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
Censorship: 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

Comment

13005101(former UVA) is identical with this course description.

Entry requirements

MM549 og MM845 or similar courses.

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.

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

See Blackboard for syllabus lists and additional literature references.

Examination regulations

Exam element a)

Timing

Autumn

Tests

Mandatory assignments

EKA

N310002102

Censorship

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 examination form for re-examination may be different from the exam form at the regular exam. Reexamination according to the rules approved by the study board.

Indicative number of lessons

42 hours per semester

Teaching Method

Lectures will introduce general concepts and theory and exercise
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.

Teacher responsible

Name E-mail Department
David Kyed dkyed@imada.sdu.dk Institut for Matematik og Datalogi, Matematik

Additional teachers

Name E-mail Department City
Wojciech Szymanski szymanski@imada.sdu.dk Institut for Matematik og Datalogi, Matematik

Timetable

31
Monday
02-08-2021
Tuesday
03-08-2021
Wednesday
04-08-2021
Thursday
05-08-2021
Friday
06-08-2021
08 - 09
09 - 10
10 - 11
11 - 12
12 - 13
13 - 14
14 - 15
15 - 16
Show full time table

Administrative Unit

Institut for Matematik og Datalogi (matematik)

Team at Registration & Legality

NAT

Recommended course of study