Isometric isomorphisms of Beurling algebras

Let G and H be locally compact groups with continuous weights ω1 and ω2 respectively, such that ωi(ei)=1, i=1,2. In this article we show that if M(G,ω1) (the weighted measure algebra on G) is isometrically algebra isomorphic to M(H,ω2), then the underlying weighted groups are isomorphic, i.e. there exists an isomorphism of topological groups ϕ:G→H such that ω1ω2∘ϕ is multiplicative. Similarly, we show that any weighted locally compact group (G,ω) is completely determined by its Beurling group algebra L1(G,ω), LUC(G,ω−1)⁎ and L1(G,ω)⁎⁎, when the two last algebras are equipped with an Arens product.