Type III Noncommutative Geometry and KK-theory

点击翻译按钮获取中文摘要

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.