A Note on Standard Closed Ideals in Weighted Discrete Abelian Semigroup Algebras

Abstract

Let $S$ be an abelian semigroup and $\omega$ be a weight on $S$. If $T$ is a semigroup ideal in $S$, then the closed linear subspace $\ell_T^1(S, \, \omega) := \{ f \in \ell^1(S, \, \omega) : \textrm{supp}\,f \subseteq T\}$ is a closed ideal in $\ell^1(S, \, \omega)$. Such ideals including $\{0\}$ and $\ell^1(S, \, \omega)$ are \emph{standard closed ideals}; while the others are \emph{non-standard closed ideals} in $\ell^1(S, \, \omega)$. The weight $\omega$ on $S$ is an \emph{unicellular weight} if every closed ideal in $\ell^1(S, \, \omega)$ is a standard closed ideal. In the case where $S = {\mathbb Z}_+$, it has been extensively studied by several mathematicians. In this article, we intend to study the standard closed ideals in the case where $S = {\mathbb Z}_+^2$.