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

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 $Φ: {\mathcal M}_2(\mathbb F)\to {\mathcal M}_2(\mathbb F)$ with range containing all rank one matrices satisfies that $[Φ(A),Φ(B)]_k = [A,B]_k$ 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 $λ\in{\mathbb F}$ with $λ^{k+1} = 1$ such that $Φ(A) = λA + h(A)I$ for all $A \in{\mathcal M}_2(\mathbb F)$.

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