Nuclearity, group C*-algebras and II_1 factors

研究的主线将集中在II_1因子的结构与单C*-代数之间出现的类比。例如，现在有了McDuff定理和Connes的“内射蕴含McDuff”结果的C*-类似结果(它在内射因子的分类中发挥了基础作用)。研究人员希望探索这些类比，最终目标是对所谓有限核维度的简单C*-代数进行分类。在另一个方向上，研究人员将继续研究与群和相关空间相关的C*-代数，这些C*-代数是最近在与Erik Guentner的合作中引入的。最后，我们将继续研究与某些同态产生的II_1因子相关的动力系统。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近似，然后到达一个极限来学习复杂对象。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。

The primary thread of research will focus on emerging analogies between the structure of II_1 factors and simple C*-algebras. For example, there are now C*-analogues of McDuff's Theorem and Connes's "Injective implies McDuff" result (which played a fundamental role in classifying injective factors). The investigator hopes to explore these analogies with the ultimate goal of classifying simple C*-algebras of so-called finite nuclear dimension. In another direction, the investigator will continue working on C*-algebras associated to groups and related spaces that were recently introduced in joint work with Erik Guentner. Finally, we will continue our study of dynamical systems associated to II_1 factors which arise from certain homomorphisms.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.