The Structure of Crossed Product C*-algebras

叉积C*-代数的结构

• 批准号: 0070776
• 负责人: Norman Phillips
• 金额: $ 9.89万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070776&HistoricalAwards=false
• 关键词: Structure; Crossed; Product; algebras

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.

Ampactphillips主要研究者最近证明，通过最小的差异性跨型歧管的交叉产物可以表示为递归下均匀的C* - 代理系统的直接限制，没有尺寸生长。他建议通过放松不同的假设（许多有趣的最小同构同态不平稳，甚至在流形上也不是）来试图概括这一结果，并考虑了更多的一般组（例如，整数的几个副本的实数或直接总和）。他还建议通过试图证明对这种直接限制的分类定理应用直接极限分解，并通过使用结果暗示了K理论的研究，以研究动力学与C*-Algebras之间的连接，以获取有趣的最小同源性同态。首席研究者进一步建议使用自由概率方法研究与自由组减少的C* - 代数有关的同构问题。该项目涉及简单C*-ergebras的分类。 C* - 代数本身就是数学的迷人而美丽的一部分，而且它们在数学的部分（例如几何学和拓扑结构），甚至是物理学（例如量子力学和统计机制）中都具有令人惊讶的应用。 由于这些原因以及其他原因，一个人想识别并描述所有C*-Algebras。没有一些局限性，此任务目前是没有希望的，我的项目专注于“简单”的C* - 代数（也就是说，将它们分解成较小的零件的通常方式），并且在其他几种技术感觉中也“不太大”。在这些限制下，已经证明了一些分类定理。也就是说，人们可以识别，标记和描述某些类别的C* - 代数中的对象，就像元素的周期表一样，或者像生物种类的命名一样。在一个特定的有趣情况下（“稳定有限”的情况），已经证明了由一种特定的结构（“直接限制”）引起的简单C* - 代数证明了分类结果，但最有趣的示例来源是不同的结构（“交叉产品”）。从某种意义上说，错误的代数已被分类。该项目的目的是表明，更有趣的类型的某些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，以至于可能仍可以应用直接限制的旧分类方法；并将这些方法扩展到足够远的地方，以实际对更有趣的类型的代数进行分类。