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

TL;DR

This paper classifies all maps on matrix algebras over algebraically closed fields that satisfy a mixed Jordan-power identity, showing they are either constant or conjugate to field automorphisms with specific forms.

Contribution

It provides a complete classification of maps preserving mixed Jordan-power identities, revealing their structure as either constant or conjugate to field automorphisms with roots of unity.

Findings

Nonconstant solutions are additive.

Solutions are either constant or conjugate to field automorphisms.

Classification hinges on preserving $(k+1)$-potents.

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 $\phi : M_n(\mathbb{F}) \to M_n(\mathbb{F})$ satisfying the mixed Jordan-power identity $$ \phi(A^{k} \circ B) = \phi(A)^{k} \circ \phi(B), \quad \text{for all } A,B \in M_n(\mathbb{F}), $$ 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 $\omega : \mathbb{F} \to \mathbb{F}$, and a $k$-th root of unity $\varepsilon \in \mathbb{F}$ such that $\phi$ takes one of the forms $$ \phi(X) = \varepsilon\, T\, \omega(X)\, T^{-1} \quad \text{ or } \quad \phi(X) = \varepsilon\, T\, \omega(X)^{t}\, T^{-1}, $$ where $\omega(X)$ denotes the matrix obtained by applying $\omega$ 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.