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Jordan *-Derivations of Finite-DimensionalSemiprime Algebras

Published online by Cambridge University Press:  20 November 2018

Ajda Fošner  and
Tsiu-Kwen Lee
Ajda Fošner
Affiliation:
Faculty of Management, University of Primorska, Cankarjeva 5, SI-6104 Koper, Slovenia e-mail: ajda.fosner@fm-kp.si
Tsiu-Kwen Lee*
Affiliation:
Department of Mathematics, National Taiwan University, Taipei 106, Taiwan e-mail: tklee@math.ntu.edu.tw
*
Corresponding author: Tsiu-Kwen Lee. T.-K. Lee was supported by NSC and NCTS/TPE of Taiwan.
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Abstract

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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.

Keywords

16W1016N6016W25semiprime algebrainvolution(inner) Jordan *-derivationelementary operator
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 1 , 14 March 2014 , pp. 51 - 60
Copyright
Copyright © Canadian Mathematical Society 2014

References

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