Type III Noncommutative Geometry and KK-theory

非对易几何试图通过将流形上的几何分析方法应用于C*-代数来分析与各种几何情况有关的C*-代数，或具有动力系统的情况，如几何空间的对称性的复杂群作用。广义的想法是，对于许多这样的情况，我们知道如何构造C*-代数，并且这反过来可以使用K-理论从拓扑上(就像它是一个空间一样)进行分析，并且还可以使用由希尔伯特空间上的C*-代数的表示和在经典情况下扮演流形上狄拉克算子角色的无界算子D组成的“谱对”的想法来进行分析。谱对产生K-理论上的映射，A.Connes和合著者的局部指数公式为其提供了一个公式。这个公式用与三元组相关的某些Zeta函数的残数来描述K理论映射，以一种非常有趣的‘局部’、高度几何的方式，在哲学上类似于在光滑流形上积分微分形式的方式。对于谱对，在某种意义上，需要算子D与C*-代数A交换到更低的阶项。但许多重要的情况，如双曲群的边界作用，产生了具有某种分形性(它们是纯无限的)的C*-代数，这一概念不适用于这种概念，因为适当定义的谱对产生密集定义的迹，而这些例子不允许有迹。在这个建议中，我们的目标是遵循A.Connes的一个更新的想法来纠正这个问题：我们的目标是使用量子统计力学的一些想法来研究谱对概念的一个变体，现在允许A在Hilbert空间H上的两个作用，一个只定义为A的稠密子代数，但现在只需要算子等变为从H到H的映射具有原始作用，到H具有扭曲作用，直到低阶算子。事实证明，定义的这种扭曲在上同调上没有影响(对陈氏特征标)，但对扩展‘积分’的障碍(纯无限代数的迹不存在)不再存在，因为它变成了一种扭曲积分，与KMS态相对应，量子热力学中的一个概念--这些例子中的KMS态确实存在，并且有关于它们的极其有趣和丰富的理论。我的目标是构造与双曲群的边界作用有关的扭曲谱三元组，我已经广泛研究了这些系统，并在其他几个例子和样本族中，将它们与K-理论联系起来，研究相应的指数映射，并更广泛地研究KMS态与K-理论之间的联系，这似乎隐含在扭曲谱三元组的框架中。

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.