NMAK23012U Operator Algebras

Volume 2023/2024

MSc Programme in Mathematics
MSc Programme in Mathematics with a minor subject


Commutative Banach algebras, C*-algebras, commutative C*-algebras, continuous function calculus, states and representations, GNS representations, polar decomposition, non-unital C*-algebras and approximate units. Key examples of C*-algebras. Von Neumann algebras. The bicommutant theorem and Kaplansky's density theorem. Borel function calculus. 

Learning Outcome

The participants are expected to acquire the knowledge listed above in the course description with an emphasis on function calculus.

The participants are expected to be able to understand and apply the Gelfand transform and the GNS-construction, they must understand basic facts about order. They must have some familiarity with important examples of C*-algebras. They must understand the basics of von Neumann algebras. 

The participants are expected to master the most fundamental concepts and constructions for C*-algebras which are are used in further studies in operator algebras and in non-commutative geometry.


Kehe Zhu: An introduction to operator algebras (or equivalent), along with handout notes.

Functional Analysis (FunkAn) and Lebesgueintegralet og målteori (LIM) - alternatively Analyse 2 (An2) from previous years or similar introductory courses on functional analysis and analysis

Academic qualifications equivalent to a BSc degree is recommended.
5 hours of lectures plus 3 hours of tutorials per week in 8 weeks (last assignment is due in week 9).
  • Category
  • Hours
  • Lectures
  • 40
  • Preparation
  • 112
  • Theory exercises
  • 24
  • Exam
  • 30
  • Total
  • 206

Individual written feedback on mandatory exercises. Individual or collective feedback on solutions presented by students at the exercise sessions.

7,5 ECTS
Type of assessment
Continuous assessment
Type of assessment details
There will be given 3 assignments, each of which will count equally towards the final grade.
All aids allowed
Marking scale
7-point grading scale
Censorship form
No external censorship
One internal examiner.

30 minutes oral examination with 30 min. preparation time during which all aids are allowed. Several internal examiners.

Criteria for exam assesment

The student should convincingly and accurately demonstrate the knowledge, skills and competences described under Intended learning outcome.