Abstract
In this article we study the embeddability of cones in a Banach space X. First we prove that c 0 is embeddable in X if and only if its positive cone \({c_0^+}\) is embeddable in X and we study some properties of Banach spaces containing c 0 in the light of this result. So, unlike with the positive cone of ℓ 1 which is embeddable in any non-reflexive space, \({c_0^+}\) has the same behavior as the whole space c 0. In the second part of this article we give a characterization of Grothendieck spaces X according to the geometry of cones of X*. By these results we give a partial positive answer to a problem of J.H. Qiu concerning the geometry of cones.
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This article is dedicated to the memory of I.A. Polyrakis’s Friend and Collaborator C.D. Aliprantis.
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Polyrakis, I.A., Xanthos, F. Cone characterization of Grothendieck spaces and Banach spaces containing c 0 . Positivity 15, 677–693 (2011). https://doi.org/10.1007/s11117-010-0103-7
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DOI: https://doi.org/10.1007/s11117-010-0103-7