Linear Maps on *-Algebras Acting on Orthogonal Elements Like Derivations or Anti-Derivations

Let U be a unital (*)-algebra and delta : 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 delta 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.