A characterization of discreteness for locally compact groups in terms of the Banach algebras $A_{p}(G)$
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Abstract:
The Banach algebra $A$ is said to have the bounded power property if for any $x \in A$, with $||x|{|_{sp}} = {\lim _{n \to \infty }}||{x^n}|{|^{1/n}} \leqslant 1$, one has ${\sup _n}||{x^n}|| < \infty$. 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].References
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Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 54 (1976), 189-192
- DOI: https://doi.org/10.1090/S0002-9939-1976-0387954-3
- MathSciNet review: 0387954