Strong k-commutativity preserving maps on 2×2 matrices

2016/03/15 by Liu, Meiyun, Hou, Jinchuan
#47B47 #47B49 #FOS: Mathematics #Functional Analysis (math.FA) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1603.08414

Abstract

Let \mathcal M₂(\mathbb F) be the algebra of 2×2 matrices over the real or complex field \mathbb F. For a given positive integer k≥ 1, the k-commutator of A and B is defined by [A,B]_k=[[A,B]_k-1,B] with [A,B]₀=A and [A,B]₁=[A,B]=AB-BA. The main result is shown that a map Φ: \mathcal M₂(\mathbb F)→ \mathcal M₂(\mathbb F) with range containing all rank one matrices satisfies that [Φ(A),Φ(B)]_k = [A,B]_k for all A, B∈\mathcal M₂(\mathbb F) if and only if there exist a functional h :\mathcal M₂(\mathbb F) → \mathbb F and a scalar λ∈\mathbb F with λ^k+1 = 1 such that Φ(A) = λA + h(A)I for all A ∈\mathcal M₂(\mathbb F).

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