New directions in C*-algebras, Groupoids, and Inverse Semigroups

冯·诺伊曼引入了算子代数，用数学语言来表达量子力学。C*-代数是一类易于处理且丰富的算子代数，它们的研究具有几何色彩——它们被cones描述为“非交换几何”。几何是由其美丽的对称性所支配的，我的程序探索了C*代数在多大程度上是由“部分对称性”所支配的，这种“部分对称性”是通过称为类群和逆半群的结构来实现的。群样是几何空间上的一类部分对称的集合。对于每一个类群，都可以构造一个由它构造的C*代数。我的长期目标是确定是否每个C*-代数都可以通过这种方式从部分对称中获得；或者如果这是不可能的，描述用这种方式可以得到的最大的一类C*-代数。如果所有或大部分都能以这种方式获得，它将在算子代数、动力系统和半群理论之间建立新的研究联系。1976年，加拿大数学家乔治·艾略特（George Elliott）有了一个重大发现：某些类别的C*代数完全由它们的k理论群决定。C*-代数的k理论群是它的（可能是无限维的）子空间的一种列表，按照它们的（广义）维数排序。从那以后，扩大艾略特的结果所适用的类别一直是一个持续的项目——“艾略特分类项目”。给出计算类群C*-代数的k -理论群的有效方法对这个程序有很大的价值，特别是如果我们的长期目标表明大多数C*-代数都是用这种方法得到的。因此，开发这种方法并将其应用于最近感兴趣的例子是一个短期目标，包括混沌动力系统和有限自动机产生的例子。

Von Neumann introduced operator algebras to express quantum mechanics in a mathematical language. C*-algebras are a tractable and rich class of operator algebras, and their study has a geometric flavour---they have been described by Connes as "noncommutative geometry". Geometry is governed by its beautiful symmetries, and my program explores the extent to which C*-algebras are governed by "partial symmetries" through structures called groupoids and inverse semigroups. A groupoid is certain type of set of partial symmetries on some geometric space. To every groupoid, one may construct a C*-algebra built from it. My long-term goal is to determine whether every C*-algebra is obtainable from partial symmetries in this way; or if this is not possible, describe the largest class of C*-algebras obtainable in this way. If all or most are obtainable in this way, it would foster new research connections between the theories of operator algebras, dynamical systems, and semigroups. In 1976, the Canadian mathematician George Elliott made a major discovery: that some classes of C*-algebras are completely determined by their K-theory group. The K-theory group of a C*-algebra is a kind of listing of its (possibly infinite-dimensional) subspaces arranged in order according to their (generalized) dimension. Expanding the classes for which Elliott's result hold has been an ongoing program ever since---the "Elliott Classification Program." Giving efficient methods for computing the K-theory groups of groupoid C*-algebras would be of great value to this program, especially if work on our long term goal shows that most C*-algebras are obtained this way. It is then a short-term goal to develop such methods, and apply them to examples of recent interest, including those arising from chaotic dynamical systems and finite automata.