Approximation Properties for Operator Algebras and Groups (Approx)

Course content

This course aims at providing a comprehensive treatment of a number of approximation properties for countable groups and their corresponding counterparts for von Neumann algebras and C*-algebras. This will include the following topics: amenable groups, nuclear C*-algebras, injective von Neumann algebras, exactness for C*-algebras and groups, the Haagerup property (property H), and Kazhdan's property T for groups and von Neumann algebras.


MSc Programme in Mathematics

Learning outcome


After completing the course, the students will have:

Knowledge of the material mentioned in the description of the content.

Skills to to read and understand research papers concerning topics discussed in lectures.

The following competences:


  • Have a good overview and understanding of the various approximation properties for groups and their associated von Neumann algebras, respectively, group C*-algebras discussed in lectures. In particular, understand how these approximation properties for the group reflect into corresponding properties for the associated operator algebras.
  • Master (at a satisfactory level) the fundamental results covered in the lectures, to the extent of understanding their proofs and be able to interconnect various results.
  • Have a good understanding and be able to work with completely positive maps (respectively, completely bounded maps), which are the natural morphisms in the setting of the course.
  • Handle complex results connecting various topics within the area of von Neumann algebras and C*-algebras, as well as approximation properties of discrete groups.

4 hours lectures, 2 hours exercises/discussion per week for 8 weeks.

qualifications: Introduction to operator algebras.

Continuous feedback during the course of the semester
7,5 ECTS
Type of assessment
Continuous assessment
Each student will give a presentation (up to 2 x 45 min) of material (not covered in lectures) relevant to the topic of the course, coming either from a research paper or from the textbook itself.
All aids allowed
Marking scale
7-point grading scale
Censorship form
No external censorship
One internal examiner
Criteria for exam assessment

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

Single subject courses (day)

  • Category
  • Hours
  • Lectures
  • 32
  • Preparation
  • 138
  • Theory exercises
  • 16
  • Exam
  • 20
  • English
  • 206