Approximation Theory and C*-algebras

调查人员将研究一些问题，粗略地说，属于表征理论。 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）更好地理解某些拟对角C*-代数的有限因子表示，B）研究迹的逼近性质，c）研究前两个主题在算子代数中各种开放问题的应用。 在这一点上，我们希望这项工作将揭示埃利奥特的分类程序和康纳斯的嵌入问题。 然而，可能有更多的，因为我们最近注意到，这些想法有K-同调的影响，可以用来证明一个一般存在的结果有限部分的方法（从数值分析），我们不会感到惊讶，如果连接几何群论很快制定了。 虽然这些想法肯定是在他们的婴儿期，他们有坚实的历史基础（例如Connes'唯一性定理有限内射因子），我们相信该理论显示的承诺。数学中一个非常成功的思想是，复杂物体的问题有时可以用简单物体的近似来解决。 例如，在微积分中，我们教学生计算曲线下的面积，首先应该用矩形近似，因为矩形的面积很容易计算。 算子代数（通常）是无限维的对象，它为量子物理中的许多问题提供了自然的框架。 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。

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.