Almost periodic vectors and representations in quasi-complete spaces

Volume 8, Issue 2 (2023), pp. 203–222

Mahmoud Filali

^{ }Mehdi Sangani Monfared^{ }
Pub. online: 26 July 2023
Type: Article

Received

27 February 2023

27 February 2023

Accepted

16 April 2023

16 April 2023

Published

26 July 2023

26 July 2023

#### Notes

Dedicated to professor A. T.-M. Lau with much appreciation for his lifetime contributions to mathematics

#### Abstract

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.