Ditkin sets for some functional spaces and applications to grand Lebesgue spaces

Abstract

Let $G$ be a locally compact Abelian group with dual group $\widehat G $ and Haar measures $d\mu$ and $\hat d\mu$ respectively. In this work we have proved that if $X$ is an essential Banach ideal in Beurling algebra $ L^1_{\omega}(G),$ then a closed subset $E\subset \widehat G$ is a Ditkin set for $X$ if and only if $E$ is a Ditkin set for $ L^1_{\omega}(G).$ Next, as an application we have investigated the Ditkin sets for grand Lebesgue space $L^{p),\theta}(G)$ and Ditkin sets for $[L^p(G)]_{L^{p),\theta }}$, where $[L^p(G)]_{L^{p),\theta }}$ is the closure of the set $C_c(G)$ in $L^{p),\theta}(G)$.