Type III Noncommutative Geometry and KK-theory

非交换几何试图分析与各种几何情形相关的C*-代数，或具有动力系统的情形，如几何空间对称性的复杂群作用，通过调整流形上的几何分析方法来工作于C*-代数。广义的思想是，对于许多这样的情况，我们知道如何构造C*-代数，而这反过来又可以进行拓扑分析（好像它是一个空间），使用K-理论，也是几何，使用“谱对”的想法，由希尔伯特空间上的C*-代数的表示和无界算子D组成，在流形上扮演狄拉克算子的角色，在经典案例中。** 一个谱对产生一个K理论上的映射，其中A的局部指数公式。Connes和合著者提供了一个公式。这个公式描述了K-理论地图的残留物的某些zeta功能相关联的三重，在一个非常有趣的“本地”，高度几何的方式，在数学上类似的方式之一，集成了一个微分形式在一个光滑的流形。** 对于谱对，在某种意义上，要求算子D与C*-代数A交换直到低阶项。但许多重要的情形，如双曲群的边界作用，产生具有某种分形性质的C*-代数（它们是纯粹无限的），这种概念是不合适的，因为适当定义的谱对诱导密集定义的迹，而这些例子不承认迹。 在这个建议中，我们的目标是遵循A.康纳斯纠正这一点：我们的目标是使用量子统计力学的一些思想来研究谱对思想的一种变体，现在允许A在希尔伯特空间H上的两个作用，一个只定义为A的稠密子代数，但是算子现在只需要是等变的，作为从具有原始作用的H到具有扭曲作用的H的映射，直到低阶算子。事实证明，这种定义的扭曲在上同调上没有影响（关于陈省身的性格），但阻碍（对于纯无限代数，迹不存在）扩展“积分”不再存在，因为相反，它变成了一种扭曲的积分，对应于KMS态，一个来自量子热力学的概念--这些例子中的KMS态确实存在，有一个非常有趣和丰富的理论。** 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，我已经广泛研究了系统，并在其他几个例子和例子家族中，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状态和K理论之间的联系，这似乎是扭曲谱三元组框架所暗示的。***

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. ***