Commutativity-Preserving Maps

• Commutativity-Preserving Maps are transformations on rings, algebras, or operator groups that maintain basic and higher commutator relations to preserve structural properties.
• They are characterized by canonical forms—such as λA + h(A)I or conjugations—across various settings including matrix, operator, and nonassociative algebras.
• Their study informs preserver problems, spectral theory, and quantum information, offering key insights into rigidity phenomena and integrability in algebraic systems.

A commutativity-preserving map is a transformation on an algebraic structure—such as a ring, algebra, operator algebra, or matrix group—that preserves the commutator structure, often in both directions and sometimes for higher commutator powers or skew commutators. This property is central in the "preserver problem" literature, where one seeks complete characterizations of maps that rigidly respect underlying algebraic relations. The theory has broad impact in functional analysis, quantum information, nonassociative ring theory, Lie theory, and matrix analysis.

1. Fundamental Definitions and Notions

Let $A$ denote an algebra, ring, or group over a field $F$. The basic commutator is $[X, Y] = X Y - Y X$. For $k\geq 1$, the $k$-commutator is defined recursively: $[A,B]_0 = A$, $[A,B]_1 = [A,B]$, and $[A,B]_k = [[A,B]_{k-1}, B]$ for $k\geq2$ (Liu et al., 2016). A map $\Phi: A \to A$ is (strong) commutativity-preserving if $[\Phi(A), \Phi(B)] = [A,B]$ for all $A,B$, or $[\Phi(A), \Phi(B)]_k = [A,B]_k$ for all $A,B$, possibly for specific $k$.

Variants include:

• Skew commutativity preservers: Maps $\Phi$ ensuring $[\Phi(A), \Phi(B)]^* = [A, B]^*$, with $[A, B]^* = AB - BA^*$ for an involutive algebra (Qi et al., 2012).
• Operator commutativity preservers: Maps preserving both commutativity and operator Jordan associator structure as in JBW*-algebras (Escolano et al., 2024).
• Spectrum-commutativity preservers: Maps preserving both spectrum and commutativity, relevant for matrix and operator theory (Chirvasitu, 26 May 2025, Chirvasitu et al., 12 Jan 2025, Chirvasitu, 3 Jan 2026).

Preservation may be required "in both directions" (strong preservation): $[A,B]=0$ iff $[\Phi(A), \Phi(B)] = 0$, or in the presence of additional algebraic or topological constraints.

2. Core Classification Theorems and Rigidity Phenomena

A central result for matrix algebras over $F=\mathbb{R}$ or $\mathbb{C}$ is the Liu–Hou theorem for $2 \times 2$ matrices with strong $k$-commutativity preservation (Liu et al., 2016):

• For $\Phi: M_2(F) \to M_2(F)$ with range containing all rank-one matrices, $[\Phi(A),\Phi(B)]_k = [A,B]_k$ for all $A,B$ iff there exists $\lambda \in F$ with $\lambda^{k+1}=1$ and a linear functional $h: M_2(F) \to F$ so that $\Phi(A) = \lambda A + h(A)I$. Specializing to $k=1$ yields the classical form: $\Phi(A) = \pm A + h(A)I$.

Analogous results exist for standard operator algebras and higher commutator powers (Liu et al., 2016). Specifically, surjective strong $3$-commutativity preservers on $\mathcal{A} \subset \mathcal{B}(X)$ are of the form $\Phi(A) = \lambda A + h(A)I$, $\lambda^4=1$.

For von Neumann algebras without central summands of type $I_1$, strong skew commutativity preservers are exactly central involutive multiplications: $\Phi(A) = Z A$ with $Z^*=Z$, $Z^2=I$ (Qi et al., 2012).

In unitriangular groups $UT(n, F)$, commutator-preserving bijections are either automorphisms or central multiplications; for $n=\infty$, they are automorphisms (Holubowski et al., 2015).

For JB*-algebras and JBW*-algebras, linear commutativity-preservers (in both directions) have canonical forms: $\Phi(x)=z_0 \circ J(x) + \beta(x)$, with $z_0$ invertible central, $J$ a Jordan isomorphism, and $\beta$ a linear map into the center (Escolano et al., 2024).

3. Role of Spectral and Topological Constraints

Spectrum-preserving and commutativity-preserving maps on matrix algebras, Lie groups, and operator spaces are sharply constrained by continuity and topological structure (Chirvasitu et al., 12 Jan 2025, Chirvasitu, 3 Jan 2026). For $n\times n$ matrices ($n \geq 3$) and continuous maps $\phi$ with $\mathrm{sp}(\phi(X)) \subseteq \mathrm{sp}(X)$, the only possibilities are conjugations or transpose-conjugations: $\phi(X) = T X T^{-1}$ or $T X^t T^{-1}$ (Chirvasitu et al., 12 Jan 2025, Chirvasitu, 3 Jan 2026).

