Approximation Theory and C*-algebras

研究者将研究一些问题，粗略地说，这些问题属于表象理论。然而，这些问题不同于经典理论(例如，不可约表示的分类)。其目的是a)更好地理解某些拟对角线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公开问题中的应用。在这一点上，我们希望这项工作将有助于阐明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，这些概念具有K-同调含义，可以用来证明有限截面法(从数值分析)的一个普遍存在结果，如果很快与几何群论建立联系，我们也不会感到惊讶。虽然这些想法还处于初级阶段，但它们有坚实的历史基础(例如康尼斯关于有限内射因子的唯一性定理)，我们相信这一理论是有希望的。数学中一个非常成功的想法是，关于复杂物体的问题有时可以用更简单的物体的近似来解决。例如，在微积分中，我们教学生，要计算曲线下的面积，应该首先用矩形逼近，因为矩形的面积很容易计算。算子代数是(通常)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，人们还发现了与其他数学领域，如几何学、拓扑学和概率学的深刻和意想不到的联系。因此，对算子代数的结构有一个坚实的理解是很重要的。由于感兴趣的对象是无限维的，所以使用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来更好地理解一些基本的无限维物体的既定传统。

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.