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Almost periodic vectors and representations in quasi-complete spaces. (English) Zbl 07735727

Summary: 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 \(E=\bigoplus_{\theta\in\widehat{G}}E_\theta\), 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, \(1\le p<\infty\)), the left regular representation is almost periodic if and only if \(G\) is compact.

MSC:

43A07 Means on groups, semigroups, etc.; amenable groups
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
46E40 Spaces of vector- and operator-valued functions
47B07 Linear operators defined by compactness properties
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