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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 \((\mathbb N,\cdot)\).

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          Author and article information

          Journal
          2008-11-24
          Article
          0811.3987

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

          Custom metadata
          43A20; 22A20
          Semigroup Forum, 80, 61--78, 2010.
          17 pages, LaTeX
          math.FA

          Functional analysis

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