Linear Mappings Characterized by Action on Zero Products or Unit Products

Abstract

Let $$\mathcal {A}$$ be a unital algebra and $$\mathcal {M}$$ be a unital $$\mathcal {A}$$ -bimodule. We characterize the linear mappings $$\delta $$ and $$\tau $$ from $$\mathcal {A}$$ into $$\mathcal {M}$$ , satisfying $$\delta (A)B+A\tau (B)=0$$ for every $$A,B \in \mathcal {A}$$ with $$AB=0$$ when $$\mathcal {A}$$ contains a separating ideal $$\mathcal {T}$$ of $$\mathcal {M}$$ , which is in the algebra generated by all idempotents in $$\mathcal {A}$$ . We apply the result to $$\mathcal {P}$$ -subspace lattice algebras, completely distributive commutative subspace lattice algebras, and unital standard operator algebras. Furthermore, suppose that $$\mathcal {A}$$ is a unital Banach algebra and $$\mathcal {M}$$ is a unital Banach $$\mathcal {A}$$ -bimodule, we give a complete description of linear mappings $$\delta $$ and $$\tau $$ from $$\mathcal A$$ into $$\mathcal M$$ , satisfying $$\delta (A)B+A\tau (B)=0$$ for every $$A,B\in \mathcal {A}$$ with $$AB=I$$ .