Multiplicative Lie-type derivations on standard operator algebras

Abstract

Abstract Let 𝔄 {\mathfrak{A}} be a standard operator algebra on a complex Banach space 𝔛 {\mathfrak{X}} , dim ⁡ 𝔛 > 1 {\dim\mathfrak{X}>1} , and p n ⁢ ( T 1 , T 2 , … , T n ) {p_{n}(T_{1},T_{2},\dots,T_{n})} the ( n - 1 ) {(n-1)} th-commutator of elements T 1 , T 2 , … , T n ∈ 𝔄 {T_{1},T_{2},\dots,T_{n}\in\mathfrak{A}} . Then every map ξ : 𝔄 → 𝔄 {\xi:\mathfrak{A}\rightarrow\mathfrak{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 ) {\xi(p_{n}(T_{1},T_{2},\dots,T_{n}))=\sum_{i=1}^{n}p_{n}(T_{1},T_{2},\dots,T_{% i-1},\xi(T_{i}),T_{i+1},\dots,T_{n})} for all T 1 , T 2 , … , T n ∈ 𝔄 {T_{1},T_{2},\dots,T_{n}\in\mathfrak{A}} is of the form ξ = Ω + Γ {\xi=\Omega+\Gamma} , where Ω : 𝔄 → 𝔄 {\Omega:\mathfrak{A}\rightarrow\mathfrak{A}} is an additive derivation and Γ : 𝔄 → ℂ ⁢ I {\Gamma:\mathfrak{A}\rightarrow\mathbb{C}I} is a map that vanishes at each ( n - 1 ) {(n-1)} th-commutator p n ⁢ ( T 1 , T 2 , … , T n ) {p_{n}(T_{1},T_{2},\dots,T_{n})} for all T 1 , T 2 , … , T n ∈ 𝔄 {T_{1},T_{2},\dots,T_{n}\in\mathfrak{A}} . In addition, if the map ξ is linear and satisfies the above relation, then there exist an operator S ∈ 𝔄 {S\in\mathfrak{A}} and a linear map Γ : 𝔄 → ℂ ⁢ I {\Gamma:\mathfrak{A}\rightarrow\mathbb{C}I} satisfying Γ ⁢ ( p n ⁢ ( T 1 , T 2 , … , T n ) ) = 0 {\Gamma(p_{n}(T_{1},T_{2},\dots,T_{n}))=0} for all T 1 , T 2 , … , T n ∈ 𝔄 {T_{1},T_{2},\dots,T_{n}\in\mathfrak{A}} , such that ξ ⁢ ( T ) = [ T , S ] + Γ ⁢ ( T ) {\xi(T)=[T,S]+\Gamma(T)} for all T ∈ 𝔄 {T\in\mathfrak{A}} .