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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A characterization of discreteness for locally compact groups in terms of the Banach algebras $ A\sb{p}(G)$


Author: Edmond E. Granirer
Journal: Proc. Amer. Math. Soc. 54 (1976), 189-192
MathSciNet review: 0387954
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Abstract: The Banach algebra $ A$ is said to have the bounded power property if for any $ x \in A$, with $ \vert\vert x\vert{\vert _{sp}} = {\lim _{n \to \infty }}\vert\vert{x^n}\vert{\vert^{1/n}} \leqslant 1$, one has $ {\sup _n}\vert\vert{x^n}\vert\vert < \infty $. It has been shown by B. M. Schreiber [9, Theorem (8.6)] that, if $ G$ is a locally compact abelian group, then the Fourier algebra $ A(G) = {L^1}{(\Gamma )^ \wedge }$ has the bounded power property, if and only if $ G$ is discrete. We improve this result in the Theorem. Let $ G$ be an arbitrary locally compact group and $ 1 < p < \infty $. Then $ {A_p}(G)$ has the bounded power property if and only if $ G$ is discrete. Our proof, even for abelian $ G$ and $ p = 2$ (then $ {A_2}(G) = A(G)$ is the usual Fourier algebra of $ G$), is much simpler and entirely different from that of [9].


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DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0387954-3
PII: S 0002-9939(1976)0387954-3
Article copyright: © Copyright 1976 American Mathematical Society