# Multiplicative Lie triple derivations on standard operator algebras

Multiplicative Lie triple derivations on standard operator algebras AbstractLet 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications in Mathematics de Gruyter

# Multiplicative Lie triple derivations on standard operator algebras

, Volume 29 (3): 13 – Dec 1, 2021
13 pages

/lp/de-gruyter/multiplicative-lie-triple-derivations-on-standard-operator-algebras-SNIEt05Cwd
Publisher
de Gruyter
eISSN
2336-1298
DOI
10.2478/cm-2021-0012
Publisher site
See Article on Publisher Site

### Abstract

AbstractLet 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.

### Journal

Communications in Mathematicsde Gruyter

Published: Dec 1, 2021

Keywords: Multiplicative Lie derivation; multiplicative Lie triple derivation; standard operator algebra; 47B47; 16W25; 47B48