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

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.