Twisted Sums , Fenchel - Orlicz Spaces

A twisted sum Z of two Banach spaces X and Y is defined (see [11]) through a short exact sequence: 0 −→ X −→ Z −→ Y −→ 0. These short exact sequences in the category of (quasi-)Banach spaces are considered naturally in the investigation of three space properties (a property P in the category of quasi-Banach spaces is called a three space property if for every short exact sequence as above, Z has property P whenever X and Y have it). The roots of this theory go to Enflo, Lindenstrauss and Pisier’s solution [3] to Palais’ problem: the property of being isomorphic to a Hilbert space is not a three space property. The first systematic study of twisted sums of quasi-Banach spaces appears in [11]. In that paper twisted sums of quasi-Banach spaces X and Y are associated to quasilinear maps from Y to X and the Banach spaces Zp, 1 < p < ∞, are studied as examples of twisted sums of lp’s. In particular, Z2 is a reflexive Banach space with a basis which has a closed subspace X isometric to l2 with Z2/X also isometric to l2. Z2 is isomorphic to its dual, yet Z2 is not isomorphic to l2. Furthermore, Z2 has no complemented subspace with an unconditional basis, in particular it has no complemented subspace isomorphic to l2. Z2 has an unconditional finite dimensional Schauder decomposition into two dimensional spaces (2-UFDD), yet Johnson, Lindenstrauss and Schechtman [6] showed that it fails to have local unconditional structure (l.u.st.). Twisted sums appear also in a natural way in complex interpolation [9]. There are several open problems on twisted sums and in particular on Z2, (see [8]), which make the study of these spaces very interesting.