‎Linear maps on $\star$-algebras acting on orthogonal elements like derivations or anti-derivations

Abstract

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