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

TL;DR

This paper characterizes Ditkin sets in certain Banach ideals within Beurling algebras and applies these results to analyze Ditkin sets in grand Lebesgue spaces and their closures.

Contribution

It establishes an equivalence of Ditkin sets between Banach ideals in Beurling algebras and the algebras themselves, and applies this to grand Lebesgue spaces.

Findings

Ditkin sets in Banach ideals correspond to those in Beurling algebras.

Characterization of Ditkin sets in grand Lebesgue spaces.

Application to closures of compactly supported functions.

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 $L^{p),\theta}(G)$.