Characterization of generalized skew Lie derivation on $\ast$-algebras

Résumé

Let $\mathcal{A}$ be a $\ast$-algebra over the complex field $\mathbb {C}$. For $A, B \in \mathcal{A}, [A, B]_\ast=AB-BA^\ast$ denotes the skew Lie product of $A, B$. In this article, it is shown that under mild assumptions a map $\xi$ : $\mathcal{A} \rightarrow \mathcal{A}$ (not necessarily linear) satisfies $\xi([A, B]_\ast)=[\xi(A),B]_\ast+[A, \xi(B)]_\ast$ for all $A, B \in \mathcal{A}$ if and only if $\xi$ is an additive $\ast$-derivation. Further, we introduce the notion of a nonlinear generalized skew Lie derivation on $\ast$-algebra and proved that every nonlinear generalized skew Lie derivation $D$ : $\mathcal{A} \rightarrow \mathcal{A}$ is of the form $D(T)=ZT+\xi(T)$ for all $T \in \mathcal{A}$, where $Z\in \mathcal{Z}(\mathcal{A})$ and $\xi$ : $\mathcal{A} \rightarrow \mathcal{A}$ is an additive $\ast$-derivation. As an applications, we apply our main results to some special classes of unital $\ast$-algebras such as prime $\ast$-algebra, factor von Neumann algebra and von Neumann algebra with no central summands of type $I_1$.