The Structure of Crossed Product C*-algebras

Phillips最近证明了紧致流形与极小次同态的交叉积可以表示为无维数增长的递归次齐次C ~*-代数系统的直接极限。他建议尝试推广这一结果，通过放松可微性假设（许多有趣的极小同胚不是光滑的，甚至不是流形上的），并考虑更一般的群（如真实的数或整数的几个副本的直和）。他还建议应用直接极限分解，试图证明一个分类定理，这种直接限制，并通过使用的结果，它意味着约K理论调查之间的联系动态和C*-代数有趣的最小同胚。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*-代数有关的C*-代数的同构问题。 C*-代数本身就是数学中迷人而美丽的一部分，而且它们对数学的其他部分（如几何学和拓扑学），甚至对物理学的某些部分（如量子力学和统计力学）都有着令人惊讶的应用。 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，这个任务目前是无望的，我的项目集中在C*-代数，这是“简单的”（也就是说，通常的方式把它们分成更小的部分不工作），也“不太大”在其他几个技术意义上。在这些限制条件下，一些分类定理得到了证明。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，有点像元素周期表或生物物种的命名。在一个特殊的情况下ofinterest（“稳定有限”的情况下），分类结果已经证明了简单的C*-代数产生于一个特定的结构（“直接限制”），但最有趣的例子来源是一个不同的结构（“交叉产品”）。在某种意义上，错误的代数已经被分类。这个项目的目的是证明一些更有趣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，以至于旧的直接极限分类方法仍然可以应用;并将这些方法扩展到足够远，以实际分类更有趣类型的代数。

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.