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