Commutativity & Structure-Preserving Encodings

• Commutativity and structure-preserving encodings are defined as mappings that maintain essential algebraic, spectral, and categorical relationships, ensuring invariant transformations.
• They restrict allowed operations to central shifts, Jordan isomorphisms, and controlled perturbations, thereby enforcing exact algebraic compatibility.
• Applications span operator algebras, quantum channels, and combinatorial frameworks, demonstrating robust invariance in signal processing, machine learning, and optimization.

Commutativity and structure-preserving encodings are fundamental concepts governing algebraic, spectral, and categorical transformations in operator algebras, quantum information, signal processing, and combinatorial settings. A commutativity-preserving map typically sends commuting elements of an algebraic structure to commuting elements in another, imposing severe restrictions on admissible transformations and often forcing exact algebraic compatibility. Structure-preserving encodings generalize this paradigm, demanding that intrinsic relations—commutativity, orthogonality, spectral multiplicities, categorical monoid structures—are maintained under the encoding. This article synthesizes prevailing rigorous characterizations, classifying the structural form of commutativity-preservers and identifying corresponding encoding principles across operator algebras, incidence algebras, quantum operations, compositional frameworks in machine learning, and categorical quantum data representation.

1. Algebraic Commutativity and Operator Algebra Preservers

In nonassociative operator contexts, notably Jordan algebras and their von Neumann analogues (JBW$^*$-algebras), commutativity is determined by associators. For $a,b,c\in J$, $[a,b,c]=(a\circ b)\circ c-a\circ(b\circ c)$, encapsulating the obstruction to full associativity. Two elements $a,b$ operator-commute iff $[a,c,b]=0$ for all $c\in J$, equivalent to commutativity of their multiplication operators.

The definitive preserver theorem for JBW$^*$-algebras without central $I_1$, $I_2$ summands establishes that any linear bijection $\Phi:M\rightarrow J$ satisfying $$[x,c,y]=0\Leftrightarrow[\Phi(x),\Phi(c),\Phi(y)]=0$$ for all $x,c,y$ must have the form

$\Phi(x) = z_0 \circ J(x) + \beta(x)$

with $z_0\in Z(J)$ central invertible, $J:M\to J$ a Jordan isomorphism, and $\beta: M\to Z(J)$ central linear. If $\Phi$ is symmetric ($\Phi(x^*) = \Phi(x)^*$), $z_0$ is self-adjoint, $J$ is a Jordan $^*$-isomorphism, and $\beta$ symmetric (Escolano et al., 10 Sep 2024). This establishes the rigid template of structure-preserving encodings: only central scaling, Jordan algebra isomorphism, and central perturbations are permitted.

2. Structure-Preserving Encodings in Quantum Operations

In quantum information, structure-preserving encodings are transparently captured by channels mapping classical states (maximal commutative operator subalgebras) to classical states. Yu et al. show that local quantum channels are commutativity-preserving (CoP) if and only if they cannot create quantum correlations (discord) from classical inputs. The classification theorem for CoP channels stipulates operational and algebraic criteria: a completely positive trace-preserving map $\Phi$ is CoP iff the overlaps $K_iK_j^\dagger$ (formed from its Kraus operators) generate a commutative algebra; equivalently, certain four-copy expectation identities and structure-constant commutator constraints hold (Yu et al., 2011). These channels act as algebra homomorphisms on commutative subalgebras, and their action preserves all distance-based measures for quantum correlations. In categorical quantum encoding frameworks, a commutative classical monoid object is carried by a monoidal functor to a commutative quantum monoid object, demanding commutative diagrams for product, unit, and swap (braiding) (Parzygnat et al., 23 Dec 2024).

3. Commutativity Preservation in Operator Theory and Matrix Transformation

Bijective transformations of operator algebras preserving commutativity (in both directions) are tightly constrained. On conjugacy classes of finite-rank self-adjoint operators of fixed spectrum and rank (with maximal eigenvector dimensionalities distinct), any commutativity-preserving transformation is implemented via either a unitary or anti-unitary conjugation: $f(A) = UAU^*$ (Pankov, 2019). For self-adjoint operators on Hilbert space, maps preserving triadic relations involving differences and commutators are of the form $cUAU^*+dI$ or $cUA^TU^*+dI$ for $c\neq0$, $U$ unitary or anti-unitary (Karder et al., 14 Feb 2024). Strong $k$-commutativity preservers on standard operator algebras necessarily take the form $\Phi(A)=\lambda A + h(A)I$, with $\lambda^{k+1}=1$ and $h$ scalar functional (Liu et al., 2016). These paradigms reiterate that commutativity preservation is fundamentally rigid, admitting only linear or conjugate-linear automorphisms up to central shifts.

