amenability of weighted measure algebras

let g be a locally compact group, and let ω be a weight on g. we show that the weightedmeasure algebra m(g,ω) is amenable if and only if g is a discrete, amenable group andsup{ω(g) ω(g−1) : g ∈ g} < ∞, where ω(g) ≥ 1 (g ∈ g) .