Fundamentals of the theory of operator algebras

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Department of Mathematics

Synopsis:

Basic course of the theory of operator algebras aimed at the theory of C* algebras and von Neumann algebras in its concrete Hilbert space representation. The state space, GNS construction and representations are studied. Comparison theory of projections, states and representations is explained. Von Neumann algebras are classified as finite and infinite and structural types I, II, III.

Requirements:

Syllabus of lectures:

1. Direct sums and tensor products of Hilbert spaces. Bounded and unbounded Hilbert space operators.

2. Banach algebras. Gelfand spectral radius formula. Holomorphic function calculus.

3. C* algebras, positivity, states and representations. GNS construction. Pure states and irreducible representation.

4. Weak and strong operator topologies. Von Neumann algebras.

5. Double commutant theorem, Kaplansky theorem, Kadison transitivity theorem.

6. Abelian operator algebras and their characterization.

7. Projection lattices, spectral measure and spectral theorem. Unbounded operators affiliated to von Neumann algebras.

8. Comparison theory of projections on von Neumann algebras.

9. Decomposition of a von Neumann algebra into finite and infinite part. Types I, II, III.

10. Normal functionals on von Neumann algebras. Preduls. Normal weights.

11. Trace and dimension function on a von Neumannově algebra. Dixmier theorem.

12. Universal representation of a $C^\ast$-algebra and enveloping von Neumannova algebra.

13. Equivalence of states and representations of operator 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).

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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)