Abstract of a paper by Matthew Daws, Hung Le Pham, and Stuart White
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.
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alspach@fourier.math.okstate.edu