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

TL;DR

This paper characterizes surjective maps on standard operator algebras that preserve the 3-commutator, showing they are essentially scalar multiples plus a scalar functional, extending understanding of algebra automorphisms.

Contribution

It provides a complete characterization of strong 3-commutativity preserving maps on standard operator algebras, identifying their form as scalar multiples plus a scalar functional.

Findings

Surjective strong 3-commutativity preserving maps are scalar multiples plus a scalar functional.

Such maps are characterized by a scalar  with ^4=1.

The result extends the understanding of structure-preserving maps in operator algebras.

Abstract

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 $\Phi:{\mathcal A} \rightarrow {\mathcal A}$ 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 $h :{\mathcal A} \rightarrow {\mathbb F}$ and a scalar $\lambda \in{\mathbb F}$ with $\lambda^4 = 1$ such that $\Phi(A) = \lambda A + h(A)I$ for all $A \in{\mathcal A}$.