Approximation Theory and C*-algebras

研究者将研究一些问题，粗略地说，属于表征理论。然而，这些问题不同于经典理论（例如，不可约表示的分类）。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)研究迹的近似性质，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有k -同构的含义，可以用来证明有限截面方法的一般存在性结果（从数值分析），如果与几何群论的联系很快就会得到解决，我们不会感到惊讶。虽然这些想法当然还处于起步阶段，但它们有坚实的历史基础（例如Connes有限内射因子的唯一性定理），我们相信该理论显示出希望。数学中一个非常成功的观点是，关于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，多年来还发现了与几何、拓扑学和概率论等其他数学领域的深刻和意想不到的联系。因此，对算子代数结构的深刻理解是很重要的。用更简单的对象来近似的一般哲学在这里变得特别相关，因为感兴趣的对象是无限维的。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。

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.