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Weighted group algebras

Let $G$ be a locally compact Abelian group, and $w: G\to (0, \infty)$ be a Borel measurable weighted function. In this paper, the algebraic and topological properties of group algebra are studied and assessed. We show that the weighted group algebra $L^{1}(G, w)$ is regular if and only if $w$ is a nonquasianalytic weight function. Also $L^{1}(G, w)$ is Tauberian, for any Borel measurable weight function $w$ on the group $G$.