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 Ф ： A→A is said to be strong 3-commutativity preserving if [Ф（A）, Ф（B）]a = [A, B]3 for all A, B C .4, 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 Ф is a surjective map on A, then Ф is strong 3-commutativity preserving if and only if there exist a functional h ： A→F and a scalar λ∈F with λ^4 = 1 such that Ф（A） = λA ＋ h（A）I for all A ∈A.