Idempotent maximal ideals and independent sets

Abstract

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