Translation-finite sets, and weakly compact derivations from \(\lp{1}(\Z_+)\) to its dual

• Published in: arXiv.org
• Main Authors: Choi, Yemon, Heath, Matthew J
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 29.03.2010
• Subjects: Combinatorial analysis; Convolution
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.0811.4432

We characterize those derivations from the convolution algebra \(\ell^1({\mathbb Z}_+)\) to its dual which are weakly compact. In particular, we provide examples which are weakly compact but not compact. The characterization is combinatorial, in terms of "translation-finite" subsets of \({\mathbb Z}_+\), and we investigate how this notion relates to other notions of "smallness" for infinite subsets of \({\mathbb Z}_+\). In particular, we show that a set of strictly positive Banach density cannot be translation-finite; the proof has a Ramsey-theoretic flavour.