Prerequisites:
A course on von Neumann algebras.
Recommended Books: The following books contain some of the background
material for 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) 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:
This course is a continuation of Introduction to von Neumann
algebras which I taught in spring 2005. I will continue
the discussion of the coupling constant and then discuss the
Jones index for subfactors. I will prove Jones' rigidity theorem
and present Jones' braid group representation and his knot
invariant, the Jones polynomial. I will then move on to some
special topics from the theory of subfactors, for instance
planar algebras. 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.
Grading: The course grade will be based on attendance and a
presentation. I will give out optional homework problems. There will
be no exams.