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 C0(G) of M(G) and C ∗(G) of A(G) are uniquely determined. In both cases we consider a natural coassociative multiplication and show that the canonical predual gives rise to the unique weak∗-topology making both the multiplication separately weak∗-continuous and the coassociative multiplication weak∗-continuous. In particular, dual cohomological properties of these algebras are well defined with this additional structure.