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1. CMB 2012 (vol 57 pp. 51)
Jordan $*$-Derivations of Finite-Dimensional Semiprime Algebras 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 Categories:16W10, 16N60, 16W25 |