A new convolution on Beurling algebras

Let G be a locally compact group or a discrete group and let (omega) be a weightfunction on G. We define a new convolution on the Beurling algebra L1(G, (omega))and show that it is also a Banach algebra with this new convolution. We showthat there is a one to one correspondence between the unitary representations of Gand the non-degenerate *-representations of L1(G, Buerling algebras, weight functions, representations, amenability.). We show that L1(G, (omega)) isamenable if and only if G is amenable. This improves earlier results which require(omega) to be diagonally bounded.