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