Nonlinear Generalized Bi-skew Jordan n-Derivations on ∗-Algebras

Let A be a ∗ -algebra over the complex field C . For any T 1 , T 2 , … , T n ∈ A , define q 1 ( T 1 ) = T 1 , q 2 ( T 1 , T 2 ) = T 1 ⋄ T 2 = T 1 T 2 ∗ + T 2 T 1 ∗ and q n ( T 1 , T 2 , … , T n ) = q n - 1 ( T 1 , T 2 , … , T n - 1 ) ⋄ T n for all integers n ≥ 2 . In this article, it is shown that under certain assumptions a map D : A → A is a nonlinear bi-skew Jordan n -derivation (that is, D satisfies D ( q n ( T 1 , T 2 , … , T n ) ) = ∑ i = 1 n q n ( T 1 , … , T i - 1 , D ( T i ) , T i + 1 , … , T n ) for all T 1 , T 2 , … , T n ∈ A ) if and only if D is an additive ∗ -derivation. Further, we introduce the notion of a nonlinear generalized bi-skew Jordan n -derivation of a ∗ -algebra and prove that every nonlinear generalized bi-skew Jordan n -derivation G : A → A is of the form G ( T ) = Z T + D ( T ) for all T ∈ A , where Z ∈ Z ( A ) and D : A → A is an additive ∗ -derivation. As applications, we apply our main results to some special classes of ∗ -algebras such as prime ∗ -algebras, factor von Neumann algebras, von Neumann algebras with no central summonds of type I 1 .