Approximation Theory and Operator Algebras

研究人员将继续研究算子代数的逼近性质，集中于C*-和W*-代数理论中的基本问题。例如，当罗丹和汤姆构造反例时，埃利奥特的分类程序发生了戏剧性的转变。然而，温特最近的工作强烈地表明，“有限维”(在适当的非交换拓扑意义下)的代数的分类应该是可分类的。我们将研究这些代数的分类问题，但使用了一个不同的不变量--它在功能上与经典的Elliott不变量等价，但形式上携带的信息要多得多。在W*方向，我们将继续研究与允许微态的II_1因子有关的拓扑空间(在Voulescu意义下)。这些拓扑不变量还没有被系统地研究过，所以研究它们是很自然的。数学中一个非常成功的想法是，我们可以通过用更简单的物体近似，然后达到一个极限来学习复杂的物体。例如，在微积分中，我们使用矩形近似来计算曲线下的面积，然后一遍又一遍地改进近似。例如，算符代数是(通常)为量子力学提供自然框架的无限维对象。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。

BrownThe investigator will continue studying approximation properties of operator algebras, concentrating on fundamental questions arising in C*- and W*-algebra theory. For example, Elliott's classification program took a dramatic turn when counterexamples were constructed by Rordam and Toms. However, recent work of Winter strongly suggests that the classification of algebras which are "finite dimensional" (in a suitable noncommutative topological sense) should be classifiable. We will investigate the classification problem for these algebras, but utilizing a different invariant -- one that turns out to be functorially equivalent to the classical Elliott invariant, but formally carries much more information. In a W*-direction, we will continue our study of topological spaces associated to II_1 factors which admit microstates (in the sense of Voiculescu). These topological invariants haven't yet been systematically investigated, so it's quite natural to explore them. One very successful idea in mathematics is that we can learn about complicated objects by approximating with simpler objects, then passing to a limit. For example, in calculus we compute the area under a curve using rectangular approximations, then refining the approximations over and over. Operator algebras are (usually) infinite dimensional objects which provide the natural framework for quantum mechanics, for example. 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.