Prerequisites: A first year course in real analysis and topology and some basic complex analysis. Functional analysis, basic operator theory and spectral theory, some basic facts about C*-algebras. Continuous and Borel functional calculi.

Recommended Books: The following books contain part of what I plan to cover in the course:
1) Jacques Dixmier, Von Neumann Algebras, North Holland, 1981.
2) Kehe Zhu, An Introduction to Operator Algebras, CRC Press, 1993.
3) Vaughan Jones, V. Sunder, Introduction to Subfactors, Cambridge University Press, 1997.
4) Masamichi Takesaki, Theory of Operator Algebras I, II, III, Springer-Verlag 2002.
5) David Evans, Y. Kawahigashi, Quantum Symmetries on Operator Algebras, Oxford University Press, 1998.
6) Gert Pedersen, Analysis Now, Springer Verlag, GTM 118, 1988 (revised edition).
7) Richard Kadison, John Ringrose, Fundamentals of the Theory of Operator Algebras, I, II, III, IV, AMS, 1997, 1997, 1991, 1992.
8) Serban Stratila, Laszlo Zsido, Lectures on Von Neumann Algebras.

Additional references will be given throughout the course.

Syllabus: Von Neumann algebras are certain algebras of operators on Hilbert space which arise naturally as algebras of symmetries of quantum physical systems. They can be viewed as noncommutative measure spaces. I will give an introduction to these von Neumann algebras and plan to discuss topics from the theory of II₁ factors and their subfactors. This may include a discussion of Jones' braid group representation and Jones' knot invariant, the Jones polynomial (it there is enough time). Other possible topics are applications of rigidity phenomena (property (T)) and L²-Betti numbers to the structure theory of II₁ factors, including recent solutions to some longstanding problems in the theory of II₁ factors due to Popa.

This course will be an excellent preparation for the Third Annual Spring Institute in Noncommutative Geometry and Operator Algebras, which will feature several mini-courses on current research in the area of von Neumann algebras, see http://www.math.vanderbilt.edu/~ncgoa05.

Grading: The course grade will be based on attendance. I will give out optional homework problems on a regular basis. Each enrolled student will be required to type up approximately two weeks worth of lectures (in TeX). There will be no exams.

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