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; 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.

研究的主要线索将集中于II_1因子的结构与简单C* - 代数之间的新兴类比。 例如，现在有麦克杜夫定理和康纳斯的“射击性”结果的c* - 分析（在对注射因素分类中起着基本作用）。研究人员希望以最终的目的探索这些类比，以对所谓有限核维度的简单C* - 代数进行分类。 在另一个方向上，研究人员将继续研究与群体和相关空间相关的C*代数，这些空间最近与Erik Guentner联合合作。最后，我们将继续研究与II_1相关的动态系统，这些系统是由某些同构化引起的。一个非常成功的数学想法是，我们可以通过与较简单的对象近似，然后传递到限制来了解复杂的对象。例如，在微积分中，我们使用矩形近似值计算曲线下的区域，然后一遍又一遍地完善近似值。操作员代数是（通常）无限的尺寸对象，例如，为量子力学提供了自然框架。此外，多年来发现了与数学，拓扑和概率等数学领域的深层和意外联系。因此，对操作员代数的结构有牢固的理解很重要。通过简单对象使用近似值的一般理念在这里特别相关，因为感兴趣的对象是无限的维度。研究人员将继续建立的既定传统，即尝试使用有限的维近似值，以更好地理解一些基本的无限尺寸对象。