Multiplicative Lie-type derivations on standard operator algebras

Abstract

Let 𝔄 be a standard operator algebra on a complex Banach space 𝔛 , dim ⁡ 𝔛 > 1 , and p n ⁢ ( T 1 , T 2 , … , T n ) the ( n - 1 ) th-commutator of elements T 1 , T 2 , … , T n ∈ 𝔄 . Then every map ξ : 𝔄 → 𝔄 (not necessarily linear) satisfying ξ ⁢ ( p n ⁢ ( T 1 , T 2 , … , T n ) ) = ∑ i = 1 n p n ⁢ ( T 1 , T 2 , … , T i - 1 , ξ ⁢ ( T i ) , T i + 1 , … , T n ) for all T 1 , T 2 , … , T n ∈ 𝔄 is of the form ξ = Ω + Γ , where Ω : 𝔄 → 𝔄 is an additive derivation and Γ : 𝔄 → ℂ ⁢ I is a map that vanishes at each ( n - 1 ) th-commutator p n ⁢ ( T 1 , T 2 , … , T n ) for all T 1 , T 2 , … , T n ∈ 𝔄 . In addition, if the map ξ is linear and satisfies the above relation, then there exist an operator S ∈ 𝔄 and a linear map Γ : 𝔄 → ℂ ⁢ I satisfying Γ ⁢ ( p n ⁢ ( T 1 , T 2 , … , T n ) ) = 0 for all T 1 , T 2 , … , T n ∈ 𝔄 , such that ξ ⁢ ( T ) = [ T , S ] + Γ ⁢ ( T ) for all T ∈ 𝔄 .