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。

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.