Operator algebras on Hilbert spaces

mathematics
physics

elective monographs

Functional Analysis 1000-135AF

Functional Analysis 1000-135AF

Classroom

Theory of operator algebras is concerned with studying families of operators on a Hilbert space. It grew out of von Neumann's attempt at providing a mathematical description of quantum mechanics. We will start the course with an introduction to the general theory, showing links with measure theory. Afterwards we will focus on examples, mainly coming from group theory. Many notions that arise in this context can be generalised to general operator algebras. Motivated by this we will carry out a deeper study of von Neumann algebras, a class of operator algebras closely connected to measure theory and ergodic theory.

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.

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

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.

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.