Approximation Theory and Operator Algebras

逼近论和算子代数

• 批准号: 0856197
• 负责人: Nathanial Brown
• 金额: $ 21.66万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-15 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0856197&HistoricalAwards=false
• 关键词: Approximation; Theory; Operator; Algebras

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.

Brown 研究员将继续研究算子代数的近似性质，重点关注 C*- 和 W*- 代数理论中出现的基本问题。 例如，当罗丹和汤姆斯构建反例时，艾略特的分类程序发生了戏剧性的转变。 然而，Winter 最近的工作强烈表明，“有限维”（在适当的非交换拓扑意义上）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，但使用不同的不变量——它在功能上等同于经典的艾略特不变量，但形式上携带更多信息。 在 W* 方向上，我们将继续研究与允许微观状态（在 Voiculescu 意义上）的 II_1 因子相关的拓扑空间。 这些拓扑不变量尚未得到系统研究，因此探索它们是很自然的。数学中一个非常成功的想法是，我们可以通过用更简单的对象进行近似，然后传递到极限来了解复杂的对象。例如，在微积分中，我们使用矩形近似计算曲线下的面积，然后一遍又一遍地改进近似值。例如，算子代数（通常）是无限维对象，它为量子力学提供了自然框架。此外，多年来还发现了与几何、拓扑和概率等其他数学领域的深刻且意想不到的联系。因此，对算子代数结构的深入理解非常重要。由于感兴趣的对象是无限维的，因此使用更简单的对象进行近似的一般哲学在这里变得特别相关。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。