Strong $k$-commutativity preserving maps on 2$\times$2 matrices

TL;DR

This paper characterizes maps on 2x2 matrices that preserve the k-commutator structure, showing they are essentially scalar multiples plus a linear functional, under certain range conditions.

Contribution

It provides a complete description of strong k-commutativity preserving maps on 2x2 matrices, extending understanding of algebra automorphisms and derivations.

Findings

Maps are of the form $oxed{ ext{scalar} imes A + ext{linear functional} imes I}$

Preservation of k-commutator implies specific algebraic structure

Range condition on rank-one matrices is crucial for the characterization

Abstract

Let ${\mathcal M}_2(\mathbb F)$ be the algebra of 2$\times$2 matrices over the real or complex field $\mathbb F$. For a given positive integer $k\geq 1$, the $k$-commutator of $A$ and $B$ is defined by $[A,B]_k=[[A,B]_{k-1},B]$ with $[A,B]_0=A$ and $[A,B]_1=[A,B]=AB-BA$. The main result is shown that a map $\Phi: {\mathcal M}_2(\mathbb F)\to {\mathcal M}_2(\mathbb F)$ with range containing all rank one matrices satisfies that $[\Phi(A),\Phi(B)]_k = [A,B]_k $ for all $A, B\in{\mathcal M}_2(\mathbb F)$ if and only if there exist a functional $h :{\mathcal M}_2(\mathbb F) \rightarrow {\mathbb F}$ and a scalar $\lambda \in{\mathbb F}$ with $\lambda^{k+1} = 1$ such that $\Phi(A) = \lambda A + h(A)I$ for all $A \in{\mathcal M}_2(\mathbb F)$.