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

We develop a systematic functional-analytic framework for Hom–Lie Banach algebras, introducing bounded $\alpha$-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, $\alpha$-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.