Weighted measure algebras and uniform norms
Let ω be a weight on an LCA group G. Let M (G, ω) consist of the Radon measures µ on G such that ωµ is a regular complex Borel measure on G. It is proved that: (i) M(G, ω) is regular iff M(G, ω) has unique uniform norm property (UUNP) iff L 1 (G, ω) has UUNP and G is discrete; (ii) M (G, ω) has a minimum uniform norm iff L 1 (G, ω) has UUNP: (iii) M 00 (G, ω) is regular iff M 00 (G, ω) has UUNP iff L 1 (G, ω) has UUNP, where M 00 (G, ω) : = {µ ∈ M (G, ω) : µ ^ - 0 on Δ (M(G, ω)) \ Δ (L 1 (G, ω))}.