Almost periodic vectors and representations in quasi-complete spaces

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.

Keywords

almost periodic vectors, vector-valued almost periodic functions, quasi-complete spaces, topological groups, equicontinuous representations, vector-valued means

2010 Mathematics Subject Classification

Primary 43A07, 43A60, 43A65. Secondary 46E40, 47B07.

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

Received 27 February 2023

Accepted 16 April 2023

Published 26 July 2023