Fall Semester 2010, Von Neumann Algebras
(Math 390A)



Instructor: Dietmar Bisch
Lecture: TuTh, 9:35am-10:50am, SC 1313
Office: SC 1405, (615) 322-1999 or SC 1334, (615) 322-4168 (Chair's office)
Office hours: TuTh 10:50am-11:50am, Fr 3:00-4:00pm
Mailbox: SC 1326

Prerequisites: Basic operator algebra theory (such as the material of Peterson's introduction to operator algebras, spring 09). Work through Kehe Zhu's An Introduction to Operator Algebras, CRC Press, 1993, if you have not taken Peterson'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: The contents of the course will depend on the background of the audience. As of now, I plan to 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 and the Murray-von Neumann coupling constant. I will then discuss Jones' theory of subfactors and will try to get as quickly as possible to the forefront of research in subfactors and planar algebras. I plan to include a discussion of 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.