# NMAK15008U  C*Topics: K-theory for C*-algebras

Volume 2015/2016
Education

MSc programme in Mathematics

Content

K-theory is one of the most important tools in the study of C*-algebras. It associates to a given C*-algebra A two abelian groups, called K_0(A) and K_1(A), in a functorial way. These groups contain much information about the structure of the individual C*-algebra and there exists a powerful machinery which allows a computation of the K-groups in many cases.

In this course we focus on methods to compute the K-groups for C*-algebras and illustrate those by numerous examples and applications. More precisely, we will cover the following topics:

• Quick repetition of the functors K_0, K_1 and their basic properties

• Order structure on the K_0-group and Elliott’s classification of AF-algebras

• The 6-term exact sequence with an explicit description of the boundary maps

• The Pimsner-Voiculescu exact sequence for the K-theory of crossed products

Learning Outcome

Knowledge: The student will obtain knowledge of the elements mentioned in the description of the content.

Skills: Calculate K-groups, use basic classification results

Rørdam, Larsen and Laustsen – An introduction to K-theory for C*-algebras

Basic knowledge on C*-algebras and topological K-theory, e.g., Introduction to operator algebras (IntroOpAlg) and Introduction to K-theory (K-Theory).
4 hours of lectures and 2 hours of exercises per week for 9 weeks.
Credit
7,5 ECTS
Type of assessment
Continuous assessment
Pass/fail based on 3 compulsory handins during the course.
Aid
All aids allowed
Marking scale
passed/not passed
Censorship form
No external censorship
One internal examiner
Re-exam

30 minutes oral exam without preparation time, several internal examiners, pass/ fail.

##### Criteria for exam assesment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.

• Category
• Hours
• Lectures
• 36
• Exercises
• 18
• Project work
• 60
• Preparation
• 92
• Total
• 206