Korean J. Math. Vol. 26 No. 2 (2018) pp.285-291
DOI: https://doi.org/10.11568/kjm.2018.26.2.285

# A characterization of additive derivations on $C^*$-algebras

## 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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