MM819: An introduction to Operator Algebras
Study Board of Science
Teaching language: Danish or English depending on the teacher, but English if international students are enrolled
EKA: N310002102
Assessment: Second examiner: Internal
Grading: 7-point grading scale
Offered in: Odense
Offered in: Autumn
Level: Master's level course approved as PhD course
STADS ID (UVA): N310002101
ECTS value: 5
Date of Approval: 07-04-2021
Duration: 1 semester
Version: Approved - active
Entry requirements
Academic preconditions
Students taking the course are expected to:
- Have a basic knowledge of topology and functional analysis, corresponding the contents of the courses MM549 and MM845.
- 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 MM549 (Topology and complex analysis) and MM845 (Functional analysis) and gives the student the necessary
prerequisites to specialize in operator algebra theory during their
master’s studies.
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 MM549 (Topology and complex analysis) and MM845 (Functional analysis) 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.
Content
The following main topics are contained in the course:
- The Gelfand transform
- Representation theory for C*-algebras
- The Gelfand Naimark theorem
- The spectral theorem for selfadjoint operators
Literature
Examination regulations
Exam element a)
Timing
Autumn
Tests
Mandatory assignments
EKA
N310002102
Assessment
Second examiner: Internal
Grading
7-point grading scale
Identification
Full name and SDU username
Language
Normally, the same as teaching language
Examination aids
To be announced during the course
ECTS value
5
Additional information
The re-exam is changed to an oral exam if there are 4 or fewer students enrolled. At the oral re-exam, the students draw a random assignment among the mandatory assignments (which make up the ordinary exam) and must now present the assignment at the blackboard.
Indicative number of lessons
Teaching Method
At the faculty of science, teaching is organized after the three-phase model ie. intro, training and study phase.
In order to enable students to achieve the learning objectives for the course, the teaching is organised in such a way that there arexx8 lectures, class lessons, etc. on a semester.
These teaching activities are reflected in an estimated allocation of the workload of an average student as follows:
- Intro phase (lectures, class lessons) - 28 hours
- Training phase: 14 hours
Introphase: Lectures will introduce general concepts and theory and exercise
sessions will be devoted to learn material in depth. Interactive
teaching will be used.
sessions will be devoted to learn material in depth. Interactive
teaching will be used.
- The students are expected to familiarize themselves with the material covered in the lectures.
- To acquire knowledge of selected topics independently.