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

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

2008-11-28

arxiv.org/abs/0811.4432v4

Bull. London Math. Soc. 42 (2010), no. 3, 429--440

Mathematics

Functional Analysis

v1: 14 pages LaTeX (preliminary). v2: 13 pages LaTeX, submitted. Some streamlining, renumbering and minor corrections. v3: app

Scientific paper

10.1112/blms/bdq003

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.

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