Type III Noncommutative Geometry and KK-theory

非交换几何试图分析与各种几何情况相关的C*-代数，或者具有动力系统的情况，如几何空间对称性的复杂群作用，通过调整流形的几何分析方法来适用于C*-代数。大致的想法是，在这些情况下，我们知道如何构造一个C*-代数，这反过来可以从拓扑上分析（就像它是一个空间一样），使用k理论，也可以从几何上分析，使用‘谱对’的思想，由希尔伯特空间上的C*-代数的表示和一个无界算子D组成，在经典情况下，在流形上扮演狄拉克算子的角色。******一个谱对在k理论上产生一个映射，A. cones及其合作者的局部指数公式提供了一个公式。这个公式用与三元组相关的ζ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交换到低阶项。但在许多重要的情况下，如双曲群的边界作用，所产生的C*-代数具有一种分形性质（它们是纯粹无限的），这种概念是不适合的，因为正确定义的谱对会产生密集定义的迹，而这些例子不承认迹。在这个提议我们旨在遵循最近的康涅狄格州整流这个:我们的目标是使用一些想法从量子统计力学研究的一个变体一双光谱的想法,现在允许两个动作的希尔伯特空间H,一只定义一个密集的子代数,但运营商现在只需要等变化从H与原操作地图,H和扭曲的行动,到低阶操作符。事实证明，这种定义的扭曲在上同调上（对陈氏特征）没有影响，但扩展“积分”的障碍（纯无限代数的迹不存在）不再存在，因为相反，它变成了一种扭曲积分，对应于KMS状态，量子热力学的一个概念——这些例子中的KMS状态确实存在，并且有一个非常有趣和丰富的理论。******我的目标是构建与双曲群边界作用有关的扭曲谱三元组，我已经广泛研究过的系统，以及其他几个例子和例子族，将它们与k理论联系起来，研究相应的索引映射，更广泛地说，研究KMS状态与k理论之间的联系，这似乎是由扭曲谱三元组的框架所暗示的。＊＊＊

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. ***