A characterization of discrete groups

Ranieri, Giovanni

doi:10.1090/S0002-9939-03-07359-3

A characterization of discrete groups
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by Giovanni Ranieri

Proc. Amer. Math. Soc. 132 (2004), 1845-1848

DOI: https://doi.org/10.1090/S0002-9939-03-07359-3

Published electronically: December 23, 2003

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Abstract:

The purpose of this article is to prove the following result. Let $G$ be a locally compact group, $\mathcal {A}(G)$ the Fourier algebra of $G,$ and $\mathcal {S} (G)=\{ u\in \mathcal {A}(G) :~\exists ~c>~0$ such that $\parallel uv\parallel _{\mathcal {A}(G)}\leq ~c\parallel v\parallel _{\infty }\hspace {0.2 cm}\forall ~ v\in \mathcal {A}(G)\}$. Then $G$ is a discrete group $\Longleftrightarrow \mathcal {S} (G)~\neq \{0\}$.

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