W*-bundle techniques and the structure of simple C*-algebras

C*-代数是从量子力学的严格化中产生的数学对象。每个C*-代数是一组从希尔伯特空间到自身的连续线性映射，在一些自然代数和解析运算下闭合。在它们诞生之初，人们很快意识到C*-代数可以从许多其他数学对象中以规范的方式创建，建模诸如对称性，时间演化系统和大型数据集等。一次又一次地，被输入对象的性质和所得到的C*-代数的性质之间的有趣关系被证明，一段时间以来，人们已经很清楚，不同的构造可以产生相同的C*-代数;这在外部是令人感兴趣的，其中它可能意味着不同输入数据之间的深刻关系，而在内部，它允许用不同的构造方法来研究单个C*-代数，但是，什么样的输入条件会产生不同的C*-代数输出还有待于彻底阐明。实现这一目标相当于对C*-代数进行分类：证明合适的、可计算的不变量（主要是K-理论）足够敏感，总是能区分不同的C*-代数。最近变得明显的是，要对C*-代数进行分类，人们应该研究C*-代数的正则性--C*-代数的某些性质表明它们不那么复杂，更容易处理。正则C*-代数是具有低（拓扑）维数的代数-以一种精确地推广空间维数的方式。正如低维空间更容易形象化一样，证明它们的东西也更容易，以至于某些对所有低维空间都成立的东西在高维空间中不再成立。这一点可以推广到C*-代数：低维C*-代数比高维C*-代数有更多更好的证明。回到分类问题，在许多情况下，只要C *-代数具有低维，其不变量取相同值的C*-代数就自动地相同（或同构）。我一直参与有关正则性的研究，并发现最近的一种称为W*-丛的工具显示出巨大的前景，尽管它的基本理论还有待发展。从一个C*-代数中，我们产生了一个W*-丛，并将其作为一个工具，这是可行的，因为：（i）W*-丛有更多的结构，而且似乎证明它比证明C*-代数更容易;（ii）W*-丛与C*-代数有一种非常特殊的关系--它以一种特殊的方式包含它--所以关于W*-丛的事实可以对C*-代数有重要的影响。本项目的目的是通过发展W*-丛的理论来加深我们对C*-代数的结构和分类的理解。

C*-algebras are mathematical objects that arose from the rigourisation of quantum mechanics. Each C*-algebra is a set of continuous linear maps from a Hilbert space to itself, closed under a few natural algebraic and analytic operations. Upon their inception, it was quickly realised that C*-algebras can be created in canonical ways from many other mathematical objects, modelling such things as symmetries, time-evolving systems, and large data sets. Time and again, interesting relationships have manifested between properties of the object being input and those of the resulting C*-algebra.For some time, it has been quite clear that different constructions can produce the same C*-algebra; this is interesting externally, where it may imply a profound relationship between the differing input data, and internally, where it allows single C*-algebras to be studied using the different techniques available from each different way of constructing it. However, a thorough elucidation of what conditions on the input objects produce different C*-algebra outputs has yet to be achieved. Achieving this goal amounts to classifying C*-algebras: showing that suitable, computable invariants (primarily, K-theory) are sufficiently sensitive to always distinguish different C*-algebras.It has recently become apparent that to classify C*-algebras, one should study regularity properties of the C*-algebras - certain properties of C*-algebras that indicate they are less complex and more tractable. Regular C*-algebras are ones that have low (topological) dimension - in a way that exactly generalises dimension of a space. Just as low dimensional spaces are easier to visualise, it is often easier to prove things about them, to the extent that certain things that are true of all low-dimensional spaces are no longer true in higher dimensions. This carries forward to C*-algebras: more and better things can be proven about low dimensional C*-algebras than high dimensional ones. Returning to classification, it has been shown in many cases that C*-algebras whose invariants take the same value are automatically the same (or isomorphic), provided that the C*-algebras have low dimension.I have been involved in research concerning regularity, and have found that a certain recent tool called W*-bundles shows tremendous promise, although its fundamental theory has yet to be developed. From a C*-algebra, one produces a W*-bundle, and uses this as a tool.This works because:(i) the W*-bundle has more structure, and it seems that it should be easier to prove things about it than about the C*-algebra;(ii) the W*-bundle has a very special relationship to the C*-algebra - it contains it in a special way - so that facts about the W*-bundle can have important implications for the C*-algebra.The aim of this project is to further our understanding of structure and classification of C*-algebras, by developing the theory of W*-bundles.