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# Preduals of semigroup algebras

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### 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$$^*$$-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$$^*$$-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 $$(\mathbb N,\cdot)$$.

### Author and article information

###### Journal
2008-11-24
###### Article
0811.3987