A note on Lie product preserving maps on \(M_n(\mathbb R)\) (Q2826353)

One of the most active and fertile subjects in matrix theory during the past one hundred years are the linear preserver problems. Such problems arise in most parts of mathematics. In fact, it turns out that in many cases the corresponding results provide important information on the automorphisms of the underlying structures.NEWLINENEWLINELet \(n>3\) be a positive integer. In the present paper, the author consider Lie product preserving maps on \(M_n(\mathbb F)\) (i.e., the set of all \(n\times n\) matrices over the field \(\mathbb F=\mathbb R, \mathbb C\)). In particular, it is shown that, for every injective, continuous, Lie product preserving map \(\phi \: M_n(\mathbb R)\to M_n(\mathbb R)\), there exist an invertible matrix \(T\in M_n(\mathbb R)\) and a continuous function \(\psi \: M_n(\mathbb R)\to \mathbb R\), where \(\psi (A)=0\) for all matrices of trace zero, such that either \(\phi (A) = TAT^{-1}+\psi (A)I\) for all \(A \in M_n(\mathbb R)\), or \(\phi (A) = -TA^tT^{-1}+\psi (A)I\) for all \(A \in M_n(\mathbb R)\). Here, \(A^t\) denotes the transpose of a matrix \(A\). Moreover, the analogous result for \(\mathbb F=\mathbb C\) is proven.