Jordan Homomorphisms

Updated 21 October 2025

Jordan homomorphisms are additive maps that preserve quadratic structures by satisfying f(x²) = (f(x))² and the symmetrized product condition.
They enable decompositions into homomorphic and antihomomorphic components, as illustrated by the Jacobson–Rickart theorem and related splitting results.
Their study connects ring theory, functional analysis, and geometry, providing practical insights into T-ideals, operator algebras, and preserver problems.

A Jordan homomorphism is an additive map between (possibly noncommutative) rings or algebras that preserves the quadratic structure, namely $f(x^2) = (f(x))^2$ , and, more generally, respects the symmetrized (Jordan) product $x \circ y = xy + yx$ . Jordan homomorphisms naturally generalize both homomorphisms and antihomomorphisms and play a central role in ring theory, functional analysis, and geometry, particularly in the paper of structural preserver problems and invariant theory. Their paper encompasses deep connections with projective geometry, the theory of preserver maps, operator and Banach algebra theory, polynomial and functional identities, and the modern machinery of commutator ideals and universal enveloping constructions.

1. Fundamental Definitions and Algebraic Properties

Let $A$ and $B$ be associative algebras over a field $F$ , $\operatorname{char}(F) \neq 2$ . An additive map $\varphi\colon A \to B$ is a Jordan homomorphism if

$\varphi(x^2) = (\varphi(x))^2\ \text{for all } x\in A.$

If $A$ is unital, one often also requires $\varphi(1_A) = 1_B$ . Any homomorphism or antihomomorphism satisfies this relation. A more general characterization, suitable for noncommutative $A$ , involves the “sandwich” identity: $\varphi(xyx) = \varphi(x)\varphi(y)\varphi(x),$ or, equivalently (when 2 is invertible in $B$ ), $\varphi(x \circ y) = \varphi(x) \circ \varphi(y)$ , with $x \circ y = xy + yx$ (Brešar et al., 19 Oct 2025).

Elementary consequences derived via polarization and induction from the defining identity include:

$\varphi(xy+yx) = \varphi(x)\varphi(y)+\varphi(y)\varphi(x)$ ,
$\varphi(x^n) = (\varphi(x))^n$ , $n\in\mathbb{N}$ ,
$\varphi(xyx) = \varphi(x)\varphi(y)\varphi(x)$ .

For special Jordan algebras—e.g., the “plus-algebra” $A^{(+)}$ or the set of symmetric elements $H(A,*)$ under an involution—Jordan homomorphisms are natural structure-preserving maps. The notion also extends to $n$ -Jordan homomorphisms, defined by $\varphi(a^n) = (\varphi(a))^n$ , and to further generalizations such as mixed and pseudo $n$ -Jordan homomorphisms (Neghabi et al., 2019).

2. Structural Theorems: Standard Decomposition and Commutator Ideals

A central theme is whether a Jordan homomorphism $\varphi$ can be decomposed as a sum of a homomorphism and an antihomomorphism on some (large) ideal of $A$ . The classical Jacobson–Rickart theorem asserts that for $M_n(S)$ ( $n\ge2$ ), every Jordan homomorphism splits: there exists a central idempotent $e$ so that $e\varphi$ is a homomorphism and $(1-e)\varphi$ is an antihomomorphism (Brešar et al., 19 Oct 2025).

This paradigm is extended through the commutator ideal $K(A)$ , generated by all $[x,y]=xy-yx$ . The main recent result states:

If $A$ is unital and $A = K(A)$ , then every Jordan epimorphism $\varphi\colon A\to B$ is the sum of a homomorphism and an antihomomorphism, with the images of the two maps orthogonal: $\varphi_1(A)\varphi_2(A) = \{0\}$ (Brešar et al., 10 Aug 2025).

This decomposition is not global in general: in several families of algebras (e.g., certain Grassmann or triangular algebras), there exist nonstandard Jordan automorphisms that fail to admit such a global splitting but still restrict to a splitting on $K(A)$ or a large $T$ -ideal (Brešar et al., 19 Oct 2025, Bezushchak, 21 Sep 2025).

