, xy = yx = 0 and xy* = y*x = 0. We characterize the map d when U is a zero product determined algebra. Special characterizations are obtained when our results are applied to properly infinite W*-algebras and unital simple C*-algebras with a non-trivial idempotent.
