Regularity and dimension for C*-algebras

算符代数是数学的一个分支，是在量子力学的精密化中发展起来的。它主要涉及两类：C*代数和冯·诺伊曼代数，它涉及纯数学中的跨学科技术，特别是在功能分析、拓扑和代数方面。除了它们的量子力学起源，C*-代数理论变得越来越重要，因为人们很快意识到，C*-代数可以用来捕获其他数学对象的关键信息。C*代数可以用来编码诸如对称性、大数据集、网络和动态系统之类的东西。这些结构遵循一个共同的模式：输入一个数学对象（比如一组对称），输出一个C*-代数。它们提出了一个重要的问题：输出C*-代数的性质告诉我们关于输入数学对象的什么？对该问题的系统分析表明，关键在于C*-代数的分类。分类的动机是不同的输入可能产生相同的C*-代数，尽管很难看出它们实际上是相同的。分类法通过证明当C*-代数在某些不变量上一致时，它们是相同的来解决这个问题。这里，不变量是与C*-代数相关的一段信息；C*-代数分类中使用的不变量是确定的不变量，有许多技术可以计算它们。近年来，C*-代数的分类问题有了惊人的进展，某些C*-代数显然不能用传统的不变量进行分类，即这些不变量不够敏感。对为什么会出现这种情况的彻底检查表明，分类障碍与高拓扑维度有关。可以考虑C*-代数的拓扑维数这一事实是由Gelfand的一个经典定理激发的：C*-代数具有一种称为交换性的特殊性质，它只是拓扑空间的编码。如果研究一个空间只看连续函数的代数就不会丢失任何信息。特别地，空间的维数可以用这个代数的性质来表示。使用正确的公式，维数可以推广到非交换的C*-代数，为每个C*-代数提供一个数字，即它的维数。C*-代数维数的研究表明：(i)有限维数是一个鲁棒的概念，由于神秘的原因等价于其他性质（称为正则性），这是分类所必需的（在某种程度上是充分的）；（ii）对于C*-代数结构，维数的特定值在某些情况下与已知的数值不变量有关，而在其他情况下，它产生了一个新的有趣的性质。这些主要是经验层面的观察结果（它们适用于某些例子或特殊情况，这表明它们应该更普遍）；这个项目的基础是系统地调查它们，以表明它们在更深和更普遍的层面上是正确的。tom和Winter的一个主要猜想抓住了(i)点，我们的目标是证明这个猜想，并使原因不那么神秘。对于（ii），我们的目标是加深我们对C*-代数维数对于C*-代数结构意味着什么的理解，使维数与其他不变量之间的关系更加透明。迄今为止，对维度的研究已经开辟了许多新的研究方向，这个项目也将追求一些有可能产生重大影响的方向。这些将：-在C*代数和拓扑学的现代研究之间产生新的联系，-为C*代数的分类提供新的视角和进展。

Operator algebras is a branch of mathematics that developed in the rigourisation of quantum mechanics. It primarily concerns two categories: C*-algebras and von Neumann algebras, and it involves interdisciplinary techniques within pure mathematics, drawing particularly on functional analysis, topology, and algebra.Beyond their quantum mechanical origins, C*-algebra theory gained importance as it was quickly realised that C*-algebras can be constructed to capture key information about other mathematical objects. C*-algebras can be used to encode such things as symmetries, large data sets, networks, and dynamical systems. These constructions follow a common pattern: a mathematical object (such as a group of symmetries) is input and a C*-algebra is output. They suggest an important problem: what do properties of the output C*-algebra tell us about the input mathematical object?Systematic analysis of this problem reveals that the key is in the classification of C*-algebras. Classification is motivated by the fact that different inputs may produce the same C*-algebra, although it can be difficult to see that they are actually the same. Classification solves this by proving that when C*-algebras agree on certain invariants, they are the same. Here, an invariant is a piece of information associated to a C*-algebra; the invariants used in the classification of C*-algebras are certain ones with many techniques available to compute them.Recently, the problem of classifying C*-algebras has taken a spectacular turn, in which it has become apparent that certain C*-algebras cannot be classified with the traditional invariants, i.e., these invariants are not sensitive enough. Thorough examinations of why this is the case reveal that the obstructions to classification are related to high topological dimension.The fact that topological dimension can be considered for C*-algebras is motivated by a classical theorem of Gelfand: that C*-algebras enjoying a special property called commutativity are simply encodings of topological spaces. No information is lost if one studies a space by looking only at this algebra of continuous functions. In particular, dimension of the space can be formulated in terms of properties of this algebra. Using the right formulation, dimension can then be generalised to noncommutative C*-algebras, providing each C*-algebra with a number, its dimension.The study of C*-algebra dimension has revealed that:(i) Finite dimension is a robust notion, equivalent to other properties (called regularity properties) for mysterious reasons, which is necessary for classification (and to some extent, sufficient);(ii) For C*-algebra constructions, the particular value of dimension relates, in some cases, to known numerical invariants, while in others it produces a new interesting property.These are observations largely at an empirical level (they hold for certain examples or special cases, suggesting that they should hold more generally); the basis of this project is to systematically investigate them, to show that they are true at deeper and more general levels. Point (i) is captured in a major conjecture of Toms and Winter, and we aim to prove this conjecture, and make the reasons less mysterious. For (ii), we aim to deepen our understanding of what C*-algebraic dimension means for C*-algebra constructions, making the relationships between dimension and other invariants more transparent.Investigations into dimension thus far have opened up a number of new research directions, and this project will also pursue some directions that have potential for major impact.These will:- Produce new connections between the modern studies of C*-algebras and topology, and- Provide new perspectives and inroads into the classification of C*-algebras.