Complete continuity properties of Banach spaces associated with subsets of a discrete abelian group (Q2731114)

The authors introduce and study three different types of \(\Lambda\)-complete continuity properties on a Banach space. Let \(\Lambda\) be a subset of the dual of a comapct metrizable Abelian group \(G\). A \(\Lambda\)-measure (resp. \(\Lambda\)-function) with values in a Banach space \(X\) is a countable additive measure of bounded variation (resp. a Bochner integrable function) for the normalized Haar measure on \(G\) such that its Fourier coefficients at every element of the set \(\Lambda\) vanish. A Banach space \(X\) has the I-\(\Lambda\) complete continuity property (resp. the II-\(\Lambda\)-CCP) if every \(X\)-valued \(\Lambda\)-measure of bounded average range (resp. of bounded variation) has relatively compact range. These properties can be seen as the completely continuous counterpart of the types of Radon-Nikodým properties studied by Edgar and Dowling. The properties are characterized in terms of good approximate identities in \(G\) and Cauchy sequences for the Pettis norm. A third type of \(\Lambda\)-complete continuity property is introduced. If \(\Lambda\) is a Riesz set, the third property coincides with the second one for a Banach space. If \(\Lambda\) is not a Riesz set, then it coincides with the usual complete continuity property (i.e. every bounded linear operator from \(L^1(G)\) into \(X\) is continuous). The three properties are stable by \(G_\delta\)-embeddings.