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）=λ A+h（A）I for all A ∈ A.