A NOTE ON NONLINEAR SKEW-JORDAN TYPE DERIVATIONS ON ∗-RINGS

Abstract

Let A be a 2-torsion free unital ∗-ring containing non-trivial symmetric idempotent. For µ1 , µ2 ∈ A, a product µ1 ◦ µ2 = µ1µ2 + µ2µ ∗ 1 is called skew-Jordan product. In this article, it is shown that if a map φ : A → A (not Compulsorily linear) fulfills φ(Pn(ν1 , ν2, . . . , νn)) = ∑ n i=1 Pn(ν1 , . . . , νi−1 , φ(νi ), νi+1 , . . . , νn) for all ν1 , ν2, . . . , νn ∈ A, then φ is additive. Moreover, if φ(I) is self-adjoint then φ is ∗-derivation. As an ointment, our main result is applied to several special classes of unital ∗-rings and unital ∗-algebras such as prime ∗-ring, prime ∗-algebra, factor von Neumann algebra.