Journal Details
Format
Journal
First Published
16 Jun 2010
Publication timeframe
2 times per year
Languages
English
Open Access

# Multiplicative Lie triple derivations on standard operator algebras

###### Accepted: 06 Jun 2019
Journal Details
Format
Journal
First Published
16 Jun 2010
Publication timeframe
2 times per year
Languages
English

Let χ be a Banach space of dimension n > 1 and 𝒰 ⊂ ℬ(χ) be a standard operator algebra. In the present paper it is shown that if a mapping d : 𝒰 → 𝒰 (not necessarily linear) satisfies d([[U,V],W])=[[d(U),V],W]+[[U,d(V),W]]+[[U,V],d(W)]d\left( {\left[ {\left[ {U,V} \right],W} \right]} \right) = \left[ {\left[ {d\left( U \right),V} \right],W} \right] + \left[ {\left[ {U,d\left( V \right),W} \right]} \right] + \left[ {\left[ {U,V} \right],d\left( W \right)} \right] for all U, V, W ∈ 𝒰, then d = ψ + τ, where ψ is an additive derivation of 𝒰 and τ : 𝒰 → 𝔽I vanishes at second commutator [[U, V ], W ] for all U, V, W ∈ 𝒰. Moreover, if d is linear and satisfies the above relation, then there exists an operator S ∈ 𝒰 and a linear mapping τ from 𝒰 into 𝔽I satisfying τ ([[U, V ], W ]) = 0 for all U, V, W ∈ 𝒰, such that d(U) = SU − US + τ (U) for all U ∈ 𝒰.

#### MSC 2010

[1] W. Cheung: Lie derivations of triangular algebras. Linear Multilinear Algebra 51 (2003) 299–310.Search in Google Scholar

[2] L. Chen, J.H. Zhang: Nonlinear Lie derivations on upper triangular matrices. Linear Multilinear Algebra 56 (6) (2008) 725–730.Search in Google Scholar

[3] M.N. Daif: When is a multiplicative derivation additive?. International Journal of Mathematics and Mathematical Sciences 14 (3) (1991) 615–618.Search in Google Scholar

[4] P. Halmos: A Hilbert space Problem Book, 2nd ed.. Springer-Verlag, New York (1982).Search in Google Scholar

[5] W. Jing, F. Lu: Lie derivable mappings on prime rings. Linear Multilinear Algebra 60 (2012) 167–180.Search in Google Scholar

[6] P. Ji, R. Liu and Y. Zhao: Nonlinear Lie triple derivations of triangular algebras. Linear Multilinear Algebra 60 (2012) 1155–1164.Search in Google Scholar

[7] P.S. Ji, L. Wang: Lie triple derivations of TUHF algebras. Linear Algebra Appl. 403 (2005) 399–408.Search in Google Scholar

[8] F. Lu: Additivity of Jordan maps on standard operator algebras. Linear Algebra Appl. 357 (2002) 123–131.Search in Google Scholar

[9] F. Lu: Lie triple derivations on nest algebras. Math. Nachr. 280 (8) (2007) 882–887.Search in Google Scholar

[10] F. Lu, W. Jing: Characterizations of Lie derivations of ℬ(χ). Linear Algebra Appl. 432 (1) (2010) 89–99.Search in Google Scholar

[11] F. Lu, B. Liu: Lie derivable maps on ℬ(χ). Journal of Mathematical Analysis and Applications 372 (2010) 369–376.Search in Google Scholar

[12] M. Mathieu, A. R. Villena: The structure of Lie derivations on C*-algebras. J. Funct. Anal. 202 (2003) 504–525.Search in Google Scholar

[13] W.S. Martindale III: When are multiplicative mappings additive?. Proc. Amer. Math. Soc. 21 (1969) 695–698.Search in Google Scholar

[14] C.R. Mires: Lie derivations of von Neumann algebras. Duke Math. J. 40 (1973) 403–409.Search in Google Scholar

[15] C.R. Mires: Lie triple derivations of von Neumann algebras. Proc. Am. Math. Soc. 71 (1978) 57–61.Search in Google Scholar

[16] P. Šemrl: Additive derivations of some operator algebras. llinois J. Math. 35 (1991) 234–240.Search in Google Scholar

[17] A.R. Villena: Lie derivations on Banach algebras. J. Algebra 226 (2000) 390–409.Search in Google Scholar

[18] W. Yu, J. Zhang: Nonlinear Lie derivations of triangular algebras. Linear Algebra Appl. 432 (11) (2010) 2953–2960.Search in Google Scholar

[19] J.H. Zhang, B.W. Wu, H.X. Cao: Lie triple derivations of nest algebras. Linear Algebra Appl. 416 (2-3) (2006) 559–567.Search in Google Scholar

[20] F. Zhang, J. Zhang: Nonlinear Lie derivations on factor von Neumann algebras. Acta Mathematica Sinica. (Chin. Ser) 54 (5) (2011) 791–802.Search in Google Scholar

• #### A note on the volume of ∇-Einstein manifolds with skew-torsion

Recommended articles from Trend MD