NMAK13030U Approximation Properties for Operator Algebras and Groups (Approx)
MSc Programme in Mathematics
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.
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.
- 7,5 ECTS
- Type of assessment
- Continuous assessmentEach 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
- 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.
- Theory exercises