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## Translation-finite sets and weakly compact derivations from $\ell^1(\mathbb Z_+)$ to its dual

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Journal publication date 2010 Bulletin of the London Mathematical Society 3 42 12 429-440 Published English

### Abstract

We characterize those derivations from the convolution algebra ℓ1(ℤ+) to its dual that are weakly compact, providing explicit examples that are not compact. The characterization is combinatorial, in terms of ‘translation-finite’ subsets of ℤ+, and we investigate how this notion relates to other notions of ‘smallness’ for infinite subsets of ℤ+. In particular, we prove that a set of strictly positive Banach density cannot be translation-finite; the proof has a Ramsey-theoretic flavour.