The structure of $k$-potents and mixed Jordan-power preservers on matrix algebras

Abstract

Let $M_n(\mathbb{F})$ denote the algebra of $n \times n$ matrices over an algebraically closed field $\mathbb{F}$ of characteristic different from $2$. For $n \ge 2$, we classify all maps $ϕ: M_n(\mathbb{F}) \to M_n(\mathbb{F})$ satisfying the mixed Jordan-power identity $$ ϕ(A^{k} \circ B) = ϕ(A)^{k} \circ ϕ(B), \quad \text{for all } A,B \in M_n(\mathbb{F}), $$ where $\circ$ denotes the (normalized) Jordan product $A \circ B := \tfrac{1}{2}(AB + BA)$ and $k \in \mathbb{N}$. We show that every such map is either constant, taking a fixed $(k+1)$-potent value, or there exist an invertible matrix $T \in M_n(\mathbb{F})$, a ring monomorphism $ω: \mathbb{F} \to \mathbb{F}$, and a $k$-th root of unity $\varepsilon \in \mathbb{F}$ such that $ϕ$ takes one of the forms $$ ϕ(X) = \varepsilon\, T\, ω(X)\, T^{-1} \quad \text{ or } \quad ϕ(X) = \varepsilon\, T\, ω(X)^{t}\, T^{-1}, $$ where $ω(X)$ denotes the matrix obtained by applying $ω$ entrywise to $X$, and $(\cdot)^{t}$ denotes matrix transposition. In particular, every nonconstant solution is necessarily additive. The classification relies fundamentally on the preservation of $(k+1)$-potents and their intrinsic structural properties.