Annals of Mathematical Sciences and Applications

Let $G$ be a topological group and $E$ a quasi-complete space. We study approximation properties of almost periodic vectors of continuous, equicontinuous representations $\pi : G \to \mathcal{B}(E)$. We extend an approximation theorem of Weyl and Maak from isometric Banach space representations to representations on quasi-complete spaces. We prove that if $\pi$ is almost periodic, then $E$ has a generalized direct sum decomposition $E = \oplus_{ \theta \in \widehat{G}} E_\theta$, where each $E_\theta$ is linearly spanned by finite-dimensional, $\pi$-invariant subspaces. We show that on left translation invariant, quasi-complete subspaces of $L^p(G)$ ($G$ locally compact, $1 \leq p \lt \infty$), the left regular representation is almost periodic if and only if $G$ is compact.