The first aim of the seminar is to introduce the ideas of topological K-theory for noncommutative algebras. We first plan to study the ordinary (monovariant) K-theory for Banach and C*-algebras. We will introduce bivariant K-theories through the example of the recently developed bivariant K-theory for locally convex algebras. Other bivariant K-theories (such as in particular the ``classical'' Kasparov theory on the category of C*-algebras) have exactly analogous properties, but require more sophisticated techniques for their definition.
A second aim is the discussion of some typical applications of these techniques. We plan a discussion of bivariant versions of the Atiyah-Singer index theorem. The most general version of this theorem determines the K-homology class determined by the extension defined by the algebra of pseudodifferential operators on a compact manifold. It contains significantly more information than the classical index theorem. Another topic that we plan to discuss is the so-called ``twisted K-theory'' that has received much attention recently among mathematical physicists, and which has a very natural interpretation using the K-theory of certain noncommutative C*-algebras.
should be sent as hard copy or by e-mail (.ps or .pdf file) to:
Prof. Dr. Gert-Martin Greuel
67663 Kaiserslautern, Germany