Are all of compact support functions of $A(G)$ in its abstract Segal algebras?

2011-08-17T09:48:13Z

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.

Are all of compact support functions of $A(G)$ in its abstract Segal algebras? - MathOverflow

Are all of compact support functions of $A(G)$ in its abstract Segal algebras?