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Spectral Theory of Almost Periodic Banach--Malcev Algebras and Applications to Moufang Dynamics

Marwa Ennaceur

TL;DR

This work extends Bohr’s almost periodicity to non‑associative Banach–Malcev algebras by using the relative compactness of adjoint orbits, yielding the spectral constraint $\sigma(ad(x))\subset i\mathbb{R}$ and a continuous functional calculus. It develops a rigorous framework for almost periodic Malcev algebras, introduces the intrinsic non‑associative defect $S(x,y)$, and analyzes how this defect governs non‑Lie corrections in dynamics via structural actions, with a concrete octonionic example on $S^7$. The paper then constructs a dynamical picture on Moufang loops, proves spectral properties for the Malcev Laplacian, and provides explicit results for octonionic dynamics, including strictly periodic flows on $S^7$ and finite‑dimensional eigenspace representations. The results establish a mathematically solid foundation for almost periodicity beyond Lie theory, enabling spectral analysis and dynamical studies on Malcev manifolds, with potential but speculative connections to non‑associative gauge models. Overall, the work broadens harmonic analysis and spectral geometry into the non‑associative realm, notably through the canonical octonionic example and its Malcev Laplacian spectrum.

Abstract

We introduce almost periodic Banach--Malcev algebras as a non-associative extension of Bohr's classical theory. Our framework is based on the relative compactness of adjoint orbits $\{e^{t\,\mathrm{ad}(x)}(y)\}$, which yields the spectral characterization $σ(\mathrm{ad}(x)) \subseteq i\mathbb{R}$, uniform boundedness of orbit closures in the strong operator topology, and a continuous functional calculus for almost periodic derivations. Compact Malcev algebras -- most notably the imaginary octonions $\mathrm{Im}(\mathbb{O})$ -- provide canonical finite-dimensional examples, and their associated Moufang loops carry strictly periodic flows. We also analyze structural actions on eigenspaces of the Malcev Laplacian as a concrete case study, where the bounded defect operator $S(x,y) \in \mathcal{B}(M)$ quantifies the non-associative correction. While speculative links to non-associative gauge theory are noted, they lie beyond the established mathematical scope. The recent convergence control of the BCH series for special Banach--Malcev algebras \cite{Athmouni2025} provides analytic justification for the local Moufang structure used throughout.

Spectral Theory of Almost Periodic Banach--Malcev Algebras and Applications to Moufang Dynamics

TL;DR

This work extends Bohr’s almost periodicity to non‑associative Banach–Malcev algebras by using the relative compactness of adjoint orbits, yielding the spectral constraint and a continuous functional calculus. It develops a rigorous framework for almost periodic Malcev algebras, introduces the intrinsic non‑associative defect , and analyzes how this defect governs non‑Lie corrections in dynamics via structural actions, with a concrete octonionic example on . The paper then constructs a dynamical picture on Moufang loops, proves spectral properties for the Malcev Laplacian, and provides explicit results for octonionic dynamics, including strictly periodic flows on and finite‑dimensional eigenspace representations. The results establish a mathematically solid foundation for almost periodicity beyond Lie theory, enabling spectral analysis and dynamical studies on Malcev manifolds, with potential but speculative connections to non‑associative gauge models. Overall, the work broadens harmonic analysis and spectral geometry into the non‑associative realm, notably through the canonical octonionic example and its Malcev Laplacian spectrum.

Abstract

We introduce almost periodic Banach--Malcev algebras as a non-associative extension of Bohr's classical theory. Our framework is based on the relative compactness of adjoint orbits , which yields the spectral characterization , uniform boundedness of orbit closures in the strong operator topology, and a continuous functional calculus for almost periodic derivations. Compact Malcev algebras -- most notably the imaginary octonions -- provide canonical finite-dimensional examples, and their associated Moufang loops carry strictly periodic flows. We also analyze structural actions on eigenspaces of the Malcev Laplacian as a concrete case study, where the bounded defect operator quantifies the non-associative correction. While speculative links to non-associative gauge theory are noted, they lie beyond the established mathematical scope. The recent convergence control of the BCH series for special Banach--Malcev algebras \cite{Athmouni2025} provides analytic justification for the local Moufang structure used throughout.

Paper Structure

This paper contains 19 sections, 11 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Malcev algebras and the Malcev identity
  4. Derivations and automorphisms
  5. Analytic and topological structures
  6. Quantifying the Deviation from the Lie Property
  7. Almost Periodic Malcev Algebras
  8. Definition and basic properties
  9. Spectral properties and functional calculus
  10. Dynamical Systems and Almost Periodic Flows on Malcev Manifolds
  11. Malcev manifolds and infinitesimal actions
  12. Recurrent and quasi-periodic Malcev dynamics
  13. Preliminary Framework for Non-Lie Actions
  14. Applications
  15. Spectral analysis on almost periodic Malcev manifolds
  16. ...and 4 more sections

Key Result

Proposition 2.3

For all $x, y \in \mathfrak{M}$, the commutator of adjoint operators satisfies where $S(x, y) \in B(\mathfrak{M})$ is the bounded linear operator defined by The operator $S(x, y)$ vanishes identically if and only if the Jacobian $J$ is zero, i.e., when $\mathfrak{M}$ is a Lie algebra.

Theorems & Definitions (45)

  • Definition 2.1
  • Example 2.1
  • Remark 2.2
  • Definition 2.2
  • Proposition 2.3
  • proof : Proof of the identity for $S(x,y)$
  • Definition 2.3
  • Definition 2.4
  • Remark 2.4
  • Definition 2.5
  • ...and 35 more