Strong 3-Commutativity Preserving Maps on Standard Operator Algebras

2016/01/24 by Liu, Meiyun, Hou, Jinchuan
#47B47 #47B49 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1601.06336

Abstract

Let X be a Banach space of dimension ≥ 2 over the real or complex field \mathbb F and \mathcal A a standard operator algebra in \mathcal B(X). A map Φ:\mathcal A → \mathcal A is said to be strong 3-commutativity preserving if [Φ(A),Φ(B)]₃ = [A,B]₃ for all A, B∈\mathcal A, where [A,B]₃ is the 3-commutator of A,B defined by [A,B]₃=[[[A,B],B],B]. The main result in this paper is shown that, if Φ is a surjective map on \mathcal A, then Φ is strong 3-commutativity preserving if and only if there exist a functional h :\mathcal A → \mathbb F and a scalar λ∈\mathbb F with λ⁴ = 1 such that Φ(A) = λA + h(A)I for all A ∈\mathcal A.

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