MM819: An introduction to Operator Algebras

MM543 or an equivalent course.

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.

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.

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.

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.