Nonlinear Lie n-centralizers at zero-products on unital algebras

Let A be a unital algebra with nontrivial idempotents. In this article, it is shown that under certain conditions if a map (not necessarily linear) δ : A → A satisfies δ(p_n(a_1, a_2, , a_n)) = p_n(δ(a_1), a_2, , a_n) for all a_1, a_2, , a_n ∈ A with a_1 a_2 ⋯ a_n = 0 , then δ(a + b) - δ(a) - δ(b) ∈ Z(A) for all a, b ∈ A . Moreover, δ is of the form δ(a) = λ a + τ(a) for all a ∈ A , where λ ∈ Z(A) and τ : A → Z(A) is an almost additive map such that τ(p_n(a_1, a_2, , a_n)) = 0 for all a_1, a_2, , a_n ∈ A with a_1 a_2 ⋯ a_n = 0 . Moreover, this result can also be applied to triangular algebras, von Neumann algebras without central summands of type I_1 and so on.