The structure of C*-algebras

这个研究计划是通过k理论和非交换维等工具来理解简单的可服从C*-代数，建立在PI最近在拟对角性方面的高调发展之上。C*-代数多年的分类研究在最近的一次突破中达到高潮，即分类定理（见提案），PI在其中发挥了主要作用。这个程序及时地使用了这个定理，探索了它可以被使用和更好地理解的方法，其广泛的目标是在冯诺依曼代数中实现Connes Fields获奖工作的C*代数类比。为实现这一目标而出现的自然问题为HQP培训提供了丰富的机会，HQP培训是本研究计划的核心组成部分。

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.