The structure of C*-algebras

这个研究计划是通过K理论和非交换维数等工具来理解简单的顺从C*-代数，建立在PI在准对角性方面最近的高调发展之上。多年的分类C*-代数最近达到高潮的突破，分类定理（见建议），其中PI发挥了重要作用。这个程序及时地利用了这个定理，探索了它可以被使用和更好地理解的方法，其广泛目标是实现Connes' Fields在冯诺依曼代数中的获奖工作的C*-代数类比。算子代数理论是泛函分析中一个非常活跃的领域，与量子力学、动力系统、逻辑学、调和分析、数论和几何群论有着密切的联系。两个领域奖章（康纳斯和琼斯）已被授予研究算子代数。简单顺从C*-代数的分类是一个在过去20年中被各种国际研究人员积极追求的问题，在过去的3年中取得了重大突破，最终形成了分类定理。我们需要启用分类定理，建立与其他领域的联系，如动力学。我们需要从C*-代数模理论的角度更好地理解非单位C*-代数。我们需要建立一个更强大的形式的分类定理的假设。研究计划通过以下目标解决这些问题。通过分类定理的透镜，研究动力学中产生的C*-代数。让HQP看看这个问题的特殊情况：哪些可分类的C*-代数可以从动力学中产生？2.将有单位元的C*-代数的结果和概念推广到非有单位元的情形，重点是在C*-代数模理论中变得重要的例子。这是一个极好的机会，HQP获得专业知识的想法和技术中使用的单位理论，同时建立新的结果感兴趣的专家在分类。解决Toms-Winter猜想，该猜想预测了分类定理的主要假设之一的鲁棒特征。此外，使用PI在Toms-Winter猜想的工作中开发的概念，以新的方式分析顺从C*-代数的结构。

This research program is to understand simple amenable C*-algebras through tools such as K-theory and noncommutative dimension, building on recent high-profile developments by the PI in quasidiagonality. Many years of the classification of C*-algebras recently culminated in a breakthrough, the Classification Theorem (see Proposal), to which the PI played a major role. This program makes timely use of this theorem, exploring ways it can be used and better understood, with the broad goal of achieving C*-algebraic analogies of Connes' Fields Medal-winning work in von Neumann algebras. Natural problems that arise to achieve this goal are rich with opportunities for HQP training, which is a central component of this research program.Operator algebra theory is an extremely active area in functional analysis, with connections to quantum mechanics (from which the subject originally arose), dynamical systems, logic, harmonic analysis, number theory, and geometric group theory. Two Fields medals (Connes and Jones) have been awarded for research in operator algebras. The classification of simple amenable C*-algebras is a problem that has been actively pursued by a variety of international researchers over the last 20 years, with major breakthroughs in the past 3 years, culminating in the Classification Theorem.We need to enable use of the Classification Theorem, building connections to other areas such as dynamics. We need to better understand nonunital C*-algebras with a view to C*-algebraic modular theory. We need to establish a more robust form of the hypotheses of the classification theorem.The research program addresses these through the following objectives.1. Study C*-algebras arising from dynamics, through the lens of the Classification Theorem. Have HQP look at special cases of the question: which classifiable C*-algebras can arise from dynamics?2. Extend results and concepts about unital C*-algebras to the nonunital setting, with an emphasis on examples which will become important in C*-algebraic modular theory. This is an excellent opportunity for HQP to gain expertise in the ideas and techniques used in the unital theory, while establishing new results of interest to experts in classification.3. Resolve the Toms-Winter conjecture, which predicts a robust characterization of one of the main hypotheses of the Classification Theorem. Moreover, use concepts developed by the PI in work on the Toms-Winter conjecture to analyze the structure of amenable C*-algebras in new ways.