Abstract
Let X be a Banach space of dimension ≥ 2 over the real or complex field F and A a standard operator algebra in B(X). A map Φ:A→A is said to be strong 3-commutativity preserving if [Φ(A), Φ(B)]3 = [A,B]3 for all A,B ∈ A, where [A,B]3 is the 3-commutator of A,B defined by [A,B]3 = [[[A,B],B],B] with [A,B] =AB−BA. The main result in this paper is shown that, if Φ is a surjective map on A, then Φ is strong 3-commutativity preserving if and only if there exist a functional h:A→F and a scalar λ ∈ F with λ4 = 1 such that Φ(A) = λA + h(A)I for all A ∈ A.
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Supported by Natural Science Foundation of China (Grant No. 11671294)
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Liu, M.Y., Hou, J.C. Strong 3-commutativity preserving maps on standard operator algebras. Acta. Math. Sin.-English Ser. 33, 1659–1670 (2017). https://doi.org/10.1007/s10114-017-6145-z
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DOI: https://doi.org/10.1007/s10114-017-6145-z