Structure and Classification of C* Algebras

“C*-代数的结构和分类”项目福尔斯EPSRC数学分析研究领域。这个项目的目的是获得一般性的结果，这将通过检查关键的测试案例和利用最近的发展tilingtheory顺从groups.This是很大的动机所做的工作在70年代由Alain Connes谁证明了thehyperfiniteness条件冯诺依曼代数是等价的，需要内射性。准确地说，他得到了一个强大的内部近似性质。因此，他能够给出一个几乎完整的分类内射因素。从那时起，使用的想法，从他的工作，theC* 代数社会试图获得一个分类的C* 代数。通过巧妙的类比，它现在是已知的，可分离的作用内射因子有一个类似的类在C*-设置，即简单的可分离单位核C* 代数。通过许多研究者的工作，通过附加两个条件，UCT和Z稳定性，我们有一个完整的分类这些代数使用K-理论的数据。然而，这仍然是不令人满意的，因为这两个额外的条件还没有完全理解。本项目的一个目标是分析简单可分核单位元C* 代数的Z-稳定性条件。特别是与本项目相关的主要内容，我们将专注于理解这些条件在基本操作下的表现，其中一个例子是离散群的交叉乘积。C* 代数和群之间有着丰富的联系，许多深层次的条件，如顺从性，起着重要的作用。人们可以采取不同的例子群，看看他们如何作用于一个给定的核C* 代数，从而有希望获得抽象的条件，对集团和作用，这将确保Z-稳定性条件成立。一个同样有趣的话题是确定什么样的群体，这失败了，什么是可能的影响。目前，在这个方向上的最先进的结果都有相当严格的条件上的两个traces和如何组的作用traces.Given的交叉产品产生的变化，我们希望保留尽可能多的信息，从潜在的代数，所以迹的研究应证明是必不可少的。由于C* 代数上的每一条迹都通过GNS构造导出一个有限的冯诺依曼代数，并且II 1因子是单射的当且仅当每条迹都是顺从的，因此通过采用Connes的技术，我和我的导师Stuart白色教授希望通过探索顺从迹的影响来理解新形成的代数的结构。任何拟对角迹都是可修正的，这是自洽的，但其匡威则未知。通过对这一蕴涵的进一步探讨，有可能扩展拟对角C ~* 代数的理论。一个关键的测试案例是超有限II 1因子，它有一个独特的顺从的痕迹。这条迹是否也是拟对角迹，至今仍是一个悬而未决的问题

The project "Structure and Classification of C*-algebras" falls within the EPSRCMathematical Analysis research area. This project aims to obtain general results and this willbe pursued both through examining key test cases and utilising recent developments in tilingtheory for amenable groups.This is heavily motivated by the work done in the '70s by Alain Connes who proved that thehyperfiniteness condition for von Neumann algebras is equivalent to requiring injectivity.Precisely he obtained a strong internal approximation property. Thus, he was able to give analmost complete classification of injective factors. Since then, using ideas from his work, theC* algebras community tried to obtain a classification of C* algebras.By making clever analogies, it is now known that separably acting injective factors have ananalogue class in the C*-setting, namely simple separable unital nuclear C* algebras. Through the work of many researchers, by additionally imposing two conditions, the UCT and Zstability, we have a complete classification of these algebras using K-theoretic data. However, this is still unsatisfactory since the two extra conditions are not totally understood.One aim of this project would be to analyse the Z-stability condition for simple separablenuclear unital C* algebras. In particular and of primary relevance to this project, we wouldfocus on understanding how these conditions behave under fundamental operations and onesuch example is taking crossed products with discrete groups. There's a rich connectionbetween C* algebras and groups, with many deep conditions, such as amenability, playing animportant role. One can take different examples of groups and see how they act on a givennuclear C* algebra, thus hopefully obtaining abstract conditions on the group and on theaction which would ensure that the Z-stability condition holds true. An equally interestingtopic is determining for what kind of groups this fails and what are the possible implications.Right now state of the art results in this direction has quite restrictive conditions on bothtraces and how the group acts on traces.Given the change produced by the crossed product, we want to retain as much information as possible from the underlying algebra, so the study of traces shall prove indispensable. Since every trace on a C* algebra induces a finite von Neumann algebra through the GNSconstruction and a II1 factor is injective if and only if every trace is amenable, by adaptingConnes' techniques, me and my supervisor, Prof. Stuart White, are hoping to understand thestructure of the newly formed algebra by exploring the effects on amenable traces. It istautological that any quasidiagonal trace is amenable, but the converse is not known. Byexploring more on this implication, it is possible to expand on the theory of quasidiagonal C*algebras. A key test case is the hyperfinite II1 factor, which has a unique amenable trace. It is still an open question whether this trace is also quasidiagonal or not