Strong 3-commutativity preserving maps on standard operator algebras

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.