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Characterizations of $ B(G)$ and $ B(G)\cap AP(G)$ for locally compact groups


Author: Kari Ylinen
Journal: Proc. Amer. Math. Soc. 58 (1976), 151-157
MSC: Primary 43A60
DOI: https://doi.org/10.1090/S0002-9939-1976-0425517-1
MathSciNet review: 0425517
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Abstract: Given a locally compact (and possibly non-Abelian) group $ G$, we denote by $ B(G)$ the set of linear combinations of continuous positive-definite functions on $ G$ and by $ AP(G)$ the set of continuous almost periodic functions on $ G$. In this paper the sets $ B(G)$ and $ B(G) \cap AP(G)$ are characterized in terms of convolutions with measures. Specifically, let $ U$ consist of those measures $ \mu \in M(G)$ for which $ \vert\vert\pi (\mu )\vert\vert \leq 1$, whenever $ \pi $ is a continuous unitary representation of $ G$. It is proved that a function $ f \in {L^\infty }(G)$ belongs to (i.e. is equal locally almost everywhere to a function in) $ B(G)$ if and only if the convolutions $ \mu \ast f,\mu $ ranging over $ U$, form a relatively weakly compact set in $ {L^\infty }(G)$. The same holds if we confine our attention to either the finitely supported or the absolutely continuous measures in $ U$. Moreover, it is shown that any of these three sets of convolutions is relatively norm compact if and only if $ f$ belongs to $ B(G) \cap AP(G)$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0425517-1
Keywords: Locally compact groups and their continuous unitary representations, $ {C^ \ast }$-algebras, positive-definite functions, relatively (weakly) compact sets
Article copyright: © Copyright 1976 American Mathematical Society

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