Twisted Sums and Unconditional Structure for Banach Spaces

摘要：本文的目的是研究一个Banach空间被另一个Banach空间扩展的概念以及扩展与无条件结构之间的关系。本文试图解决一个猜想，即如果Banach空间的每一个最小扩展都是平凡的，那么这个空间就有有限的共型，并尝试对这些空间进行分类，使Hilbert空间的每一个扩展都是平凡的。这两个问题都可以用其他术语重新表述，并且与扩展理论无关，具有相当大的意义。本文还将研究无条件结构理论中的基本问题，即具有无条件基的Banach空间的补子空间是否也具有无条件基。可拓理论可以用以下术语来考虑。假设我们有一个三维或多维的中心对称实体，我们只允许计算它在某些方向上的截面和物体在垂直方向上的阴影。这使我们能够计算关于固体的某些信息，但不能完全重建它。该项目的目的是在某些附加条件下获得关于身体的更完整的信息。事实证明，在数学分析和应用中，许多重要的问题，虽然没有以这种形式表示，但可以被可视化为这种性质的问题，可能涉及无限维。特定函数类中的无条件基是相对简单函数的集合，我们可以用它来近似该类中的每一个成员；这种近似对于许多实际情况非常重要，例如在工程应用中。该项目的目的之一是确定可以构建基础的一般条件。可拓理论和无条件基础理论之间存在相互作用，本项目将进一步探讨这一点。

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.