Twisted Sums and Unconditional Structure for Banach Spaces

提案：DMS-9870027主要研究者：奈杰尔J. Kalton 摘要： 这个建议的目的是研究一个Banach空间的另一个扩展的概念相关的一些问题，以及扩展之间的关系 和无条件的结构。 它建议试图解决的猜想，如果每一个最小的扩展的Banach空间是平凡的，那么空间有有限的cotype，并试图分类这些空间，使每一个扩展的希尔伯特空间是平凡的。这两个问题都可以用其他术语重新表述，并且独立于扩张理论具有相当大的意义。 本文还将研究无条件结构理论中的一个基本问题，即是否每个有补 子空间 一个Banach 有无条件基的空间也有无条件基。 扩张理论可以用下列术语来考虑。 假设我们有一个三维或多维的中心对称固体，我们只允许通过在某些方向上的切片和物体在垂直方向上投射的阴影来计算它的横截面。这使我们能够计算有关固体的某些信息，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该机构的更完整的信息。事实证明，数学分析和应用中的许多重要问题，尽管没有以这种形式表达，但可以被可视化为这种性质的问题，也许涉及无限维。在一个特定的函数类中，无条件基是一个相对简单的函数的集合，我们可以用它来近似该类中的每个成员;这样的近似对于许多实际情况都非常重要， 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。有扩张理论和无条件的基础，本项目将进一步探讨之间的相互作用。

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.