Operator algebras on Hilbert spaces 1000-1M24APH
The plan may be modified according to the interests of the participants.
1. Spectral theorem for self-adjoint operators.
2. Functional calculus
3. Definition of a C*-algebra.
4. Examples of C*-algebras: group algebras, Cuntz algebras.
5. Gelfand transform and commutative C*-algebras.
6. Gelfand-Naimark theorem: equivalence of the concrete and abstract definitions of C*-algebras.
7. Weak topologies and von Neumann algebras.
8. The bicommutant theorem.
9. Traces on von Neumann algebras. Crossed products.
More advanced topic that we might cover if time permits:
10. Conditional expectations.
11. L^p-spaces.
12. Injectivity of von Neumann algebras and its relationship with amenability of groups.
Main fields of studies for MISMaP
physics
Type of course
Mode
Requirements
Prerequisites
Course coordinators
Learning outcomes
After finishing the course "Operator algebras on Hilbert spaces" the student knows basic definitions of C*-algebras and can appreciate usefulness of different approaches. They understand the analogies between topology/measure theory and the theory of operator algebras. They can name examples of C*-algebras arising in different areas of mathematics.
Assessment criteria
The final result will be based mainly on the activity during the tutorials. At the end of the semester each student will be asked to give a short presentation.
Bibliography
1. W. Arveson "An invitation to C*-algebras".
2. K. Davidson "C*-algebras by example".
3. C. Anantharaman, S. Popa "An introduction to II_1 factors" https://www.math.ucla.edu/~popa/Books/IIun.pdf
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
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