Preduals of semigroup algebras
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 C0(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 ℓ1(S) also carries a coproduct. In this paper we examine preduals for ℓ1(S) making both the product and the coproduct weak*-continuous. Under certain conditions on S, we show that ℓ1(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 ℓ1(S) when S is either ℤ+×ℤ or (ℕ,⋅).
KeywordsUnique predual Semigroup algebra Semitopological semigroup
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