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

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<mark>Journal publication date</mark>2010
<mark>Journal</mark>Bulletin of the London Mathematical Society
Issue number3
Volume42
Number of pages12
Pages (from-to)429-440
Publication StatusPublished
<mark>Original language</mark>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.