Preduals of semigroup algebras

Abstract

For a locally compact group $G$, the measure convolution algebra $M(G)$ carries a natural coproduct. In previous work, we showed that the canonical predual $C_0(G)$ of $M(G)$ is the unique predual which makes both the product and the coproduct on $M(G)$ weak${}^{\ast }$-continuous. Given a discrete semigroup $S$, the convolution algebra ${\ell }^1(S)$ also carries a coproduct. In this paper we examine preduals for ${\ell }^1(S)$ making both the product and the coproduct weak${}^{\ast }$-continuous. Under certain conditions on $S$, we show that ${\ell }^1(S)$ has a unique such predual. Such $S$ include the free semigroup on finitely many generators. In general, however, this need not be the case even for quite simple semigroups and we construct uncountably many such preduals on ${\ell }^1(S)$ when $S$ is either ${\mathbb{Z}}_{+}\times \mathbb{Z}$ or $(\mathbb{N},\cdot )$.

17 pages, LaTeX

References in corpus (1)

• Conditions implying the uniqueness of the weak${}^{\ast }$-topology on certain group algebras