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.

研究人员将研究许多问题，这些问题大致属于代表理论。 然而，这些问题与经典理论不同（例如，不可约说明的分类）。 目的是a）更好地了解某些准核对角c* - 代数的有限因子表示，b）研究轨迹的近似特性和c）在操作员代数中的各种开放式问题上的应用程序上的应用。 在这一点上，我们希望这项工作将阐明埃利奥特的分类计划和康涅斯的嵌入问题。 但是，正如我们最近注意到的那样，可能会有更多的想法具有k个学的含义，可以用来证明有限部分方法的一般存在结果（来自数值分析），如果很快就能确定与几何群体理论的联系，我们不会感到惊讶。 尽管这些想法肯定还处于起步阶段，但它们具有扎实的历史基础（例如，Connes的独特性定理是有限的注射因素），我们认为该理论表现出了希望。数学中的一个非常成功的想法是，关于复杂对象的问题有时可以使用近似对象来解决。 例如，在微积分中，我们告诉学生，要在曲线下计算该区域，应该首先用矩形近似，因为矩形的区域很容易计算。 操作员代数是（通常）无限的尺寸对象，可为量子物理学中的许多问题提供自然框架。 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。