Jordan *-Derivations of Finite-DimensionalSemiprime Algebras

Fošner, Ajda ; Lee, Tsiu-Kwen

Canadian mathematical bulletin, Tome 57 (2014) no. 1, pp. 51-60

Résumé

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 $\delta :\,R\,\to \,R$ be a Jordan $*$ -derivation. Then there exists a $*$ -algebra decomposition $R\,=\,U\,\oplus \,V$ such that both $U$ and $V$ are invariant under $\delta $ . Moreover, $*$ is the identity map of $U$ and $\delta {{|}_{U}}$ is a derivation, and the Jordan $*$ -derivation $\delta {{|}_{V}}$ is inner. We also prove the following. Let $R$ be a noncommutative, centrally closed prime algebra with involution $*$ , char $R\,\ne \,2$ , and let $\delta $ be a nonzero Jordan $*$ -derivation of $R$ . If $\delta $ is an elementary operator of $R$ , then ${{\dim}_{C}}\,R\,<\,\infty $ and $\delta $ is inner.

Détail

DOI : 10.4153/CMB-2012-024-2

Mots-clés : 16W10, 16N60, 16W25, semiprime algebra, involution, (inner) Jordan *-derivation, elementary operator