论文标题
在正交对处三重衍生的地图上的线性保存问题
A linear preserver problem on maps which are triple derivable at orthogonal pairs
论文作者
论文摘要
jb $^*$上的线性映射$ t $ - 三重在正交对中称为三重衍生，如果对于每个$ a，b，c \ in e $ a $ a \ perp b $ a \ perp b $，我们都有$ 0 = \ {t（t（a），b，b，b，b，b，c \}证明，对于JB $^*$ - 代数$ a $的每个有界线性映射$ t $，以下断言是等效的： $（a）$ $ t $在零时是三重推导； $（b）$ $ t $在正交元素上是三重衍生的； $（c）$存在Jordan $^*$ - 衍生$ D： \ hbox {all} a \ in; $$ $（d）$存在三重推导$δ：a \ to a^{**} $和对称元素的$ a^{**} $中的对称元素$ s $，因此$ t =δ+s $。 即使在C $^*$ - 代数的情况下，结果也是新的。接下来，我们在JBW $^*$ - 三重的三个线性地图上建立了新的表征，这是在Peirce 2-Subspaces上良好的本地行为而言，这是三重派生。我们还证明，假设在JBW $^*$ - 三重$ m $上使用一些额外条件，以下陈述是$ m $上的每个有界线性映射$ t $的等效词： $（a）$ $ t $在正交对中是三重衍生的； $（b）$存在三个推导$δ：m \至m $和$ m $ centroid的操作员$ s $，因此$ t =δ+ s $。 \ end {枚举}
A linear mapping $T$ on a JB$^*$-triple is called triple derivable at orthogonal pairs if for every $a,b,c\in E$ with $a\perp b$ we have $$0 = \{T(a), b,c\} + \{a,T(b),c\}+\{a,b,T(c)\}.$$ We prove that for each bounded linear mapping $T$ on a JB$^*$-algebra $A$ the following assertions are equivalent: $(a)$ $T$ is triple derivable at zero; $(b)$ $T$ is triple derivable at orthogonal elements; $(c)$ There exists a Jordan $^*$-derivation $D:A\to A^{**}$, a central element $ξ\in A^{**}_{sa},$ and an anti-symmetric element $η$ in the multiplier algebra of $A$, such that $$ T(a) = D(a) + ξ\circ a + η\circ a, \hbox{ for all } a\in A;$$ $(d)$ There exist a triple derivation $δ: A\to A^{**}$ and a symmetric element $S$ in the centroid of $A^{**}$ such that $T= δ+S$. The result is new even in the case of C$^*$-algebras. We next establish a new characterization of those linear maps on a JBW$^*$-triple which are triple derivations in terms of a good local behavior on Peirce 2-subspaces. We also prove that assuming some extra conditions on a JBW$^*$-triple $M$, the following statements are equivalent for each bounded linear mapping $T$ on $M$: $(a)$ $T$ is triple derivable at orthogonal pairs; $(b)$ There exists a triple derivation $δ: M\to M$ and an operator $S$ in the centroid of $M$ such that $T = δ+ S$. \end{enumerate}
