Conditions implying the uniqueness of the weak${}^{\ast }$-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${}^{\ast }$-continuous so that these algebras are dual Banach algebras. In this paper we find additional conditions under which the preduals $C_0(G)$ of $M(G)$ and $C^{\ast }(G)$ 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${}^{\ast }$-topology making both the multiplication separately weak${}^{\ast }$-continuous and the coassociative multiplication weak${}^{\ast }$-continuous. In particular, dual cohomological properties of these algebras are well defined with this additional structure.