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

TL;DR

This paper develops a spectral theory for almost periodic Banach--Malcev algebras, extending classical concepts to non-associative structures, and explores applications to Moufang loops and non-associative dynamics.

Contribution

It introduces almost periodic Banach--Malcev algebras, providing spectral characterization, functional calculus, and concrete examples like imaginary octonions, expanding the mathematical framework of non-associative algebras.

Findings

Spectral characterization of derivations with spectra in iℝ.

Existence of a continuous functional calculus for almost periodic derivations.

Finite-dimensional examples include imaginary octonions and their Moufang loops.

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 $\sigma(\mathrm{ad}(x)) \subseteq i\mathbb{R}$, 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.