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Let $G$ be a locally compact group. We know that if $G$ is abelian and $\cal F$ implies the Fourier transform, for every Segal algebra of $G$ say $S^1(G)$, ${\cal F}S^1(G)$ is an abstract Segal algebra with respect to $A(\widehat{G})$ (since $S^1(G)$ is an abstract Segal algebra of $L^1(G)$). Reiter in [1, Proposition 6.2.5] has shown that $C_c(\widehat{G})\cap A(\widehat{G}) \subseteq {\cal F}S^1(G)$.

My Question is: Can we show that for an aribtarary locally compact group $G$ (not necessarily abelian) every abstract Segal algebra of $A(G)$ contains all functions in $A(G)$ that have compact support?

[1] H. Reiter, and J. D. Stegeman, Classical harmonic analysis and locally compact groups, 2nd edn, London Mathematical Society Monographs, New series 22, Oxford university press, New York, 2000.

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Does Reiter offer any results for amenable groups? – Yemon Choi Aug 17 '11 at 10:17
My vague recollection is that he proves results along these lines for a very general class of algebras, but perhaps only for symmetric Segal subalgebras. I would suggest checking the early sections of the book more closely, paying attention to the results that are stated for general commutative Banach algebras satisfying certain conditions. The result you want might not be there, but something like it might be – Yemon Choi Aug 17 '11 at 10:22
@ Yemon: I could not find any suggestion in Reiter's books for amenable groups. About your second comment: If I can show that a general version of "Wiener-Levy Theorem" is correct for $A(G)$ when $G$ is not abelian, then easily I can follow Reiter's proof. Therefore, I can rewrite the question as this: Is there any extension of "Wiener-Levy Theorem" for other locally compact groups (non-abelian ones)? – Mahmood Alaghmandan Aug 17 '11 at 17:47

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