The Structure of Crossed Product C*-algebras

主要研究者最近证明了紧致流形的极小微分同胚交叉积可以表示为递归次齐次C*-代数系统的不增长的直接极限。他建议通过放松可微性假设(许多有趣的极小同胚不光滑，甚至不在流形上)，并通过考虑更一般的群(例如实数或整数的几个副本的直和)来尝试推广这一结果。他还建议应用直接极限分解，试图证明这种直接极限的一个分类定理，并利用它蕴含的关于K-理论的结果来研究有趣的极小同胚的动力学和C*-代数之间的联系。主要研究者进一步提出利用自由概率的方法来研究与自由群的约化C*-代数相关的C*-代数的同构问题。C*-代数本身就是数学中一个迷人而美丽的部分，而且它们在数学的其他部分(如几何和拓扑学)甚至物理的部分(如量子力学和统计力学)中都有令人惊讶的应用。由于这些原因和其他原因，人们想要识别和描述所有的C*-代数。如果没有一些限制，这项任务目前是没有希望的，我的项目专注于C*-代数，它们是“简单的”(即，通常将它们分成更小的部分的方法不起作用)，而且在其他几个技术意义上也是“不太大”的。在这些限制下，证明了一些分类定理。也就是说，人们可以在某些C*-代数类中识别、标记和描述对象，有点像元素元素周期表，或者像生物物种的命名。在一个感兴趣的特殊情况下(“稳定有限”情况)，简单C*-代数的分类结果已被证明是由一个特定结构(“直接极限”)产生的，但最有趣的例子来源是不同的结构(“交叉积”)。在某种意义上，错误的代数被分类了。这个项目的目的是证明一些更有趣类型的代数实际上与一些已经分类的代数相同，或者至少足够接近，以至于来自直接极限的旧分类方法仍然可以应用；并将这些方法扩展到足够远，以实际对更有趣类型的代数进行分类。

AbstractPhillipsThe Principal Investigator has recently proved that crossed products of compact manifolds by minimal diffeomorphisms can be represented as direct limits of systems of recursive subhomogeneous C*-algebras with no dimension growth. He proposes to attempt to generalize this result, by relaxing the differentiability assumptions (many interesting minimal homeomorphisms are not smooth, or not even on manifolds), and by considering more general groups (such as the real numbers or the direct sum of several copies of the integers). He also proposes to apply the direct limit decomposition, by attempting to prove a classification theorem for such direct limits, and by using the results it implies about K-theory to investigate the connections between dynamics and C*-algebras for interesting minimal homeomorphisms. The Principal Investigator further proposes to use methods of free probability to study isomorphism questions for C*-algebras related to the reduced C*-algebras of free groups.This project concerns the classification of simple C*-algebras. C*-algebras are a fascinating and beautiful part of mathematics in their own right, and moreover they have surprising applications toother parts of mathematics (such as geometry and topology) and even to parts of physics (such as quantum mechanics and statistical mechanics). For these reasons, and others, one wants to identify and describe all C*-algebras. Without some limitations, this task is presently hopeless, and my project focuses on C*-algebras which are "simple" (that is, the usual way of taking them apart into smaller pieces doesn't work), and also "not too large" in several other technical senses. Under these limitations, some classification theorems have been proved. That is, one can identify, label, and describe the objects in certain classes of C*-algebras, sort of like the periodic table of the elements or like the naming of the species of living things. In one particular case ofinterest (the "stably finite" case), the classification results have been proved for simple C*-algebras arising from one particular construction ("direct limits"), but the most interesting source of examples is a different construction ("crossed products"). In a sense, the wrong algebras have been classified. The aim of this project is to show that some algebras of the more interesting kind are actually the same as some of the ones already classified, or are at least close enough that the old classification methods from direct limits might still be applied; and to extend those methods far enough to actually classify the algebras of the more interesting type.