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

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Ali Taghavi
Aboozar Akbari


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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[1] K. I. Beidar, M. Bresar, M. A. Chebator and W. A. Martindale 3rd , On Hersteins Lie map conjectures II, J. Algebra 238 (2001), no. 1, 239-264. Google Scholar

[2] M. Bresar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104(1988), 1003-1006. Google Scholar

[3] M. Bresar, Jordan mappings of semiprime rings, J. Algebra. 127(1989), 218-228. Google Scholar

[4] M. Bresar, P. Semrl, Commutativity preserving linear maps on central simple algebras, Journal of algebra, 284 (2005) 102-110. Google Scholar

[5] J. Cusak, Jordan derivations on rings, Proc. Amer. Math. Soc. 53 (1975), 321-324. Google Scholar

[6] A. B. A. Essaleha, A. M.Peralta, Linear maps on C*-algebras which are derivations or triple derivations at a point, Linear Algebra and its Applications 538(2018)121. Google Scholar

[7] B. E. Johnson , Symmetric amenability and the nonexistence of Lie and Jordan derivations, Math. Proc. Camb. Phil. Soc. 120 (1996), 455-473. Google Scholar

[8] U.Haagerup and N. Laustsen, , Weak amenability of C-algebras and a theorem of Gold-stein, Banach algebras 97 (Blaubeuren), 223-243, de Gruyter, Berlin, 1998. Google Scholar

[9] I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957) 1104-1119. Google Scholar

[10] Shoichiro sakai, Operator algebras in dynamical systems, Volume 41, Cambrige University press, 2008. Google Scholar

[11] Vukman , Jordan derivations on prime rings, Bull. Austral. Math. Soc.,37 (1988), 321-322. Google Scholar