Unconditional Bases and Unconditional Finite-dimensional Decompositions in Banach Spaces

Let X he a Banach space with an unconditional finite-dimensional Schauder decomposition (En). We consider the general problem of characterizing conditions under which one can construct an unconditional basis for X by forming an unconditional basis for each En. For example, we show that if sup,, dim En < c~ and X has Gordon-Lewis local unconditional s t ructure then X has an unconditional basis of this type. We also give an example of a non-Hilbertian space X with the property that whenever Y is a closed subspace of X with a UFDD (En) such that supn dim En < oo then Y has an unconditional basis, showing that a recent result of Komorowski and Tomczak-Jacgermann cannot be improved. 1. I n t r o d u c t i o n Let X be a separable Banach space with an unconditional finite-dimensional Schauder decomposition (UFDD) (E,~). It is well-known that even if for some constant K each E,~ has a K-unconditional basis it is still possible that X may fail to have an unconditional basis. The first example of this phenomenon was given in [10] where a twisted sum of two Hilbert spaces Z2 is constructed in such a way that it has a UFDD into a two-dimensional spaces (or a 2-UFDD) En but Z2 has no unconditional basis. Later, Johnson, Lindenstrauss and Schechtman * Both authors were supported by NSF Grant DMS-9201357. Received September 28, 1994