Nonlinear Skew Lie-Type Derivations on ∗-Algebra

Let A be a unital ∗-algebra over the complex fields C . For any H 1 , H 2 ∈ A , a product [ H 1 , H 2 ] • = H 1 H 2 − H 2 H 1 * is called the skew Lie product. In this article, it is shown that if a map ξ : A → A (not necessarily linear) satisfies ξ ( P n ( H 1 , H 2 , … , H n ) ) = ∑ i = 1 n P n ( H 1 , … , H i − 1 , ξ ( H i ) , H i + 1 , … , H n ) ( n ≥ 3 ) for all H 1 , H 2 , … , H n ∈ A , then ξ is additive. Moreover, if ξ ( i e 2 ) 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 I 1 .