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

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.

💡 Research Summary

The paper investigates maps on the full matrix algebra Mₙ(F) (n ≥ 2) over an algebraically closed field F of characteristic different from 2 that preserve a mixed Jordan–power product. For a fixed natural number k, the product is defined by Aᵏ ∘ B = ½(AᵏB + BAᵏ), where ∘ denotes the normalized Jordan product. The main functional equation studied is

  ϕ(Aᵏ ∘ B) = ϕ(A)ᵏ ∘ ϕ(B) for all A, B ∈ Mₙ(F).

The authors give a complete classification of all such maps ϕ. The result, Theorem 1.1, states that any map satisfying the mixed Jordan‑power identity is either

1. a constant map whose value is a fixed (k + 1)-potent (i.e. X^{k+1}=X), or
2. of the form

  ϕ(X) = ε T ω(X) T⁻¹ or ϕ(X) = ε T ω(X)ᵗ T⁻¹,

where T ∈ GLₙ(F) is invertible, ω : F → F is a ring monomorphism (applied entrywise to matrices), ε ∈ F is a k‑th root of unity, and (·)ᵗ denotes matrix transposition. In particular, every non‑constant solution is automatically additive.

The proof proceeds through several conceptual steps:

• Preservation of (k + 1)-potents. Substituting A = I in the functional equation shows that ϕ(I)^{k} behaves as an identity for the Jordan product, forcing ϕ to map (k + 1)-potents to (k + 1)-potents. This yields strong constraints on the image of idempotents and on the rank of matrices.

• Structural tools on k‑potents. The authors introduce a partial order ⪯ on the set Potₖ(A) of k‑potents, defined by p ⪯ q ⇔ pq = qp = p², and an orthogonality relation p ⊥ q ⇔ pq = qp = 0. Lemma 2.5 proves that ⪯ is indeed a partial order and that orthogonal sums of k‑potents remain k‑potents. Lemma 2.7 establishes that the rank function r(P) is ortho‑additive on Potₖ(Mₙ(F)), i.e., r(P + Q) = r(P) + r(Q) when P ⊥ Q. Consequently, any map preserving the mixed Jordan‑power product also preserves rank and the order ⪯.

• Reduction to the case ϕ(I)=I. If ϕ(I) is not the identity, the authors construct an auxiliary map ϕ̃(X) = ε⁻¹ T⁻¹ ϕ(X) T with a suitable scalar ε and invertible T so that ϕ̃(I)=I. This normalization does not affect the functional equation and allows the use of known Jordan‑preserver results.

• Application of existing Jordan‑preserver theory. Once ϕ̃ fixes the identity and preserves ∘, the authors invoke their earlier classification of Jordan‑multiplicative maps (