Nonlinear skew commuting maps on $\ast$-rings

Abstract

UDC 512.5

Let $\mathcal{R}$ be a unital $\ast$-ring with the unit $I$. Assume that $\mathcal{R}$ contains a symmetric idempotent $P$ which satisfies $A{\mathcal{R}}P = 0$ implies $A=0$ and $A{\mathcal{R}}(I-P) = 0$ implies $A = 0$. In this paper, it is proved that if $\phi\colon\mathcal{R} \rightarrow \mathcal{R}$ is a nonlinear skew commuting map, then there exists an element $Z \in \mathcal{Z}_{S}(\mathcal{R})$ such that $\phi(X) = ZX$ for all $X \in \mathcal{R}$, where $\mathcal{Z}_{S}(\mathcal{R})$ is the symmetric center of $\mathcal{R}$.
As an application, the form of nonlinear skew commuting maps on factors is obtained.