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arXiv · 0811.4432

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

Yemon Choi·Matthew J. Heath
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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.

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Yemon Choi, Matthew J. Heath. 2010-03-29. Translation-finite sets, and weakly compact derivations from $\lp{1}(\Z_+)$ to its dual. https://doi.org/10.1112/blms%2Fbdq003

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