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# Jordan $*$-Derivations of Finite-Dimensional Semiprime Algebras

Published:2012-07-27
Printed: Mar 2014
• Ajda Fošner,
Faculty of Management, University of Primorska, Cankarjeva 5, SI-6104 Koper, Slovenia
• Tsiu-Kwen Lee,
Department of Mathematics, National Taiwan University, Taipei 106, Taiwan
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## Abstract

In the paper, we characterize Jordan $*$-derivations of a $2$-torsion free, finite-dimensional semiprime algebra $R$ with involution $*$. To be precise, we prove the theorem: Let $deltacolon R o R$ be a Jordan $*$-derivation. Then there exists a $*$-algebra decomposition $R=Uoplus V$ 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 theorem: Let $R$ be a noncommutative, centrally closed prime algebra with involution $*$, $operatorname{char},R e 2$, and let $delta$ be a nonzero Jordan $*$-derivation of $R$. If $delta$ is an elementary operator of $R$, then $operatorname{dim}_CRlt infty$ and $delta$ is inner.
 Keywords: semiprime algebra, involution, (inner) Jordan $*$-derivation, elementary operator
 MSC Classifications: 16W10 - Rings with involution; Lie, Jordan and other nonassociative structures [See also 17B60, 17C50, 46Kxx] 16N60 - Prime and semiprime rings [See also 16D60, 16U10] 16W25 - Derivations, actions of Lie algebras