Duality and Twisted Sums of Banach Spaces

We give a negative answer to the three-space problem for the Banach space properties to be complemented in a dual space and to be isomorphic to a dual space (solving a problem of Vogt [Lectures held in the Functional Analysis Seminar, Dusseldorf/Wuppertal, Jan–Feb. 1987] and another posed by Dı́az et al. in [Bull. Polish Acad. Sci. Math.40 (1992), 221–224]). Precisely, we construct an exact sequence 0→ℓ2→D→W*→0 in which W* is a separable dual and D is not isomorphic to a dual space. We also show the existence of an exact sequence 0→Y→X→Z→0 where both Y and Z are dual spaces and X is not even complemented in its bidual. To do that we perform a study of the basic questions on duality from the point of view of exact sequences of Banach spaces.