KK-Theory

Content / Literature / Recommended previous knowledge Content, short version: We will introduce K-theory and KK-theory of C*-algebras and apply them to the classification problem of C*-algebras. Content, long version: We will start with an introduction to C*-algebras and prove fundamental structure theorems for them (Gelfand representation, Gelfand-Naimark representation, classification of finite-dimensional C*-algebras). Then we will introduce K-theory, refine it by introducing an order structure on it, and use this to classify AF-algebras. We will then spend some time on the definition of the category KK via the suspension-stable homotopy category of C*-algebras, and show that KK is a triangulated category. Finally we will prove the universal coefficient theorem for KK and apply it to the classification problem of Kirchberg algebras. Required knowledge: For the first part of the seminar basic knowledge about compact operators on Hilbert spaces is needed. For the rest of the seminar basic familiarity with category theory and homology theories will be additionally needed.

Prüfungsbestandteile Vortrag, schriftliche Ausarbeitung; je nach zutreffendem Modulkatalog ist der Vortrag eine Studienleistung. Giving a presentation, writing a detailed report; depending on the applicable Modulkatalog the presentation is a Studienleistung.

Anteile der Bestandteile an der Note Je nach zutreffendem Modulkatalog basiert die Note entweder auf der schriftlichen Ausarbeitung oder auf dem Vortag. Depending on the applicable Modulkatalog: the grade is based either only on the written report or only on the presentation.