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Approximation Properties for Operator Algebras and Groups (Approx)
Provider: Faculty of Science

Activity no.: 5549-18-07-31 
Enrollment deadline: 29/08/2018
PlaceDepartment of Mathematical Sciences
Date and time03.09.2018, at: 09:00 - 11.11.2018, at: 16:00
Regular seats50
ECTS credits7.50
Contact personNina Weisse    E-mail address: weisse@math.ku.dk
Enrolment Handling/Course OrganiserMagdalena Elena Musat    E-mail address: musat@math.ku.dk
Written languageEnglish
Teaching languageEnglish
Semester/BlockBlock 1
Block noteDuration: 1 block
Scheme groupA (Tues 8-12 + Thurs 8-17)
Exam requirementsThe student must in a satisfactory way demonstrate that he/she has mastered the learning outcome
Exam formContinuous assessment
Exam detailsEach 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
Grading scalePassed / Not passed
Censorship formOne internal examiner
Exam re-examinationOral, 30 minutes with 30 minutes preparation time with all aids. Several internal examiners
Course workload
Course workload categoryHours
Lectures36.00
Preparation152.00
Theory exercises18.00
Sum206.00
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.

Literature
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.

Teaching and learning methods
4 hours lectures, 2 hours exercises/discussion per week for 9 weeks.

Remarks
Education:
MSc Programme in Mathematics
MSc Programme in Statistics
MSc Programme in Mathematics-Economics

Academic qualifications:
Qualifications: Introduction to operator algebras.

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