Mixed Jordan-Power Identity

• Mixed Jordan-Power Identity is an algebraic framework linking power operations and Jordan products to achieve deep classification results in various algebraic structures.
• It characterizes structure-preserving maps in matrix, quadratic Jordan algebras, and Jordan loops using constant and automorphism-type solutions.
• The identity supports Korovkin-type operator theorems, offering insights into fixed-point conditions, convergence properties, and rigidity in functional analysis.

The mixed Jordan-power identity unifies and generalizes a broad class of structural relationships in Jordan algebras, matrix algebra morphisms, and power identities in nonassociative settings. The identity relates power operations and Jordan products, leading to deep classification results for structure-preserving maps and revealing rigidity phenomena in operator theory, nonassociative algebra, and functional analysis.

1. Fundamental Definitions and Framework

A mixed Jordan-power identity is an algebraic assertion connecting powers and Jordan products in a given algebraic structure. For a matrix algebra $M_n(\mathbb{F})$ over an algebraically closed field $\mathbb{F}$ of characteristic $\ne 2$, the normalized Jordan product is defined as

$A \circ B := \frac{1}{2}(AB + BA).$

A map $\phi: M_n(\mathbb{F}) \to M_n(\mathbb{F})$ satisfies the $k$-th mixed Jordan-power identity if

$$\phi(A^k \circ B) = \phi(A)^k \circ \phi(B), \quad \forall\,A, B \in M_n(\mathbb{F}),$$

where $k \in \mathbb{N}$. The notion extends to general algebras and nonassociative systems, including quadratic Jordan algebras and Jordan loops. In Jordan loops, the mixed identity connects the powers and the basic Jordan law:

$(x^m y) x^n = x^m (y x^n).$

In the analytic context of Euclidean Jordan algebras, the mixed Jordan-power identity refers to the combination of fixed-point conditions for the unit $e$ and an element $p$ with distinct eigenvalues:

$T(e) = e,\quad T(p) = p,\quad T(p^2) = p^2,$

where $T$ is a positive linear operator (Gogić et al., 15 Dec 2025, Wetering, 2018, Gowda, 2022, Pula, 2010).

2. Classification and Rigidity Results in Matrix Algebras

In $M_n(\mathbb{F})$ with $n \geq 2$, all maps $\phi$ preserving the mixed Jordan-power identity admit a precise classification (Gogić et al., 15 Dec 2025):

• Constant solutions: $\phi$ is identically a fixed $(k+1)$-potent, i.e., $Q^{k+1} = Q$.
• Automorphism-type solutions: There exist $T \in GL_n(\mathbb{F})$, a field monomorphism $\omega: \mathbb{F} \to \mathbb{F}$, and a $k$-th root of unity $\varepsilon \in \mathbb{F}$ such that

$$\phi(X) = \varepsilon\,T\,\omega(X)\,T^{-1} \quad \text{or} \quad \phi(X) = \varepsilon\,T\,\omega(X)^t\,T^{-1}.$$

Here, $\omega(X)$ is the entrywise application of $\omega$ and $X^t$ is transposition.

Every nonconstant solution is additive (indeed, $\mathbb{F}$-linear up to $\omega$). This rigidity fully characterizes all structure-preserving maps of this type.

The proof synthesizes:

• Preservation of $(k+1)$-potents and their partial order.
• Diagonal and matrix-unit analysis to constrain the form of $\phi(E_{ij})$.
• Jordan block generation by repeated symmetrized $k$-th powers.

Parameter roles:

• $T$ captures inner automorphism structure.
• $\omega$ encodes field automorphism or embedding.
• $\varepsilon$ accounts for scalar $k$-fold periodicity.

Nontrivial examples include constant projectors, inner automorphisms, transposed conjugations, field-twisted variants, and root multiplications (Gogić et al., 15 Dec 2025).

3. Mixed Jordan-Power Identities in Abstract Jordan Algebras

For general (quadratic) Jordan algebras $(V,*)$ over a field of characteristic $\ne 2$, the fundamental quadratic mixed Jordan-power identity is

$Q_{Q_a b} = Q_a Q_b Q_a,$

where $Q_a(x) = 2 a*(a*x) - (a^2)*x$. In associative settings, this reduces to familiar sandwich powers. In abstract quadratic Jordan settings, $Q_{Q_a b}$ is expanded in terms of quadratic and bilinear operators, and the identity is a nontrivial polynomial law of degree six (Wetering, 2018).

The proof employs:

• Linearized commutator identities for Jordan operators.
• Polarizations and quadratic power identities.
• A normalization procedure and operator rewriting, automated by Python-driven “randomized reductions,” ultimately leading to algorithmic certifiability of such identities.

This algebraic and semi-automated approach underpins the structural theory of Jordan algebras, connecting with symmetric cones, JB-algebras, Jordan pairs, and providing a foundation for structure theorems independent of analytic machinery (Wetering, 2018).

