Schedule, syllabus and examination date

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Course content

The course gives an introduction to C*-algebras and von Neumann algebras, which are the most important classes of operator algebras. Such algebras have fundamental applications in various areas of mathematics and in quantum physics. After having covered the basic results in the theory, some important classes of examples will be studied. The choice of examples may vary from one year to another, depending on the interests of the students following the course.

Learning outcome

After completing the course you:

  • are familiar with the definitions of C*-algebras and von Neumann algebras, and with basic related concepts, such as *-homomorphisms, ideals, quotients, approximate units and multiplier algebras;
  • know about the positive cone of a C*-algebra, how it induces an order on the self-adjoint elements and how it is used to define states;
  • know how to construct the GNS-representation associated with a state and how to represent a C*-algebra faithfully as bounded operators on a Hilbert space;
  • know the connection between pure states and irreducible representations of a C*-algebra;
  • have a good understanding of several operator topologies and know how to characterize von Neumann algebras with the help of double commutants;
  • can describe C*-algebras of compact operators and have a knowledge of other important classes of C*-algebras.


PhD candidates from the University of Oslo should apply for classes and register for examinations through Studentweb.

If a course has limited intake capacity, priority will be given to PhD candidates who follow an individual education plan where this particular course is included. Some national researchers’ schools may have specific rules for ranking applicants for courses with limited intake capacity.

PhD candidates who have been admitted to another higher education institution must apply for a position as a visiting student within a given deadline.


Recommended previous knowledge

MAT4450 - Advanced Functional Analysis.

Overlapping courses

*The information about overlaps for discontinued courses may not be complete. If you have questions, please contact the Department. 


4 hours of lectures/exercises per week.

Upon the attendance of three or fewer students, the lecturer may, in conjunction with the Head of Teaching, change the course to self-study with supervision.


mandatory assignment.

Final oral examination.

In addition, each PhD candidate is expected to give a one hour oral presentation on a topic of relevance chosen in cooperation with the lecturer. The presentation has to be approved by the lecturer for the student to be admitted to the final exam.

Examination support material

No examination support material is allowed.

Language of examination

You may write your examination paper in Norwegian, Swedish, Danish or English.

Grading scale

Grades are awarded on a pass/fail scale. Read more about the grading system.

Explanations and appeals

Resit an examination

Students who can document a valid reason for absence from the regular examination are offered a postponed examination at the beginning of the next semester.

Re-scheduled examinations are not offered to students who withdraw during, or did not pass the original examination.

Withdrawal from an examination

It is possible to take the exam up to 3 times. If you withdraw from the exam after the deadline or during the exam, this will be counted as an examination attempt.

Special examination arrangements

Application form, deadline and requirements for special examination arrangements.


The course is subject to continuous evaluation. At regular intervals we also ask students to participate in a more comprehensive evaluation.

Facts about this course






Autumn 2020

Autumn 2019

Taught according to demand and resources. Contact if you are interested in this course.


Autumn 2020

Autumn 2019

The same semester as taught.

Teaching language


The course may be taught in Norwegian if the lecturer and all students at the first lecture agree to it.