Type III Noncommutative Geometry and KK-theory

III 类非交换几何和 KK 理论

基本信息

批准号：
RGPIN-2017-04718
负责人：
Emerson, Heath
金额：
$ 1.17万
依托单位：
University of Victoria
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2019
资助国家：
加拿大
起止时间：
2019-01-01 至 2020-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675796
关键词：
Type III Noncommutative Geometry KK

项目摘要

Noncommutative Geometry seeks to analyze C*-algebras associated to various geometric situations, or situations in which one has a dynamical system, like a complicated group action by symmetries of a geometric space, by adapting the methods of geometric analysis on manifolds to work for C*-algebras. The broad idea is that to many of these situations we know how to construct a C*-algebra, and this in turn can be analyzed topologically (as if it were a space), using K-theory, and also geometrically, using the idea of a `spectral pair', consisting of a representation of the C*-algebra on a Hilbert space, and an unbounded operator D, playing the role of the Dirac operator on a manifold, in the classical case. ******A spectral pair produces a map on K-theory for which the Local Index Formula of A. Connes and co-authors provides a formula. This formula describes the K-theory map in terms of residues of certain zeta functions associated to the triple, in an extremely interesting `local', highly geometric manner, philosophically analogous to the way one integrates a differential form over a smooth manifold. ******For a spectral pair one requires that the operator D commutes with the C*-algebra A up to lower order terms, in a certain sense. But many important situations, like the boundary action of a hyperbolic group, produce C*-algebras with a kind of fractal nature (they are purely infinite) for which this notion is unsuitable, because spectral pairs properly defined induce densely defined traces, and these examples admit no traces. In this Proposal we aim to follow a more recent idea of A. Connes for rectifying this: we aim to use some ideas from quantum statistical mechanics to study a variation of the idea of a spectral pair, to now allow two actions of A on the Hilbert space H, one only defined for a dense subalgebra of A, but the operator is now only required to be equivariant as a map from H with the original action, to H with the twisted action, up to lower order operators. It turns out that this twisting of the definition has no effect cohomologically (on the Chern character), but the obstruction (failure of traces to exist for purely infinite algebras) to extending `integration' no longer exists, because instead, it becomes a kind of twisted integration, corresponding to a KMS state, a concept from quantum thermodynamics -- KMS states in these examples do exist, and there is an extremely interesting and rich theory of them. ******My goal is to construct twisted spectral triples in connection with boundary actions of hyperbolic groups, systems I have already studied extensively, and in several other examples and families of examples, to connect them to K-theory, study the corresponding index maps, and more broadly, investigate the connection between KMS states and K-theory which seems to be implied by the framework of twisted spectral triples. ***

非交通性几何形状试图分析与各种几何情况相关的C* - 代数，或者通过适应几何方法的对称性，例如通过对c*-Algebras进行流形的几何分析方法，例如几何空间的对称性进行了复杂的小组动作。 The broad idea is that to many of these situations we know how to construct a C*-algebra, and this in turn can be analyzed topologically (as if it were a space), using K-theory, and also geometrically, using the idea of a `spectral pair', consisting of a representation of the C*-algebra on a Hilbert space, and an unbounded operator D, playing the role of the Dirac operator on a manifold, in the古典案例。 ******光谱对产生了K理论的地图，其本地索引公式A. Connes和合着者提供了一个公式。该公式用与三重Zeta函数的残基来描述K理论图，以一种非常有趣的“局部”，高度几何的方式，在哲学上类似于与平滑歧管上的差异形式相似的方式。 ******对于一个频谱对，就要求操作员D与C*-Algebra上下通勤，从某种意义上说，较低的术语。但是，许多重要的情况，例如双曲线群的边界作用，产生具有某种分形性质（它们纯粹是无限）的C* - 代数，因此该概念是不合适的，因为光谱对正确定义了密集定义的痕迹，并且这些示例不承认没有痕迹。在这项提议中，我们旨在遵循A. Connes的最新想法，以纠正这一点：我们的目的是使用量子统计力学的一些想法来研究光谱对的想法的变化，现在允许在Hilbert Space H上进行两个动作，而仅针对A的繁殖序列而定，但仅适用于操作员的命令，而不是按等于原始的动作，将其与原始的动作相等，该动作是由原始的绘制绘制的，该动作是由原始的绘制绘制的，该动作是由原始的绘制绘制的，该动作是由原始的绘制来列出。操作员。 It turns out that this twisting of the definition has no effect cohomologically (on the Chern character), but the obstruction (failure of traces to exist for purely infinite algebras) to extending `integration' no longer exists, because instead, it becomes a kind of twisted integration, corresponding to a KMS state, a concept from quantum thermodynamics -- KMS states in these examples do exist, and there is an extremely interesting and rich theory of them. ******我的目标是构建与双曲线组的边界动作，我已经进行了广泛研究的系统以及其他几个示例和示例家族有关的扭曲光谱三元组，以将它们连接到K理论，研究相应的索引图，以及更广泛地研究KMS状态和KMY之间的联系，这似乎是由Twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist twist。 ***