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We investigate possible preduals of the measure algebra $M(G)$ of a locallycompact group and the Fourier algebra $A(G)$ of a separable compact group. Bothof these algebras are canonically dual spaces and the canonical preduals makethe multiplication separately weak$^*$-continuous so that these algebras aredual Banach algebras. In this paper we find additional conditions under whichthe preduals $C_0(G)$ of $M(G)$ and $C^*(G)$ of $A(G)$ are uniquely determined.In both cases we consider a natural coassociative multiplication and show thatthe canonical predual gives rise to the unique weak$^*$-topology making boththe multiplication separately weak$^*$-continuous and the coassociativemultiplication weak$^*$-continuous. In particular, dual cohomologicalproperties of these algebras are well defined with this additional structure.

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