Linear Maps on Group Algebras Determined by the Action of the Derivations or Anti-derivations on a Set of Orthogonal Elements

Let $L^1\left(G\right)$ and $M\left(G\right)$ be the group algebra and the measure algebra of a locally compact group G, respectively, and $\Delta :L^1\left(G\right)\to M\left(G\right)$ be a continuous linear map. Assuming that $\Delta$ behaves like derivation or anti-derivation at orthogonal elements for several types of orthogonality conditions, our aim is to characterize such maps. Indeed, we assume that $\Delta$ is a derivation or anti-derivation through orthogonality conditions on $L^1\left(G\right)$ such as $f\ast g=0$, $f\ast g^{\star }=0$, $f^{\star }\ast g=0$, $f\ast g=g\ast f=0$ and $f\ast g^{\star }=g^{\star }\ast f=0$.