Summary: Let \(\mathcal{U}\) be an associative unital Banach algebra endowed with the Lie product \([x,y]=xy-yx (x,y\in\mathcal{U})\) and create a Lie algebra. In this article, we are going to study the additive maps on \(\mathcal{U}\) that act at idempotent-products such as centralizers on the Lie structure of \(\mathcal{U}\). More precisely, we consider the subsequent condition on an additive map \(\varphi\) on a unital Banach algebra \(\mathcal{U}\) with a non-trivial idempotent \(p\): \[ x,y \in\mathcal{U},xy=p\Longrightarrow \varphi ([x,y])=[\varphi (x),y]=[x,\varphi (y)], \] and we show under certain conditions that \(\varphi (x)=cx+\mu (x)\) for all \(x\in\mathcal{U}\), where \(c\in Z(\mathcal{U}), \mu :\mathcal{U}\rightarrow Z(\mathcal{U}) (Z(\mathcal{U})\) is the center of \(\mathcal{U})\) is an additive map in which \(\mu ([x,y])=0\) for any \(x,y \in\mathcal{U}\) with \(xy=p\). The obtained results will be used for some Banach algebras, especially, for von Neumann algebras.