Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON C*-ALGEBRAS

  • Taghavi, Ali (Department of Mathematics Faculty of Mathematical Sciences University of Mazandaran) ;
  • Akbari, Aboozar (Department of Mathematics Faculty of Mathematical Sciences University of Mazandaran)
  • Received : 2018.04.11
  • Accepted : 2018.05.28
  • Published : 2018.06.30
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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 $${\delta}({\mid}x{\mid}x)={\delta}({\mid}x{\mid})x+{\mid}x{\mid}{\delta}(x),\;{\forall}x{{\in}}{\mathcal{A}}_N$$ 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

  1. K. I. Beidar, M. Bresar, M. A. Chebotar and W. A. Martindale 3rd , On Her-steins Lie map conjectures II, J. Algebra 238 (1) (2001), 239-264. https://doi.org/10.1006/jabr.2000.8628
  2. M. Bresar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104 (1988), 1003-1006. https://doi.org/10.1090/S0002-9939-1988-0929422-1
  3. M. Bresar, Jordan mappings of semiprime rings, J. Algebra. 127 (1989), 218-228. https://doi.org/10.1016/0021-8693(89)90285-8
  4. M. Bresar, P. Semrl, Commutativity preserving linear maps on central simple algebras, Journal of algebra 284 (2005) 102-110. https://doi.org/10.1016/j.jalgebra.2004.09.014
  5. J. Cusak, Jordan derivations on rings, Proc. Amer. Math. Soc. 53 (1975), 321-324. https://doi.org/10.1090/S0002-9939-1975-0399182-5
  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).
  7. B. E. Johnson , Symmetric amenability and the nonexistence of Lie and Jordan derivations, Math. Proc. Camb. Phil. Soc. 120 (1996), 455-473. https://doi.org/10.1017/S0305004100075010
  8. U.Haagerup and N. Laustsen, , Weak amenability of C*-algebras and a theorem of Goldstein, Banach algebras 97 (Blaubeuren), 223-243, de Gruyter, Berlin, 1998.
  9. I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957) 1104-1119. https://doi.org/10.1090/S0002-9939-1957-0095864-2
  10. Shoichiro sakai, Operator algebras in dynamical systems, Volume 41, Cambrige University press, 2008.
  11. Vukman , Jordan derivations on prime rings, Bull. Austral. Math. Soc. 37 (1988), 321-322. https://doi.org/10.1017/S0004972700026927