Spaces with dual order-isomorphic to \({\ell}_1\)
It is known that each Banach space whose dual is linearly isometric to \(\ell_1\) contains a copy of \(c_0\) [\textit{V. P. Fonf}, Funct. Anal. Appl. 13, 66-67 (1979; Zbl 0426.46012)]. However, if the dual is isomorphic to \(\ell_1\) the situation is quite different. By \textit{J. Bourgain} and \textit{F. Delbaen} [Acta Math. 145, 155-176 (1980; Zbl 0466.46024)] was constructed an example of a Banach space \(E\) whose dual is isomorphic to \(\ell_1\) but \(c_0\) is not imbedable in \(E\). In this direction in the paper the author proves the following result. Theorem. For an ordered Banach space \(E\) the following statements are equivalent: (i) \(E\) is order-isomorphic to \(c_0\). (ii) \(E\) is a \(\sigma\)-Dedekind complete Riesz space and the norm dual \(E'\) of \(E\) is order isomorphic to \(\ell_1\). If statement (ii) is true, \(F\) is order-isomorphism of \(\ell_1\) onto \(E'\) and \(F'\) is the adjoint of \(F\), then the restriction of \(F'\) to \(\widehat E\) is an order-isomorphism of \(\widehat E\) onto \(c_0\).
scientific article; zbMATH DE number 1449089