Advanced theory of operator algebras

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Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

Some advanced aspects of the theory of operator algebras are treated. In particular, structure of ideals, convex structure of the state space, tensor products, cross products, and modular theory.

Requirements:

Syllabus of lectures:

1. Ideals in operator algebras. Faces in positive cones and duals.

2. Tensor products of Banach spaces and algebras.

3. Tensor products of C* algebras. Projective and injective tensor product. Takeasi theorem on minimal tensor norm.

4. Tensor products of von Neumann algebras. Types of tensor product.

5. Infinite tensor products.

6. Modular theory - mudular operator and modular group.

7. KMS states.

8. Group algebras.

9. Dynamical systems, covariance algebra.

11. Discrete and continuous cross products.

12. Cross products with modular groups. Connes spectrum.

13. Direct integrals of von Neumann algebras.

Syllabus of tutorials:

Study Objective:

Study materials:

1. R.V.Kadison and J.R.Ringrose: Fundamentals of the Theory of Operator Algebras I, II, Academic Press (1986). 2. M.Takesaki: Theory of Operator Algebras I, Berlin, Heidelberg, New York, Springer 2002. 3. J.Hamhalter: Quantum Measure Theory, Kluwer Academic Publishers, Dordrecht 2003.

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• Doctoral studies, daily studies (compulsory elective course)
• Doctoral studies, combined studies (compulsory elective course)