# NMAK13030U  Approximation Properties for Operator Algebras and Groups (Approx)

Volume 2014/2015
Education
MSc Programme in Mathematics
MSc Programme in Statistics
MSc Programme in Mathematics-Economics
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.

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.
Credit
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.
Re-exam
Oral, 30 minutes with 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.

• Category
• Hours
• Lectures
• 36
• Preparation
• 152
• Theory exercises
• 18
• Total
• 206