Characterizations of Nonlinear Lie Derivations ofB(X)

Donghua Huo,Baodong Zheng,Hongyu Liu

doi:10.1155/2013/245452

Donghua Huo, Baodong Zheng + Show 1 more

Open Access

https://doi.org/10.1155/2013/245452

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Abstract

LetXbe an infinite dimensional Banach space, andΦ:B(X)→B(X)is a nonlinear Lie derivation. Then Φ is the formδ+τwhereδis an additive derivation ofB(X)andτis a map fromB(X)into its centerZB(X), which maps commutators into the zero.

Highlights

The local actions of derivations of operator algebras are still not completely understood
Without loss of generality, we proof that ZA12 ⊆ A12 + ZB(X)
We divided the proof into several steps

Summary

Introduction

The local actions of derivations of operator algebras are still not completely understood. In [11], it was shown that letting X be a Banach space of dimension greater than 2, if δ : B(X) → B(X) is a linear map satisfying δ([A, B]) = [δ(A), B] + [A, δ(B)] for any A, B ∈ B(X) with AB = 0, δ = d + τ, where d is a derivation of B(X) and τ : B(X) → CI is a linear map vanishing at commutators [A, B] with AB = 0. In [12], it was shown that every nonlinear Lie derivation of triangular algebras is the sum of an additive derivation and a map into its center sending commutators to zero. We prove that every nonlinear Lie derivation of B(X) is the sum of an additive derivation and a map from B(X) into its center sending commutators to the zero. We want to mention here that there is no additive assumed

Result and Proof

It follows that