The concept of splittable Jordan homomorphisms further formalizes this, requiring that the two defect ideals associated to deviations from homomorphic and antihomomorphic behavior intersect trivially. For splittable Jordan homomorphisms, the sum decomposition always holds on the commutator ideal (Brešar, 9 Oct 2024).

3. Connections to Polynomial, T-Ideals, and Functional Identities

Modern analysis of Jordan homomorphisms draws on techniques from polynomial identity (PI) theory and functional identities:

T-ideals are two-sided ideals of the free associative algebra stable under all endomorphisms. If $G$ is a T-ideal such that all elements of the T-ideal of $2\times2$ matrices ( $T(M_2)$ ) are nilpotent modulo $G$ , then restrictions of Jordan homomorphisms to the subalgebra determined by $G$ satisfy standard extension properties (Brešar et al., 10 Aug 2025).
Tetrad-eating T-ideals arise in extension results: e.g., modulo some T-ideal $T$ , any Jordan homomorphism from the symmetric elements $H(A,*)$ extends to an associative homomorphism provided the subalgebra generated by the image has trivial annihilator (Brešar et al., 19 Oct 2025).
Functional identities and $d$ -free sets: Sets so that any functional identity of a given "degree" (number of variables) has a standard solution. Results show that for images of Jordan homomorphisms into $d$ -free sets, a homomorphism–antihomomorphism decomposition is mandatory.

These tools systematically unify and extend classical results by managing the obstruction to standard extension via controlled nilpotence or annihilator conditions.

4. Operator Algebras and Jordan Homomorphisms

In *-algebra and operator algebra theory, Jordan homomorphisms provide a bridge between order structure and algebraic structure:

In $C^*$ -algebras, any linear map that is a $*$ -homomorphism at the unit is a Jordan $*$ -homomorphism; if $A$ is simple and infinite, this suffices for $T$ to be a full $*$ -homomorphism (Burgos et al., 2016).
2-local Jordan $*$ -homomorphisms on JBW $^*$ -algebras are always linear and Jordan $*$ -homomorphic (Burgos et al., 2014).
For completely bounded normal Jordan $*$ -homomorphisms on von Neumann algebras, a precise structural dichotomy holds: such a homomorphism is completely bounded if and only if it splits as a sum of a $*$ -homomorphism and an anti- $*$ -homomorphism on summands, with explicit control via $n$ -minimality (Arhancet, 2020).
In function algebras valued in operator algebras, every Jordan $*$ -homomorphism on $C(X,\mathcal{A})$ or $\operatorname{Lip}(X,\mathcal{A})$ (with $\mathcal{A}$ unital) is represented as a "weighted composition operator" involving Jordan $*$ -homomorphisms on fibers parameterized by irreducible representations (Oi, 2022).

5. Geometric and Categorical Aspects

Jordan homomorphisms are deeply tied to geometric structures:

Any Jordan homomorphism $R\to R'$ gives rise to a harmonic mapping of a connected component of the projective line $\mathbb{P}(R)$ , with harmonicity understood as preservation of cross-ratio $-1$ (harmonic quadruples). When the projective line has more than one component, the mapping can be extended in various harmonic ways to the whole projective line, though such extension is non-unique and reflects the underlying ring's geometric connectivity (Blunck et al., 2013).
In the language of Jordan geometries via inversions, Jordan homomorphisms at the tangent level correspond to tangent maps of morphisms of geometric spaces equipped with inversions, tying algebraic and differential structures through the concept of Jordan pairs and the algebraic differential calculus of Weil functors (Bertram, 2013).

These perspectives clarify the origin of Jordan structure in projective, symmetric and reflection geometries and ensure a functorial relationship between geometric morphisms and algebraic homomorphisms.

