Jordan *-Derivations of Finite-Dimensional Semiprime Algebras

Source abstract

Abstract In this paper, we characterize Jordan *-derivations of a 2-torsion free, finite-dimensional semiprime algebra R with involution *. To be precise, we prove the following. Let δ : R → R be a Jordan *-derivation. Then there exists a *-algebra decomposition R = U ⊕ V such that both U and V are invariant under δ . Moreover, * is the identity map of U and δ |U is a derivation, and the Jordan *-derivation δ|V is inner. We also prove the following. Let R be a noncommutative, centrally closed prime algebra with involution *, char R ≠ 2, and let δ be a nonzero Jordan *-derivation of R . If δ is an elementary operator of R , then dim C R < ∞ and δ is inner.