$C^*$-代数上可加Jordan左$*$-导子的刻画

设$\delta$是一个$*$-代数$\mathcal A$到其左$\mathcal A$-模$\mathcal M$的可加映射, 如果对任意$A\in\mathcal A$, 有$\delta(A^2)=A\delta(A)+A^*\delta(A)$, 则称$\delta$~是一个可加Jordan左$*$-导子. 在本文中, 我们证明了复的单位$C^*$- 代数到其Banach左模的每个可加Jordan左$*$-导子都恒等于零. 设$G\in\mathcal A$, 如果对任意$A,B\in \mathcal A$, 当$AB=G$时, 有$\delta(AB)=A\delta(B)+B^*\delta(A)$, 则称$\delta$在$G$处左$*$-可导. 我们证明了复的单位$C^*$-代数到其Banach左模的在单位点处左$*$-可导的连续可加映射恒等于零.

Characterizations of Additive Jordan Left $*$-Derivations on $C^*$-Algebras

An additive mapping $\delta$ from a $*$-algebra $\mathcal{A}$ into a left $\mathcal{A}$-module $\mathcal{M}$ is called an additive Jordan left $*$-derivation if $\delta(A^2)=A\delta(A)+A^*\delta(A)$ for every $A$ in $\mathcal A$. In this paper, we prove that every additive Jordan left $*$-derivation from a complex unital $C^*$-algebra into its unital Banach left module is equal to zero. An additive mapping $\delta$ from a $*$-algebra $\mathcal{A}$ into a left $\mathcal{A}$-module $\mathcal{M}$ is called left $*$-derivable at $G$ in $\mathcal{A}$ if $\delta(AB)=A\delta(B)+B^*\delta(A)$ for each $A,B$ in $\mathcal{A}$ with $AB=G$. We prove that every continuous additive left $*$-derivable mapping at the unit element $I$ from a complex unital $C^*$-algebra into its unital Banach left module is equal to zero.