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 Diaz et al. in [Bull. Polish Acad. Sci. Math. 40 (1992), 221 224]). Precisely, we construct an exact sequence 0 --> l(2) --> 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. (C) 2000 Academic Press.