Course Catalogue

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The aim of this course is give a topical introduction into operator algebra theory assuming only a minimal background in abstract Functional Analysis.

2 hours of lectures a week, over 11 weeks.

1 hour tutorial a week over 10 weeks (or equivalent)

Mandatory Entry Requirements

Level 4 Functional Analysis

Recommended Entry Requirements

Assessment

100% Final Exam

Reassessment

In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below. 

The aim of this course is give a topical introduction into operator algebra theory assuming only a minimal background in abstract Functional Analysis. The course will be based on key examples thus making the material quite concrete.

By the end of this course students will be able to:

- Demonstrate knowledge and the ability to apply central definitions and facts in the theory of operator algebras and use these to solve problems of a numerical or logical nature. Specific topics include:

1. Basic definitions and results in abstract Banach and C*-algebras

2. Basic theory and classes of operators on Hilbert space (projections, isometries, unitaries, self-adjoint and normal operators).

3. The structure of commutative and finite dimensional C*-algebras; its use in operator theory, possibly functional calculus; examples of such structures

 4. Positivity, ideals, quotients and homomorphisms of C*-algebras, basic results about simple C*-algebras.

5. AF-C*-algebras, basic K0 theory, mentioning of classification of AF algebras.

6. Toeplitz and Cuntz algebras and their basic properties.

7. Irrational rotation algebras and their basic properties.

8. Group C∗-algebras, some basic examples thereof, connections to group representations.

9. Further constructions of C∗-algebras such as crossed products and algebras given by generators and relations.

10. Definitions and key results in K-theory for C*-algebras.

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.