Korean J. Math.  Vol 26, No 3 (2018)  pp.459-465
DOI: https://doi.org/10.11568/kjm.2018.26.3.459

A note on nonlinear skew Lie triple derivation between Prime $\ast$-algebras

Ali Taghavi, Mojtaba Nouri, Vahid Darvish

Abstract


Recently, Li et al proved that $\Phi$ which satisfies the following condition on factor von Neumann algebras
$$\Phi([[A,B]_{*},C]_{*})=[[\Phi(A),B]_{*},C]_{*}+[[A,\Phi(B)]_{*},C]_{*}+[[A,B]_{*},\Phi(C)]_{*}$$
where $[A,B]_{*}=AB-BA^{*}$ for all $A,B\in\mathcal{A}$, is additive $\ast$-derivation. In this short note we show the additivity of $\Phi$ which satisfies the above condition on prime $\ast$-algebras.


Keywords


Lie triple derivation, Prime *-algebra, additive map.

Subject classification

46J10, 47B48, 46L10.

Sponsor(s)



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References


Z. Bai and S. Du, The structure of non-linear Lie derivations on factor von Neumann algebras, Linear Algebra Appl. 436 (2012), 2701–2708. (Google Scholar)

J. Cui and C.K. Li, Maps preserving product XY −Y X∗ on factor von Neumann algebras, Linear Algebra Appl. 431 (2009), 833–842. (Google Scholar)

C. Li, F. Lu and X. Fang, Nonlinear ξ−Jordan ∗-derivations on von Neumann algebras, Linear and Multilinear Algebra. 62 (2014), 466–473. (Google Scholar)

C. Li, F. Lu and X. Fang, Nonlinear mappings preserving product XY + Y X∗ on factor von Neumann algebras, Linear Algebra Appl. 438 (2013), 2339–2345. (Google Scholar)

C. LI, F Zhao and Q. Chen, Nonlinear Skew Lie Triple Derivations between Factors, Acta Mathematica Sinica 32 (2016), 821–830. (Google Scholar)

L. Moln ́ar, A condition for a subspace of B(H) to be an ideal, Linear Algebra Appl. 235 (1996), 229–234. (Google Scholar)

A. Taghavi, V. Darvish and H. Rohi, Additivity of maps preserving products AP ± PA∗ on C∗-algebras, Mathematica Slovaca 67 (2017), 213–220. (Google Scholar)

A. Taghavi, H. Rohi and V. Darvish, Non-linear ∗-Jordan derivations on von Neumann algebras, Linear Multilinear Algebra 64 (2016), 426–439. (Google Scholar)

W. Yu and J. Zhang, Nonlinear ∗-Lie derivations on factor von Neumann algebras, Linear Algebra Appl. 437 (2012), 1979–1991. (Google Scholar)


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