Ditkin’s condition for certain Beurling algebras

Abstract

Let $G$ be a locally compact abelian group. A function $\omega :G\to [1,\infty )$ is said to be a weight if it is locally bounded, Borel measurable and submultiplicative. We call a weight $\omega$ on $G$ semi-bounded if there exist a constant $K$ and a subsemigroup $S$ with $S-S=G,$ such that \[ \omega (s)\leq K\quad \text {and}\quad \lim _{n\to \infty }\frac {\log \omega (-ns)}{\sqrt {n}}=0\] for all $s\in S.$ Using functional analytic methods, we show that all Beurling algebras $\lg$ whose defining weight $\omega$ is semi-bounded satisfy Ditkin's condition.