6. Variants and Preservation Problems

A spectrum of generalizations exists:

Weighted Jordan homomorphisms: $cT(x^2) = T(x)^2$ for a central invertible $c$ arise in preserver problems; under broad conditions (e.g., additive maps $T$ on $M_n(R)$ such that $T(x)T(y) + T(y)T(x)=0$ when $xy=yx=0$ ), $T$ must be weighted Jordan (Brešar et al., 2021).
Mixed and pseudo-Jordan homomorphisms: These involve differing or more flexible constraints on polynomial expressions involving the map and powers of its images (Neghabi et al., 2019).
Jordan triple product homomorphisms: Maps $\Phi$ satisfying $\Phi(ABA) = \Phi(A)\Phi(B)\Phi(A)$ , classified in detailed ways for Hermitian matrix algebras in terms of determinants and inertia signatures (Bukovsek et al., 2015, Bukovsek et al., 2015).
Centralizer-type generalizations: If an algebra $A$ has a right identity, every Jordan left $\alpha$ -centralizer is automatically an $\alpha$ -centralizer, and thus every Jordan homomorphism is a full homomorphism in this setting (Eisaei et al., 4 Aug 2025).

These extensions reveal that minor relaxations in the defining identity either force full multiplicativity under module or functional conditions, or lead to a precise taxonomy of Jordan homomorphic maps.

7. Open Questions, Counterexamples, and Current Trends

Despite the broad classification and extension theory developed, several directions remain actively pursued:

Nonstandard Jordan homomorphisms illustrate that outside matrix and division ring settings, the pathologies and obstruction to decomposition arise in the presence of nontrivial annihilators or nonstandard ideals; counterexamples provided in (Brešar et al., 10 Aug 2025) stress the necessity of these assumptions.
For functional analysis, the question of whether all spectrum-preserving bijections between semisimple Banach algebras must be Jordan isomorphisms remains unresolved (Brešar et al., 19 Oct 2025).
Tetrad-eating T-ideals and functional identities furnish powerful, unifying approaches for both pure algebra and operator theory, with ongoing research focused on their application to preserver problems, automorphism classification, and the structure of derivations on prime and semiprime rings.

Further, connections with Lie algebra theory—via the Tits–Kantor–Koecher construction—provide a categorical bridge whereby Jordan homomorphisms relate to Lie homomorphisms, embedding associative, Jordan, and Lie structural identities in a unified framework (Brešar et al., 19 Oct 2025).

References

(Blunck et al., 2013): Jordan homomorphisms and harmonic mappings
(Bertram, 2013): Jordan Geometries – an Approach by Inversions
(Burgos et al., 2014): A Kowalski–Słodkowski theorem for 2-local $^*$ -homomorphisms on von Neumann algebras
(Bukovsek et al., 2015, Bukovsek et al., 2015): Jordan triple product homomorphisms on Hermitian matrices
(Burgos et al., 2016): Linear maps between $C^*$ -algebras that are $^*$ -homomorphisms at a fixed point
(Yao et al., 2018): Biderivations and triple homomorphisms on perfect Jordan algebras
(Neghabi et al., 2019): Characterization of mixed $n$ -Jordan homomorphisms and pseudo $n$ -Jordan homomorphisms
(Arhancet, 2020): A characterization of completely bounded normal Jordan $^*$ -homomorphisms on von Neumann algebras
(Brešar et al., 2021): Weighted Jordan homomorphisms
(Oi, 2022): Jordan $^*$ -homomorphisms on the spaces of continuous maps taking values in $C^{*}$ -algebras
(Brešar, 9 Oct 2024): Splittable Jordan homomorphisms and commutator ideals
(Eisaei et al., 4 Aug 2025): Jordan Left $\alpha$ -centralizers on Algebras with Applications to Group Algebras
(Brešar et al., 10 Aug 2025): Jordan homomorphisms and T-ideals
(Bezushchak, 21 Sep 2025): Jordan homomorphisms of triangular algebras over noncommutative algebras
(Brešar et al., 19 Oct 2025): Jordan homomorphisms: A survey