Twisted Sums and Unconditional Structure for Banach Spaces

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Proposal: DMS-9870027 Principal Investigator: Nigel J. Kalton Abstract: The aim of this proposal is to study a number of problems related to the notion of an extension of one Banach space by another, and the relationships between extensions and unconditional structure. It is proposed to attempt to resolve the conjecture that if every minimal extension of a Banach space is trivial then the space has finite cotype, and to attempt a classification of those spaces so that every extension by a Hilbert space is trivial. Both these problems can be reformulated in other terms and would have considerable significance independent of the theory of extensions. The proposer will also work on the fundamental problem in the theory of unconditional structure of whether every complemented subspace of a Banach space with unconditional basis also has an unconditional basis. The theory of extensions can be considered in the following terms. Suppose we are given a centrally symmetric solid in three or more dimensions and we are allowed only to compute its cross-section by a slice in some directions and the shadow cast by the body in the perpendicular directions. This allows us to compute certain information about the solid, but not to reconstruct it completely. The aim of this project is to obtain more complete information about the body under certain additional conditions. It turns out that many questions of importance in mathematical analysis and applications, although not expressed in this form, can be visualized as problems of this nature, perhaps involving infinite dimensions. An unconditional basis in a particular class of functions is a collection of relatively simple functions with which we can approximate every member of the class; such approximations are very important for many practical situations, for example in engineering applications. One of the aims of this project is to decide general conditions under w hich a basis can be constructed. There is interaction between the theory of extensions and of unconditional bases which this project will explore further.