Nuclearity, group C*-algebras and II_1 factors

核性、C* 族代数和 II_1 因子

• 批准号: 1201385
• 负责人: Nathanial Brown
• 金额: $ 22万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201385&HistoricalAwards=false
• 关键词: Nuclearity; group; algebras; II_1; factors

The primary thread of research will focus on emerging analogies between the structure of II_1 factors and simple C*-algebras. For example, there are now C*-analogues of McDuff's Theorem and Connes's "Injective implies McDuff" result (which played a fundamental role in classifying injective factors). The investigator hopes to explore these analogies with the ultimate goal of classifying simple C*-algebras of so-called finite nuclear dimension. In another direction, the investigator will continue working on C*-algebras associated to groups and related spaces that were recently introduced in joint work with Erik Guentner. Finally, we will continue our study of dynamical systems associated to II_1 factors which arise from certain homomorphisms.One very successful idea in mathematics is that we can learn about complicated objects by approximating with simpler objects, then passing to a limit. For example, in calculus we compute the area under a curve using rectangular approximations, then refining the approximations over and over. Operator algebras are (usually) infinite dimensional objects which provide the natural framework for quantum mechanics, for example. Moreover, deep and unexpected connections with other areas of mathematics such as geometry, topology and probability were discovered over the years. As such, a solid understanding of the structure of operator algebras is important. The general philosophy of using approximations by simpler objects becomes especially relevant here since the objects of interest are infinite dimensional. The investigator will continue an established tradition of trying to use finite dimensional approximations to better understand some fundamental infinite dimensional objects.

研究的主线将集中在II_1因子和简单C*-代数的结构之间的新出现的类比。 例如，现在有C*-类似的McDuff定理和Connes的“内射蕴含McDuff”结果（在分类内射因子中发挥了基本作用）。研究者希望探索这些类比，最终目标是对所谓有限核维数的简单C*-代数进行分类。 在另一个方向，研究人员将继续研究与最近与Erik Guentner联合工作中引入的群和相关空间相关的C*-代数。最后，我们将继续研究与某些同态产生的II_1因子相关的动力系统。数学中一个非常成功的思想是，我们可以通过用较简单的对象逼近，然后传递到一个极限来学习复杂的对象。例如，在微积分中，我们使用矩形近似计算曲线下的面积，然后一遍又一遍地细化近似。算子代数（通常）是无限维的对象，例如，它为量子力学提供了自然的框架。此外，多年来，人们发现了与其他数学领域（如几何、拓扑和概率）的深刻而意想不到的联系。因此，对算子代数结构的深入理解是很重要的。由于感兴趣的对象是无限维的，所以使用较简单对象的近似的一般哲学在这里变得特别相关。研究人员将继续尝试使用有限维近似更好地理解一些基本的无限维对象的既定传统。