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

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

2008-04-23

arxiv.org/abs/0804.3764v1

Houston J. Math. 35 (2009), no. 1, 253--276.

Mathematics

Functional Analysis

21 pages

Scientific paper

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 $C_0(G)$ of $M(G)$ and $C^*(G)$ of $A(G)$ are uniquely determined. In both cases we consider a natural coassociative multiplication and show 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.

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