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

Abstract

We characterize those derivations from the convolution algebra 1(Z+)\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 Z+{\mathbb Z}_+, and we investigate how this notion relates to other notions of "smallness" for infinite subsets of Z+{\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.Comment: v1: 14 pages LaTeX (preliminary). v2: 13 pages LaTeX, submitted. Some streamlining, renumbering and minor corrections. v3: appendix removed. v4: Modified appendix reinstated; 14 pages LaTeX. To appear in Bull. London Math. Soc

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