A characterization of additive derivations on $C^*$-algebras
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Abstract
Let $\mathcal{A}$ be a unital $C^*$-algebra. It is shown that additive map $\delta:\mathcal{A}\rightarrow\mathcal{A}$ which satisfies
$$
\delta(|x|x)=\delta(|x|)x+|x|\delta(x),~~\forall x \in {\mathcal{A}}_{N}
$$
is a Jordan derivation on $\mathcal{A}$. Here, ${\mathcal{A}}_{N}$ is the set of all normal elements in $\mathcal{A}$. Furthermore, if $\mathcal{A}$ is a semiprime $C^*$-algebra then
$\delta$ is a derivation.
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References
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