Conditions implying the uniqueness of the weak*-topology on certain group algebras

We investigate possible preduals of the measure algebra M(G) of a locally compact group and the Fourier algebra A(G) of a separable compact group. Both of these algebras are canonically dual spaces and the canonical preduals make the multiplication separately weak-continuous so that these algebras are dual Banach algebras. In this paper we find additional conditions under which the preduals Co(G) of M(G) and C*(G) of A(G) are uniquely determined. In both cases we consider a natural comultiplication and show that the canonical predual gives rise to the unique weak*-topology making both the multiplication separately weak-continuous and the comultiplication weak-continuous. In particular, dual cohomological properties of these algebras are well defined with this additional structure.