Nonlinear η-∗-Jordan n-Derivation on ∗-Algebras

Let A be a unital ∗-algebra with the unit I over the complex field C and let η≠0, ±1 be a complex number. For any A, B∈A, A⋄ηB=AB+ηBA* is referred to as the η-Jordan ∗-product. Suppose that n≥3 is a fixed positive integer. In this study, it is shown that if a map φ: A→A satisfies φ (A1⋄ηA2⋄η⋯⋄ηAn) =∑k=1nA1⋄η⋯⋄ηAk−1⋄ηφ (Ak) ⋄ηAk+1⋄η⋯⋄ηAn for all A1, A2⋯An−3∈I, iI and An−2, An−1, An∈A, then φ is an additive ∗-derivation and φ (ηA) =ηφ (A) for all A∈A, where i is the imaginary unit. In application, characterizations of prime ∗-algebras, von Neumann algebras with no central summands of type I1 and factor von Neumann algebras are obtained.