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

Abstract

Let $X$ be a Banach space of dimension $\ge 2$ over the real or complex field $\mathbb{F}$ and $\mathcal{A}$ a standard operator algebra in $\mathcal{B}(X)$. A map $\Phi :\mathcal{A}\to \mathcal{A}$ is said to be strong $3$-commutativity preserving if $[\Phi (A),\Phi (B){]}_3=[A,B{]}_3$ 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 $\Phi$ is a surjective map on $\mathcal{A}$, then $\Phi$ is strong $3$-commutativity preserving if and only if there exist a functional $h:\mathcal{A}\to \mathbb{F}$ and a scalar $\lambda \in \mathbb{F}$ with ${\lambda }^4=1$ such that $\Phi (A)=\lambda A+h(A)I$ for all $A\in \mathcal{A}$.

arXiv comment14 pages