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.

研究的主要线索将集中于 II_1 因子结构和简单 C* 代数之间新出现的类比。 例如，现在有 McDuff 定理的 C* 类似物和 Connes 的“单射蕴含 McDuff”结果（在单射因子分类中发挥了基础作用）。研究人员希望探索这些类比，最终目标是对所谓的有限核维数的简单 C* 代数进行分类。 另一方面，研究人员将继续研究与最近与 Erik Guentner 合作引入的群和相关空间相关的 C* 代数。最后，我们将继续研究与由某些同态产生的 II_1 因子相关的动力系统。数学中一个非常成功的想法是，我们可以通过用更简单的对象进行近似，然后传递到极限来了解复杂的对象。例如，在微积分中，我们使用矩形近似计算曲线下的面积，然后一遍又一遍地改进近似值。例如，算子代数（通常）是无限维对象，它为量子力学提供了自然框架。此外，多年来还发现了与几何、拓扑和概率等其他数学领域的深刻且意想不到的联系。因此，对算子代数结构的深入理解非常重要。由于感兴趣的对象是无限维的，因此使用更简单的对象进行近似的一般哲学在这里变得特别相关。研究者将继续尝试使用有限维近似来更好地理解一些基本的无限维对象的既定传统。