Von Neumann techniques in C*-algebras

算子代数的理论起源于量子物理学和局部紧群的酉表示理论。该地区有许多连接到其他领域的数学与许多出现在最近几年。深结构的结果在20世纪70年代和新出现的应用几何和拓扑开创和浓缩的工作卡斯帕罗夫大大加快了发展。算子代数是现代数学中一个非常活跃的领域，传统上主要有两个分支：von Neumann代数理论和C ~*-代数理论，这两个分支的性质不同，但最终却有着惊人的相似之处。冯·诺依曼代数最早是由默里和冯·诺依曼在20世纪30年代和40年代与量子物理学有关的研究。它们被广泛认为是非交换测度空间，更类似于概率论，比C*-代数更灵活，后者是在大约十年后由Gelfand和Naimark引入的。C*-代数可以看作是非交换拓扑空间，对它的研究更类似于对空间和几何对象的研究。在很长一段时间里，这些子领域平行发展，它们之间的直接联系有限。其中一个主要的成就，算子代数理论是康纳斯的分类顺从冯诺依曼代数在20世纪70年代（完成Haagerup在80年代）这大致意味着这些代数可以减少到一个'名单'的已知例子。艾略特计划在80年代后期推出了雄心勃勃的目标做类似的C*-代数：分类简单的顺从C*-代数的K-理论（和痕迹）;这里K-理论是一个工具，分类空间的拓扑结构，适用于C*-代数以及。这一计划最近取得了巨大的进展，现在已经解决了一类确定的代数：那些有限的核尺寸，拓扑尺寸的概念类似于通常的空间尺寸。该计划现在的一个主要突出问题是找到有效的标准来确定哪些C*-代数具有有限的核维数，特别是在大类的突出的例子，其可分类性还不知道。最近的重大进展中出现的一个关键主题是冯诺依曼代数和C*-代数理论之间的相似之处。特别是在冯诺依曼代数的Connes-Haagerup分类中使用的许多概念在C* 世界中有类似物，不仅在概念层面上，而且足够强大，可以用于证明。这个建议的主要创新是充分理解和发展这些相似之处，并将其应用到识别有限核维数的突出问题中。我们将考虑的最重要的一类是与动力系统（即作用于空间的群，如圆的无理旋转）相关的交叉积代数。这本身就是一门重要的数学学科，与算子代数的紧密联系可以追溯到默里和冯·诺依曼的工作。可测动力学对应于冯·诺依曼交叉积，而连续动力学对应于C*-交叉积。后者提供了不可或缺的指导简单的顺从C*-代数的例子，并正在深入研究。最近，某些群体的行动取得了巨大的进展，比如整数，它们相对较小（在粗略的意义上）。我们的目标是开发新的方法，它的工作更普遍，并允许我们完全证明当简单的交叉积C*-代数有有限的核维数。为了使这一点得到广泛应用，我们所寻求的特征在本质上将是完全动态的，并且在具体的例子中很容易检查。

The theory of operator algebras has its origins in quantum physics and the theory of unitary representations of locally compact groups. The area has many connections to other fields of mathematics with many more appearing in recent years. Deep structure results in the 1970s and new emerging applications to geometry and topology pioneered and condensed in the work of Kasparov have accelerated the development considerably. Today operator algebras has grown into a vast, attractive and very active area in modern mathematics.Traditionally, there are two main sub-areas of the field: von Neumann algebra theory and C*-algebra theory which are of fairly different flavour but in the end have striking similarities. Von Neumann algebras were first studied by Murray and von Neumann in the 1930's and 40's in connection with quantum physics. They are widely regarded as non-commutative measure spaces and a more akin to probability theory, more flexible than C*-algebras which were introduced by Gelfand and Naimark about a decade later. C*-algebras can be regarded as non-commutative topological spaces and their study is more akin to the study of spaces and geometric objects. For a long time, these sub-areas developed in parallel with limited direct connections between them. One of the major achievements in operator algebra theory is Connes' classification of amenable von Neumann algebras during the 1970's (completed by Haagerup in the 80's) which roughly means that these algebras can be reduced to a `list' of known examples. The Elliott programme launched in the late 80's has the ambitious goal to do something similar for C*-algebras: classify simple amenable C*-algebras by K-theory (and traces); here K-theory is a tool for classification of spaces from topology which applies to C*-algebras as well. This programme has seen dramatic recent progress and has now been solved for a definite class of algebras: those with finite nuclear dimension, a topological dimension concept analogous to the usual dimension of spaces. A major outstanding problem of the programme now is to find effective criteria to determine which C*-algebras have finite nuclear dimension, particularly in large classes of prominent examples for which classifiability is not yet known.A key theme emerging from recent major advances is the parallels between von Neumann algebra and C*-algebra theory. In particular many concepts used in the Connes-Haagerup classification of von Neumann algebras have analogues in the C*-world, not just at the conceptual level, but strong enough to be used in proofs. The major innovation of this proposal is to understand and develop these parallels fully and to apply this to the outstanding problem of identifying finite nuclear dimension.One of the most important classes we will consider are the crossed product algebras which are associated to dynamical systems (i.e. groups acting on spaces, such as irrational rotations of the circle). This is a major mathematical discipline in its own right, and the strong connections to operator algebras date back to the work of Murray and von Neumann. Measurable dynamics correspond to von Neumann crossed products whereas continuous dynamics to C*-crossed products. The latter provide indispensable guiding examples of simple amenable C*-algebras which have and are being studied intensively. Tremendous progress has been made recently for actions of certain groups like the integers, which are relatively small (in a coarse sense). We aim to develop new methods, which work much more generally, and allow us to completely characterise when simple crossed product C*-algebras have finite nuclear dimension. To allow this to be widely used, the characterisation we seek will be entirely dynamical in nature, and readily checkable in concrete examples.