Cocycle Superrigidity and Borel Complexity

Course Description and Grading Breakdown

Course Meeting Time and Location
Tuesday and Thursday
10:30 - 11:55 am
122 Math Building (Building 15)

Course Instructor Contact Information and Office Hours
210-1 Math Building (Building 15)

Course Schedule and Textbook

 April 3 Introduction to C*-algebras. Spectrum and spectral radius. Normal, selfadjoint, and unitary elements. Gelfand duality. See Section 1 of [1].
 April 5 States on C*-algebras. The GNS construction. The strong and weak operator topologies on B(H). See Section 1 and Section 2 of [1].
 April 10Characterization of (ultra)weakly continuous functionals. Spectral measures and the spectral theorem. Borel functional calculus. Isometries and partial isometries. Polar decomposition. See Section 2 and 3 of [1].
 April 12 Von Neumann's bicommutant theorem.The predual of a von Neumann algebra. Characterization of normal states. Kaplanski's density theorem. See Section 3 of [1].
 April 17 Spectral decomposition in von Neumann algebras. Projections in von Neumann algebras. Murray-von Neumann equivalence. II_1 factors. The hyperfinite II_1 factor. See Section 3 of [1].

There is no official textbook. We list below some recommended reading containing most of the topics covered.

[1] C. Houdayer, An invitation to von Neumann algebras. Available at
[2] C. Houdayer, An introduction to II_1 factors. Available at

Course Policies
Late work - 

 Date PostedAssignment Due Date 

Midterm and Final Exam

Collaboration Table
You may consult:  
Course textbook (including answers in the back)YESYES
Other booksYESNO
Solution manualsNONO
Your notes (taken in class)YESYES
Class notes of othersYESNO
Your hand copies of class notes of othersYESYES
Photocopies of class notes of othersYESNO
Electronic copies of class notes of othersYESNO
Course handoutsYESYES
Your returned homework / examsYESYES
Solutions to homework / exams (posted on webpage)YESYES
Homework / exams of previous yearsNONO
Solutions to homework / exams of previous yearsNONO
Emails from TAsYESNO
You may:

Discuss problems with othersYESNO
Look at communal materials while writing up solutionsYESNO
Look at individual written work of othersNONO
Post about problems onlineNONO
For computational aids, you may use:


* You may use a computer or calculator while doing the homework, but may not refer to this as justification for your work.  For example, "by Mathematica" is not an acceptable justification for deriving one equation from another.  Also, since computers and calculators will not be allowed on the exams, it's best not to get too dependent on them.