This is an announcement for the paper "Preduals of semigroup algebras" by Matthew Daws, Hung Le Pham, and Stuart White.
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 $(\mathbb N,\cdot)$.
Archive classification: math.FA
Mathematics Subject Classification: 43A20; 22A20
Remarks: 17 pages, LaTeX
The source file(s), semigroups.tex: 50737 bytes, is(are) stored in gzipped form as 0811.3987.gz with size 15kb. The corresponding postcript file has gzipped size 114kb.
Submitted from: matt.daws@cantab.net
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/0811.3987
or
http://arXiv.org/abs/0811.3987
or by email in unzipped form by transmitting an empty message with subject line
uget 0811.3987
or in gzipped form by using subject line
get 0811.3987
to: math@arXiv.org.