Preduals of semigroup algebras

TL;DR

This paper investigates the uniqueness of preduals for semigroup convolution algebras, showing that under certain conditions the predual is unique, but in other cases, there are many such preduals, including for simple semigroups.

Contribution

It extends the understanding of preduals for semigroup algebras, identifying conditions for uniqueness and constructing multiple preduals in specific cases.

Findings

Unique predual for certain semigroups like free semigroup on finitely many generators.

Existence of uncountably many preduals for semigroups like a5_+ imesa5 or (a5, extperiodcentered).

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 $(\mathbb N,\cdot)$.