$\mathbb{M}_n$上的弱2-局部导子
Weak 2-local derivations on $\mathbb{M}_n$
摘要：
本文在C-$^*$ -代数$A$上引入弱2-局部导子（分别为$^*$ -导子）的概念，作为一个（非必然线性）映射$Δ: A\to A$，满足对每个$a,b\in A$和$ϕ\in A^*$，存在一个导子（分别为$^*$ -导子）$D_{a,b,ϕ}: A\to A$，依赖于$a$，$b$和$ϕ$，使得$ϕΔ(a) = ϕD_{a,b,ϕ} (a)$和$ϕΔ(b) = ϕD_{a,b,ϕ} (b)$ .证明了$M_n$上的每个弱2-局部$^*$ -导子都是线性导子.我们还证明了有限维C $^*$ -代数上的弱2-局部$^*$-导子也是如此.
英文摘要：
We introduce the notion of weak-2-local derivation (respectively, $^*$-derivation) on a C$^*$-algebra $A$ as a (non-necessarily linear) map $Δ: A\to A$ satisfying that for every $a,b\in A$ and $ϕ\in A^*$ there exists a derivation (respectively, a $^*$-derivation) $D_{a,b,ϕ}: A\to A$, depending on $a$, $b$ and $ϕ$, such that $ϕΔ(a) = ϕD_{a,b,ϕ} (a)$ and $ϕΔ(b) = ϕD_{a,b,ϕ} (b)$. We prove that every weak-2-local $^*$-derivation on $M_n$ is a linear derivation. We also show that the same conclusion remains true for weak-2-local $^*$-derivations on finite dimensional C$^*$-algebras.
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