NSCPHD1254 Approximation Properties for Operator Algebras and Groups (Approx)

Volume 2015/2016

MSc Programme in Mathematics
MSc Programme in Statistics
MSc Programme in Mathematics-Economics



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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.

Nathanial P. Brown, Narutaka Ozawa: C*-algebras and finite dimensional approximations, Graduate Texts in Mathematics Vol. 88, Amer. Math. Society, Providence, Rhode-Island, 2008, and research papers.

qualifications: Introduction to operator algebras.
4 hours lectures, 2 hours exercises/discussion per week for 9 weeks.
  • Category
  • Hours
  • Lectures
  • 36
  • Preparation
  • 152
  • Theory exercises
  • 18
  • Total
  • 206
7,5 ECTS
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
passed/not passed
Censorship form
No external censorship
One internal examiner

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

Criteria for exam assesment

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