Nonlinear Lie n-centralizers at zero-products on unital algebras with nontrivial idempotents

Abstract

Let $\mathcal{A}$ be a unital algebra with nontrivial idempotents. In the article, it is shown that under certain conditions if a nonlinear map
$\delta: \mathcal{A} \rightarrow \mathcal{A}$ satisfies

$$
\delta\left(p_{n}\left(a_{1},a_{2},\cdots,a_{n}\right)\right)=p_{n}\left(\delta(a_{1}),a_{2},\cdots,a_{n}\right)
$$
for all $a_{1},a_{2},\cdots,a_{n}\in{\mathcal A}$ with $a_{1}a_{2} \cdots a_{n}=0$, then
$\delta(a+b)-\delta(a)-\delta(b)\in\mathcal{Z}(\mathcal{A})$

for all $a,b\in\mathcal{A}$. Moreover, $\delta$ is of the form $\delta(a)=\lambda a + \tau(a)$ for all $a\in\mathcal{A}$, where $\lambda\in\mathcal{Z}(\mathcal{A})$ and $\tau : \mathcal{A} \rightarrow \mathcal{Z}(\mathcal{A})$ is an almost additive map such that $\tau\left(p_{n}\left(a_{1},a_{2},\cdots,a_{n}\right)\right)=0$

for all $a_{1},a_{2},\cdots,a_{n}\in{\mathcal A}$ with $a_{1}a_{2} \cdots a_{n}=0$. Moreover, this result can also be applied to triangular rings, von Neumann algebras without central summands of type $\uppercase\expandafter{\romannumeral1}_{1}$ and so on.