4. Mixed n-Jordan Homomorphisms and Related Notions

Within the broader family of power-preserving maps, mixed $n$-Jordan homomorphisms are defined for algebras $A$ and $B$ as linear maps $\phi: A \to B$ satisfying

$$\phi(a^n b) = \phi(a)^n \phi(b),\quad \forall\,a, b \in A,$$

with special cases for $n=2$ (mixed Jordan). These are distinct from:

• Ordinary $n$-Jordan homomorphisms: $\phi(a^n) = \phi(a)^n$ for all $a$.
• Pseudo $n$-Jordan homomorphisms: existence of a fixed $w$ such that $\phi(a^n w) = \phi(a)^n w$ for all $a$.

Mixed $n$-Jordan homomorphisms may automatically be continuous when the codomain is a commutative semisimple Banach algebra. Iterative and idempotent techniques enable transitions between mixed $(n+1)$ and mixed $n$ identities under mild conditions (Neghabi et al., 2019).

Table: Relations among Jordan-type homomorphisms

Homomorphism type	Defining identity	Flexibility of right argument
$n$-Jordan	$\phi(a^n) = \phi(a)^n$	No
Mixed $n$-Jordan	$\phi(a^n b) = \phi(a)^n\phi(b)$	Yes
Pseudo $n$-Jordan	$\phi(a^n w) = \phi(a)^n w$	For fixed $w$ only

Illustrative examples show that mixed and pseudo maps need not coincide, and subtle distinctions arise in non-simple Banach algebra contexts (Neghabi et al., 2019).

5. Mixed Jordan-Power Phenomena in Loops and Nonassociative Settings

In the context of Jordan loops—commutative loops $(Q, \cdot, e)$ satisfying the Jordan identity $(x^2 y) x = x^2(y x)$—mixed power identities take the form

$$(x^m y) x^n = x^m (y x^n), \quad \forall\,x, y \in Q,$$

for certain pairs $(m, n)$ depending on congruence and power well-definedness. Inductive proofs use the loop's commutativity and the base Jordan law to show that finite Jordan loops exhibit approximate power-associativity for surprisingly high exponents. These identities are essential in the classification of Jordan loops, as shown by the classification of order-$9$ Jordan loops, which must be associative (Pula, 2010).

Corollaries include binary expansions of powers using the mixed identities, and various cancellation/inversion relations. The scope of validity is constrained by power well-definedness and congruence conditions.

6. Operator-Theoretic Mixed Jordan-Power Identities and Korovkin-Type Results

In finite-dimensional Euclidean Jordan algebras, mixed Jordan-power phenomena govern positive linear maps $T: V \to V$ that are fixed on a minimal set of test elements—specifically, $T(e) = e$, $T(p) = p$, $T(p^2) = p^2$ for a unit $e$ and $p$ with distinct spectral components. Under these conditions, $T$ must act identically on the entire Jordan frame of $p$, and thus is doubly stochastic (positive, unital, trace-preserving) (Gowda, 2022). This is a Jordan-algebraic analog of classic Korovkin theorems in approximation theory.

Extensions include:

• Sequential (approximation) forms: convergence on test elements implies convergence on the span of the Jordan frame.
• Weak-majorization forms: strict inequalities (in Hardy–Littlewood–Pólya order) on test elements, supplemented by operator commutativity, suffice to identify automorphism behavior on the frame and force double stochasticity.

These operator-theoretic identities illustrate how minimal algebraic constraints (fixed points on just three elements) entail complete rigidity in the action on significant subspaces, even in the absence of associativity or full order structure (Gowda, 2022).

7. Theoretical Significance and Open Directions

The mixed Jordan-power identity, in its various incarnations, provides a unifying language for power-associativity, structure preservation, and automatism results in both associative and nonassociative algebraic settings. It has led to:

• Comprehensive rigidity theorems for morphisms in matrix and operator algebras.
• Algorithmic and semi-automated proofs of deep structural identities in quadratic Jordan theory, independent of analytic functional calculus (Wetering, 2018).
• Functional-analytic results on automatic continuity for mixed homomorphisms, and hierarchy development among approximate homomorphism concepts (Neghabi et al., 2019).
• Classification and combinatorial simplification in nonassociative systems, such as Jordan loops.
• Explicit operator-theoretic links between fixed-point conditions and strong forms of stochasticity and invariance (Gowda, 2022).

Open problems include extending the random-rewrite paradigm to higher-degree or fully polarized polynomial identities in Jordan theory, characterizing finite bases of identities for quadratic Jordan algebras, and further generalizing automatic rigidity phenomena to noncommutative or non-semisimple settings. The mixed Jordan-power identity thus remains an organizing principle in the ongoing development of both the structure theory and applications of Jordan-algebraic and related nonassociative systems.