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)

Office hours: by appointment

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].
 April 19     The von Neumann algebras associated with a group, a group action, a countable pmp equivalence relation, and a countable pmp gropoid. See Section 4 of [1].
 April 24Positive semidefinite functions on groups. M-N-bimodules. Completely positive maps. The correspondence between unital completely-positive maps and bimodules with a distinguished tracial unit vector. Stinespring dilation of completely positive maps. Property (T) for (pairs of) groups and von Neumann algebras. See Subsection 2.1, 2.2, and 4.1 of [2].
 April 26 Representations of groups and C*-algebras. The (reduced) group C*-algebra. Weak containment of representations. Orthogonality of representations.  The spectrum of a C*-algebra and the hull-kernel topology. 
 May 1 Faithful and essential representations and turbulence. See Lemma 2.1 in [3]
 May 3 The Gaussian action associated with a unitary representation. Nonclassifiability of free weak mixing actions of infinite groups. See Appendices B,C,D,E in [4].
 May 8 Type I groups and algebras. Nonclassifiability of irreducible representations of a nontype I algebra. The rigid action of F_2. See Theorem 2.8 in [3] and Section 16 of [4].
 May 10     Nonclassifiability of orbit equivalence of free weakly mixing actions of F_2. See the Main Result of [8], Theorem 3.12 of [10], and Lemma 7.4 and Theorem D of [11].
 May 15 Coinduced action for groups. Nonclassifiability of orbit equivalence of free weakly mixing actions of groups that contain F_2. See Section 2 of [8].
 May 17     Actions of groupoids. Coinduced action for groupoids. Groupoid von Neumann algebra. Extensions and expansions of equivalence relations. Nonclassifiability of orbit equivalence of free weakly mixing actions of nonamenable groups. See [12], Theorem 3.12 of [10], and Lemma 7.4 and Theorem D of [11].

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
[3] D. Kerr, H. Li, M. Pichot, Turbulence, representations, and trace-preserving actions, Proceedings of the London Mathematical Society
[4] A. Kechris, Global aspects of ergodic group actions, 2010
[5] S. Popa, Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups, Inventiones Mathematicae
[6] A. Furman, On Popa's cocycle superrigidity theorem, International Mathematics Research Notices
[7] E. Gardella, M. Lupini, The complexity of conjugacy, orbit equivalence, and von Neumann equivalence of actions of nonamenable groups
[8] A. Ioana, Orbit inequivalent actions for groups containing a copy of F_2, Inventiones Mathematicae
[9] A. Ioana, Cocycle superrigidity for profinite actions of property (T) groups
[10] A. Ioana, A. Kechris, T. Tsankov, Subequivalence relations and positive-definite functions
[11] L. Bowen, D. Hoff, A. Ioana, von Neumann's problem and extensions of non-amenable equivalence relations
[12] I. Epstein, Orbit inequivalent actions of non-amenable groups

End of course projects:

1) Popa's cocycle superrigidity theorem for malleable actions. References: Theorem 5.2 in  [5] and Theorem 1.3 in [6] 
2) The relation of orbit equivalence of free weak mixing actions of a countable groups containing a copy of F_2 is not Borel. Theorem 2.12 of [7]; see also Section 2 of [8].
3) Orbit equivalence superrigidity for profinite actions in the case of property (T) groups. Reference: Theorem A in [9] in the case when Gamma_0 is equal to Gamma.

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.