Introduction to Operator Algebras (IntroOpAlg)

Course content

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. AF algebras. Von Neumann algebras. The bicommutant theorem and Kaplansky's density theorem. Borel function calculus. 


MSc Programme in Mathematics

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.


5 hours of lectures plus 3 hours of tutorials per week in 8 weeks (last assignment is due in week 9).

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.


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
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.
Criteria for exam assessment

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

Single subject courses (day)

  • Category
  • Hours
  • Lectures
  • 40
  • Preparation
  • 112
  • Theory exercises
  • 24
  • Exam
  • 30
  • English
  • 206