Strong $3$-Commutativity Preserving Maps on Standard Operator Algebras

Let $X$ be a Banach space of dimension $\geq 2$ over the real or complex field ${\mathbb F}$ and ${\mathcal A}$ a standard operator algebra in ${\mathcal B}(X)$. A map $Φ:{\mathcal A} \rightarrow {\mathcal A}$ is said to be strong $3$-commutativity preserving if $[Φ(A),Φ(B)]_3 = [A,B]_3$ for all $A, B\in{\mathcal A}$, where $[A,B]_3$ is the 3-commutator of $A,B$ defined by $[A,B]_3=[[[A,B],B],B]$. The main result in this paper is shown that, if $Φ$ is a surjective map on ${\mathcal A}$, then $Φ$ is strong $3$-commutativity preserving if and only if there exist a functional $h :{\mathcal A} \rightarrow {\mathbb F}$ and a scalar $λ\in{\mathbb F}$ with $λ^4 = 1$ such that $Φ(A) = λA + h(A)I$ for all $A \in{\mathcal A}$.