Are all of compact support functions of $A(G)$ in its abstract Segal algebras? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T12:26:00Z http://mathoverflow.net/feeds/question/73041 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73041/are-all-of-compact-support-functions-of-ag-in-its-abstract-segal-algebras Are all of compact support functions of $A(G)$ in its abstract Segal algebras? Mahmood Alaghmandan 2011-08-17T09:48:13Z 2011-08-17T09:55:32Z <p>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 $C_c(\widehat{G})\cap A(\widehat{G}) \subseteq {\cal F}S^1(G)$.</p> <p>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?</p> <p>[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.</p>