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

Abstract

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.

arXiv commentv1: 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.