Idempotent Maximal Ideals and Independent Sets

Abstract

Let $E$ be a compact independent subset of a nondiscrete LCA group $G$. Let $GpE$ be the subgroup of $G$ generated algebraically by $E$. If $\mu$ is a continuous, regular, Borel measure on $GpE$ with $\mu (GpE) \ne 0$, then there exists a maximal ideal $\chi$ of the algebra $M(G)$ of regular Borel measures on $G$ such that the restriction of $\chi$ to ${L^1}(\mu ) = \{ \nu \in M(G):\nu \ll \mu \}$ is a nontrivial idempotent in ${L^\infty }(\mu )$. This result is used to give a new proof that $GpE$ has zero Haar measure.