Characterizations of ∗-Lie derivable mappings on prime ∗-rings

Abstract

Let R be a ∗-ring containing a nontrivial self-adjoint idempotent. In this paper it is shown that under some mild conditions on R, if a mapping d : R → R satisfies d([U, V ]) = [d(U), V ] + [U, d(V )] for all U, V ∈ R, then there exists ZU,V ∈ Z(R) (depending on U and V ), where Z(R) is the center of R, such that d(U +V ) = d(U) + d(V ) +ZU,V . Moreover, if R is a 2-torsion free prime ∗-ring additionally, then d = ψ+ ξ, where ψ is an additive ∗-derivation of R into its central closure T and ξ is a mapping from R into its extended centroid C such that ξ(U + V ) = ξ(U) + ξ(V ) +ZU,V and ξ([U, V ]) = 0 for all U, V ∈ R. Finally, the above ring theoretic results have been applied to some special classes of algebras such as nest algebras and von Neumann algebras.