MM819: An introduction to Operator Algebras (5 ECTS)

STADS: 13005101

Level
Master's level course approved as PhD course

Teaching period
The course is offered in the spring semester.

Teacher responsible
Email: dkyed@imada.sdu.dk

Timetable
There is no timetable available for the chosen semester.

Comment:
Ubegrænset deltagerantal.

Prerequisites:
MM543 or an equivalent course.

Academic preconditions:
Students taking the course are expected to:

• Have a basic knowledge of topology and functional analysis, corresponding the contents of the courses MM535 and MM543.
• Be able to use basic arguments from topology.
• Be able to work independently with linear algebra.
• Have a basic knowledge of the theory of groups and rings.

Course introduction
The aim of the course is to introduce the student to the basics of the theory of operator algebras, with emphasis on the theory of C*-algebras and operators on Hilbert spaces. The material covered is important in almost all aspects of modern analysis. The course builds on the knowledge acquired in the courses MM535 (Topology) and MM543 (Measure and integration and Banach spaces) and gives the student the necessary prerequisites to specialize in operator algebra theory during their master’s studies.

In relation to the competence profile of the degree it is the explicit focus of the course to:

• Give the competence to take responsibility for the academic development and specialization.
• Give the competence to develop an overview of the interplay between different mathematical disciplines.
• Give skills to work concretely with Hilbert spaces and their operators.
• Give skills to use central functional analytic tools (the Hahn Banach theorem and its consequences)
• Give knowledge and understanding of concrete examples of operator algebras.
• Give knowledge of the theory of commutative C*-algebras.
• Give knowledge of the theory of operators on Hilbert spaces.

Expected learning outcome
The learning objectives of the course are that the student demonstrates the ability to:

• Reproduce definitions and results, including their proofs, covered in the course.
• Be able to use these results to analyse concrete examples.
• Formulate and present definitions, proofs and calculations in a mathematically rigorous way.

Subject overview
The following main topics are contained in the course:

• The Gelfand transform
• Representation theory for C*-algebras
• The Gelfand Naimark theorem
• Compact operators and the spectral theorem.

Literature

Meddeles ved kursets start.

Website
This course uses e-learn (blackboard).

Prerequisites for participating in the exam
None

Assessment and marking:

1. Mandatory assignments , grades according to the Danish 7-point scale, internal examiner. (13005102).

The re-exam may be different than the ordinary exam. Reexamination according to the rules approved by the study board.

Expected working hours
The teaching method is based on three phase model.
Intro phase: 28 hours
Skills training phase: 14 hours, hereof:
 - Tutorials: 14 hours

Educational activities
Educational form


Language
This course is taught in Danish or English, depending on the lecturer. However, if international students participate, the teaching language will always be English.

Course enrollment
See deadline of enrolment.

Tuition fees for single courses
See fees for single courses.