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Spectral Theory and Almost Periodic Structures in Hom--Lie Banach Algebras

Marwa Ennaceur

TL;DR

The paper develops a functional-analytic framework for Hom--Lie Banach algebras with a bounded twisting map $\alpha$ and a twisted bracket $[a,b]_{\alpha}=\alpha(ab-ba)$. It proves a complete Bohr–Fourier spectral decomposition for inner $\alpha$-twisted derivations $\delta=\mathrm{ad}_{\alpha}(X)$ under the assumption that the generated $C_0$-group has relatively compact orbits and that $\alpha$ commutes with the group, yielding a direct sum decomposition into almost periodic and ergodic subalgebras closed under the twisted bracket. The work provides explicit Hom--Banach--Malcev constructions, several operator-algebraic applications (including a twisted Weyl algebra with a $\mathbb{Z}^2$ Bohr spectrum), and a scalable numerical illustration validating norm-convergence of the Bohr expansion. It also develops a categorical morphism framework ensuring the algebraic compatibility of the decomposed subspaces, and discusses limitations and avenues for generalization to unbounded operators, weaker notions of almost periodicity, and connections to non-associative deformation theories and quantum ergodicity.

Abstract

We develop a systematic functional-analytic framework for Hom--Lie Banach algebras, introducing bounded $α$-twisted derivations and almost periodic elements. Under natural continuity and compactness assumptions, we establish a complete Bohr--Fourier spectral decomposition of such derivations. We prove that the associated almost periodic and ergodic subspaces are not merely topological complements but closed, $α$-invariant subalgebras, stable under the twisted Lie bracket a key structural novelty that enables coherent restriction of the dynamics. We provide explicit constructions of Hom--Banach--Malcev algebras and demonstrate our theory with concrete operator-algebraic applications, including a novel twisted Weyl algebra example, analyzed via the metaplectic representation, where a non-commuting twist enriches the Bohr spectrum from a cyclic group to a two-dimensional lattice.

Spectral Theory and Almost Periodic Structures in Hom--Lie Banach Algebras

TL;DR

The paper develops a functional-analytic framework for Hom--Lie Banach algebras with a bounded twisting map and a twisted bracket . It proves a complete Bohr–Fourier spectral decomposition for inner -twisted derivations under the assumption that the generated -group has relatively compact orbits and that commutes with the group, yielding a direct sum decomposition into almost periodic and ergodic subalgebras closed under the twisted bracket. The work provides explicit Hom--Banach--Malcev constructions, several operator-algebraic applications (including a twisted Weyl algebra with a Bohr spectrum), and a scalable numerical illustration validating norm-convergence of the Bohr expansion. It also develops a categorical morphism framework ensuring the algebraic compatibility of the decomposed subspaces, and discusses limitations and avenues for generalization to unbounded operators, weaker notions of almost periodicity, and connections to non-associative deformation theories and quantum ergodicity.

Abstract

We develop a systematic functional-analytic framework for Hom--Lie Banach algebras, introducing bounded -twisted derivations and almost periodic elements. Under natural continuity and compactness assumptions, we establish a complete Bohr--Fourier spectral decomposition of such derivations. We prove that the associated almost periodic and ergodic subspaces are not merely topological complements but closed, -invariant subalgebras, stable under the twisted Lie bracket a key structural novelty that enables coherent restriction of the dynamics. We provide explicit constructions of Hom--Banach--Malcev algebras and demonstrate our theory with concrete operator-algebraic applications, including a novel twisted Weyl algebra example, analyzed via the metaplectic representation, where a non-commuting twist enriches the Bohr spectrum from a cyclic group to a two-dimensional lattice.

Paper Structure

This paper contains 20 sections, 11 theorems, 80 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Banach algebras and derivations
  4. Hom--Lie algebras and twisted brackets
  5. Twisted derivations
  6. Hom--Banach--Malcev algebras
  7. Spectral Decomposition
  8. Applications and Examples
  9. Shift Operators
  10. $C^*$-Algebra Endomorphisms
  11. Weighted Shifts
  12. A Twisted Weyl Algebra: A Heuristic Example Beyond the Commuting Framework
  13. Morphisms and Invariant Substructures
  14. Technical Remarks on Convergence and Limitations
  15. Numerical Illustration
  16. ...and 5 more sections

Key Result

Lemma 3.2

If the underlying Malcev bracket satisfies $\|[x,y]\|\le C\|x\|\|y\|$ for some $C>0$ and $\alpha$ is bounded, then the twisted bracket $[\cdot,\cdot]_\alpha$ is continuous and satisfies

Figures (3)

  • Figure 1: (a) Growth of the number of significant Bohr frequencies with system size $N$. (b) Power-law decay of the reconstruction error, confirming the stability and scalability of the spectral decomposition.
  • Figure 2: Almost periodic evolution of the $(0,1)$-entry of $A(t) = e^{t\delta}(A_0)$. Both components display recurrent quasi-patterns characteristic of Bohr almost periodicity.
  • Figure 3: Reconstruction error $\varepsilon(t) = |a(t) - a_{\mathrm{rec}}(t)|$. The error remains below $8 \times 10^{-4}$, validating the theoretical prediction of norm-convergent Bohr expansion.

Theorems & Definitions (33)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 23 more