Kursussøgning, efter- og videreuddannelse – Københavns Universitet

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Kursussøgning, efter- og videreuddannelse

Approximation Properties for Operator Algebras and Groups (Approx)

Practical information
Study year 2016/2017
Block 1
Programme level Full Degree Master
Course responsible
  • Magdalena Elena Musat (5-737b79677a4673677a6e34717b346a71)
phone +45 35 32 07 45, office 04.2.05
  • Department of Mathematical Sciences
Course number: NMAK13030U

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 completely contractive and completely bounded approximation properties (CCAP and CBAP, respectively) and the Haagerup property (property H). If time permits, Kazhdan's property T for groups and von Neumann algebras will also be discussed.

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.

Recommended prerequisites

qualifications: Introduction to operator algebras.

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MSc Programme in Mathematics


Study Board of Mathematics and Computer Science

Course type

Single subject courses (day)


1 block


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Teaching and learning methods

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


No limit.




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


Type of assessment

Continuous assessment
Each student will give a 2x45 min presentation of material (not covered in lectures) relevant to the topic of the course, coming either from a research paper or from the textbook itself.

Marking scale

7-point grading scale

Criteria for exam assessment

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

Censorship form

No external censorship
One internal examiner


Oral, 30 minutes with 30 minutes preparation time with all aids. Several internal examiners.

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