Exceptions arise on spaces such as $SU(n)$, diagonalizable, or semisimple matrices where spectrum-selection maps may appear; but when the spectrum is constrained to a simple closed curve (e.g., a circle), only conjugation or transpose-conjugation persist (Chirvasitu, 3 Jan 2026, Chirvasitu, 26 May 2025). Spectral-selection maps (diagonalizing according to orderings compatible with the spectrum's geometry) are realized on open intervals but not closed curves; in SU($n$) these reductions have a classifying combinatorial underpinning via Coxeter-Lipschitz maps and maximal torus analysis.

4. Commutativity Preservers in Nonassociative and Nonclassical Contexts

Significant advances have developed in nonassociative and nonclassical contexts:

• In semiprime $\Gamma$-rings, every strong commutativity-preserving endomorphism is "almost inner": $\sigma(x) = x + \zeta(x)$ with $\zeta$ central-valued (Xu et al., 2012).
• In incidence algebras of finite connected posets, strong commutativity-preserving bijections preserving the diagonal are decomposable into shift-type maps and quadruple-based pure maps associated to a combinatorial scalar data and order automorphisms (Fornaroli et al., 2022).

In the Heisenberg algebra—a nilpotent associative algebra—linear commuting maps exhibit rich structure well beyond the traditional central-plus-scalar maps; the full classification involves anti-transpose and skew-persymmetric parameters, demonstrating that commutativity preservation on such algebras fails to be characterizable as in the classical theory (Bounds et al., 20 Nov 2025).

5. Quantum Channels and Operator Commutativity

Quantum information theory introduces commutativity-preserving channels (CoP) as those CPTP quantum channels $\Phi$ for which input commuting states yield output commuting states (Yu et al., 2011). For CoP, operational and structure-constant (Lie bracket) criteria exist:

• The Choi-state witness provides a single observable distinguishing CoP channels (Yu et al., 2011).
• CoP channels are "discord non-creating": bipartite states remain classical–quantum under local CoP channels, formally linking commutativity preservation to quantum resource theory.

In finite dimension, CoP channels reduce to unital and semi-classical channels, with Hamiltonian channels failing CoP for $d\geq3$. Distance-based discord measures are monotonic under CoP channels, but projective discord may increase—exhibiting subtle behaviors distinguishing CoP from LOCC or entanglement-preserving channels.

6. Higher Commutators, Integrability, and Preserver Problems

Preserver problems for higher commutators ($k>1$) are significantly more rigid, and explicit classification exists only for small dimensions or under strong algebraic conditions (Liu et al., 2016, Liu et al., 2016). The preservation of $k$-commutators often forces maps to be nearly linear and central-valued perturbations with scalar constraints $\lambda^{k+1}=1$, as explicit for $2\times2$ matrices.

In dynamical systems and symplectic geometry, commuting maps arising from preservation of invariants associated with vector fields yield rich integrable systems; constructive procedures for integrable, commuting maps on the plane derive from symmetry reduction, directly connecting commutativity preservation to explicit Liouville-integrable families (Fordy et al., 2013).

7. Summary Table: Main Forms of Commutativity-Preserving Maps

Setting	Canonical Map Forms	Associated Papers
$2\times2$, $k$-commutator	$\Phi(A)=\lambda A + h(A)I$, $\lambda^{k+1}=1$	(Liu et al., 2016)
Operator algebra, $3$-commutator	$\Phi(A)=\lambda A + h(A)I$, $\lambda^4=1$	(Liu et al., 2016)
Von Neumann algebra, skew commutativity	$\Phi(A) = Z A$, $Z^2=I$ central	(Qi et al., 2012)
Matrix algebra, spectrum-commutativity	Conjugation, transpose-conjugation	(Chirvasitu et al., 12 Jan 2025, Chirvasitu, 3 Jan 2026)
JBW*-algebra, operator commutativity	$z_0 \circ J(x) + \beta(x)$ centrally	(Escolano et al., 2024)
$\Gamma$-ring, strong commutativity endomorphism	$\sigma(x) = x + \zeta(x)$, $\zeta$ central	(Xu et al., 2012)
Unitary group, spectrum selection	$T X T^{-1}$, $T X^t T^{-1}$, diagonal-spectrum	(Chirvasitu, 26 May 2025)
Heisenberg algebra	Bilinear, antitranspose, central-addition	(Bounds et al., 20 Nov 2025)
Quantum channel, CoP	Unital/semi-classical channels, Choi witness	(Yu et al., 2011)

8. Broader Context and Open Directions

Commutativity-preserving maps, especially in their strong forms, serve as a foundational lens for structural rigidity across operator, matrix, and quantum algebraic domains. Their classification interacts deeply with automorphism theory, Jordan and Lie structures, spectral theory, and quantum resources. Open problems include extensions beyond finite dimension, precise forms for higher $k$-commutators in general rings or operator algebras, and connections to geometric and combinatorial spectral constraints, as well as deeper understanding of commutativity-preserving dynamics in nonclassical algebraic systems.