Nonlinear Skew Lie-Type Derivations on ∗-Algebra

Abstract

Let A be a unital ∗-algebra over the complex fields C. For any H1,H2∈A, a product [H1,H2]•=H1H2−H2H1* is called the skew Lie product. In this article, it is shown that if a map ξ : A→A (not necessarily linear) satisfies ξ(Pn(H1,H2,…,Hn))=∑i=1nPn(H1,…,Hi−1,ξ(Hi),Hi+1,…,Hn)(n≥3) for all H1,H2,…,Hn∈A, then ξ is additive. Moreover, if ξ(ie2) is self-adjoint, then ξ is ∗-derivation. As applications, we apply our main result to some special classes of unital ∗-algebras such as prime ∗-algebra, standard operator algebra, factor von Neumann algebra, and von Neumann algebra with no central summands of type I1.