4. Categorical and Combinatorial Encodings: Incidence and Monoidal Frameworks

In combinatorial algebra, bijective linear maps of incidence algebras that strongly preserve commutativity are parameterized by poset-automorphisms, field-automorphisms, maximal-chain interval scalars, and diagonal shifts, yielding a product structure: $$\varphi(f)(x, y) = c(x, y) \sigma(f(\theta^{-1}(x), \theta^{-1}(y)))\quad \text{(off-diagonal)},\quad \varphi(f)(x, x) = f(\theta^{-1}(x),\theta^{-1}(x))$$ composed with $$\varphi_{\text{shift}}(f)(x, y) = \kappa_x f(x, y)\kappa_y^{-1}$$ (Fornaroli et al., 2022). This strictly limits commutativity-preserving encodings to concrete combinatorial and scalar manipulations.

In categorical quantum and data-theoretic settings, structure preservation is formalized via monoidal functors that carry commutative monoid objects—sets, vector spaces, or metric spaces—into commutative quantum monoids respecting product, unit, and swap diagrams (Parzygnat et al., 23 Dec 2024). Such functors guarantee that operations on classical structures translate to commuting quantum channels, fundamental for symmetry and information-theoretic invariants in quantum machine learning.

5. Directional Non-Commutative Monoidal Structures for Multidimensional Encodings

Recent developments propose directional non-commutative monoidal frameworks, where per-axis associative (but non-commutative) composition operators obey a global interchange law. In $D$ dimensions, the elements are tuples $(a; R_1^{n_1}, \ldots, R_D^{n_D})$, and composition along axis $i$ takes the form $$(a, R_i^{n_i}) \circ_i (b, R_i^{m_i}) := (a + R_i^{n_i} b, R_i^{n_i+m_i})$$, with the crucial requirement that $R_i R_j = R_j R_i$ for all $i\neq j$ (Godavarti, 30 May 2025, Godavarti, 21 May 2025). The interchange law

$$(a \circ_i b) \circ_j (c \circ_i d) = (a \circ_j c) \circ_i (b \circ_j d)$$

ensures traversal-independence and global coherence. This structure subsumes classical transforms (DFT, Hadamard, Walsh), and, when parametrized, extends to learnable transforms in machine learning and structured positional encoding in transformers. The cross-axis commutativity is precisely the algebraic principle bridging directional “local” non-commutativity and “global” structure preservation.

6. Spectral and Optimization Principles: Fan-Theobald-von Neumann Systems

Fan–Theobald–von Neumann (FTvN) systems formalize commutativity and optimization principles via spectral maps satisfying norm and trace inequalities: for $(V, W, \lambda)$, commutativity is $(x,y) = (\lambda(x), \lambda(y))$, equivalent to $\lambda(x+y) = \lambda(x) + \lambda(y)$ and $|\lambda(x) - \lambda(y)| = |x - y|$ (Gowda et al., 2022). Structure-preserving encodings are automorphisms $A$ with $\lambda(Ax) = \lambda(x)$, forming an orthogonal group fixing the system's center. In applications, these automorphisms translate optimization over orbit sets into invariant formulations and underlie commutativity and majorization in spectral cones, Jordan algebras, and matrix systems.

7. Analytical and Algorithmic Implications

In computational imaging and ML contexts, structure preservation is realized by enforcing commutativity of associated Laplacians, yielding joint diagonalizability and preserved geometric structure in transforms and embeddings (Eynard et al., 2013). Optimization objectives penalize commutator norms, and solutions preserve eigenvector alignments, guaranteeing spectral and relational structure transfer.

In semiprime $\Gamma$-rings, structure preservation is enforced by strong commutativity preserving (scp) maps, forcing derivations into the center and ensuring any scp endomorphism is an inner shift by a central element. Such rigidity generalizes to coding theory and operator-algebraic derivations, tightly coupling commutative relations with encoding forms (Xu et al., 2012).

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The results across operator, combinatorial, quantum, and categorical regimes converge on a unified principle: commutativity and higher-order relation preservation enforce exact algebraic, spectral, and categorical structure, restricting admissible transformations to automorphisms, central shifts, scalar rescalings, and specified functional symmetries. Structure-preserving encodings furnish a rigorous mechanism for invariance transfer, spectral coherence, and compatibility in operator theory, quantum information, data embeddings, and optimization frameworks.