A characterization of the Schur property by means of the Bohr topology

Let G be a MAPA group that is metrizable and satisfies Pontryagin duality; that is, it coincides with its topological bidual. We prove that the Bohr topology of G respects compactness if and only if every non-totally bounded subset contains an infinite discrete subset which is C∗-embedded in the Bohr compactification of G. This result is used to characterize the Banach spaces which respect compactness, or, with a different terminology, have the Schur property (defined below). Among other equivalent properties, we prove that a Banach space E has the Schur property if and only if every bounded basic sequence contains an infinite subsequence equivalent to a l1-basis.