Approximation Theory and C*-algebras

逼近理论和 C* 代数

• 批准号: 0554870
• 负责人: Nathanial Brown
• 金额: $ 14.75万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-15 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0554870&HistoricalAwards=false
• 关键词: Approximation; Theory; algebras

The investigator will study a number of problems which, roughly speaking, belong to representation theory. The questions, however, differ from the classical theory (e.g. classification of irreducible representations). The goal is to a) better understand finite factor representations of certain quasidiagonal C*-algebras, b) study approximation properties of traces and c) work on applications of the previous two topics to various open questions in operator algebras. At this point, our hope is that this work will shed light on Elliott's classification program and Connes' embedding problem. However, there may be more as we have recently noticed that these ideas have K-homological implications, can be used to prove a general existence result for the finite section method (from numerical analysis) and we would not be surprised if connections with geometric group theory were soon worked out. Though these ideas are certainly in their infancy, they have solid historical foundations (e.g. Connes' uniqueness theorem for finite injective factors) and we believe the theory shows promise. One very successful idea in mathematics is that problems about complicated objects can sometimes be solved using approximations by simpler objects. For example, in calculus we teach students that to compute the area under a curve one should first approximate by rectangles since the area of a rectangle is easy to compute. Operator algebras are (usually) infinite dimensional objects which provide the natural framework for many questions in quantum physics. 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.

研究者将研究一些大体上属于表征论的问题。 然而，这些问题与经典理论不同（例如不可约表示的分类）。 目标是 a) 更好地理解某些准对角 C* 代数的有限因子表示，b) 研究迹的近似性质，以及 c) 将前两个主题应用于算子代数中的各种开放问题。 此时，我们希望这项工作能够阐明 Elliott 的分类程序和 Connes 的嵌入问题。 然而，可能还有更多，因为我们最近注意到这些想法具有 K-同调含义，可以用来证明有限截面方法（来自数值分析）的一般存在性结果，如果与几何群的联系我们不会感到惊讶理论很快就得到了解决。 尽管这些想法确实还处于起步阶段，但它们具有坚实的历史基础（例如有限单射因子的康尼斯唯一性定理），并且我们相信该理论具有前景。数学中一个非常成功的想法是，有时可以使用简单对象的近似来解决有关复杂对象的问题。 例如，在微积分中，我们教学生要计算曲线下的面积，首先应该用矩形来近似，因为矩形的面积很容易计算。 算子代数（通常）是无限维对象，它为量子物理学中的许多问题提供了自然框架。 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。