Prerequisites:
Basic operator algebra theory (such as the material
of Nicoara's introduction to operator algebras, spring 07). Work
through Kehe Zhu's An Introduction to Operator Algebras,
CRC Press, 1993, if you have not taken Nicoara's course.
Recommended Books: Some of the material covered in the course
can be found in the following books:
1) Jacques Dixmier, Von Neumann Algebras, North Holland, 1981.
2) Vaughan Jones, V. Sunder, Introduction to Subfactors,
Cambridge University Press, 1997.
3) Masamichi Takesaki, Theory of Operator Algebras I, II, III,
Springer-Verlag 2002.
4) David Evans, Y. Kawahigashi, Quantum Symmetries on Operator
Algebras, Oxford University Press, 1998.
5) Serban Stratila, Laszlo Zsido, Lectures on Von Neumann Algebras.
Additional references will be given throughout the course.
Syllabus: I will start the course with some of the basics of
the theory of von Neumann algebras, such as the group measure space
construction, group von Neumann algebras, bimodules, Connes' bimodule
tensor product, the Murray-von Neumann coupling constant and perhaps
property Gamma, Haagerup's approximations property and property T
for von Neumann algebras.
I will then discuss Jones' theory of
subfactors and perhaps planar algebras if there is enough time.
In particular I will show how Jones' braid group representation and
his knot invariant, the Jones polynomial, arise naturally from subfactor
theory.
Grading:
The course grade will be based on attendance and possibly a
presentation. I will give out optional homework problems. There will be no
exams.