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

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 $k\ge 1$, 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 $h:{\mathcal{M}}_2(\mathbb{F})\to \mathbb{F}$ and a scalar $\lambda \in \mathbb{F}$ with ${\lambda }^{k+1}=1$ such that $\Phi (A)=\lambda A+h(A)I$ for all $A\in {\mathcal{M}}_2(\mathbb{F})$.

12 pages