Lie centralizers at zero products on a class of operator algebras

Abstract

Let ${\mathcal {A}}$ be an algebra. In this paper, we consider the problem of determining a linear map $\psi$ on ${\mathcal {A}}$ satisfying $a,b\in {\mathcal {A}}$, $ab=0 \Longrightarrow \psi ([a,b])=[\psi (a),b] \, (C1)$ or $ab=0 \Longrightarrow \psi ([a,b])=[a,\psi (b)] \, (C2)$. We first compare linear maps satisfying (C1) or (C2), commuting linear maps, and Lie centralizers with a variety of examples. In fact, we see that linear maps satisfying (C1), (C2) and commuting linear maps are different classes of each other. Then, we introduce a class of operator algebras on Banach spaces such that if ${\mathcal {A}}$ is in this class, then any linear map on ${\mathcal {A}}$ satisfying (C1) (or (C2)) is a commuting linear map. As an application of these results, we characterize Lie centralizers and linear maps satisfying (C1) (or (C2)) on nest algebras.