Math 259B, Operator Algebras on Hilbert Space, Winter 2013

Announcements

Syllabus

Meets: MWF, 2:00-2:50, MS 5233

Office Hours: by appointment

Requisites: Basic knowledge of C*- and W*-algebras will be assumed.

Description: This is the second part of a course on Operator Algebras. The first part was dedicated to setting up the basic results, tools, and techniques in the theory of C*-algebras with an emphasis on von Neumann algebras and finite factors. The focus this quarter will be on the classification of II₁ factors with a heavy emphasis on developments in S. Popa's deformation/rigidity theory (2001-present).

Topics covered will reflect the interests of the class. A likely selection will be:

• Ergodic actions and Borel equivalence relations
• Correspondences, weak containment, and approximation properties
• Popa's intertwining-by-bimodules technique
• OE and cocycle superrigidity (Popa, Ioana, Furman, Peterson-S.)
• C*-algebraic techniques in von Neumann algebras (Ozawa)
• L²-rigidity (Peterson)
• Uniqueness of Cartan and strong solidity (Ozawa-Popa, S., Chifan-S., Popa-Vaes, Ioana,...)
• W*-superrigidity (Peterson, Popa-Vaes, Ioana, Chifan-Peterson, Popa-Vaes,...)

Grades will be determined by (1) attendance and (2) presentation of assigned project. Possible topics for presentation are:

• The Dye/Ornstein-Weiss theorem on orbit equivalence of free, ergodic actions of amenable groups (Ian)
• Introduction to Free Probability (Jeff)
• Popa's "short" proof of the finite case of Connes' classification of injective factors (Andreas & Alin)
• Gaboriau's theory of L²-Betti numbers for equivalence relations
• Markov semigroups and Quantum Dirichlet forms
• Classification of Singular Masas (Brian)
• Weakly exact von Neumann algebras (Paul)
• Complete metric approximation for q-Gaussians
• Adams' indecomposability theorem for non-singular ergodic actions of hyperbolic groups (Brent)

Presentations

Andreas Naes Aaserud and Alin Galatan: Popa's Proof of Connes' Theorem (Injective implies hyperfinite, II₁ case).

Brent Nelson: Indecomposability of free nonsingular actions by nonamenable groups in QH_reg (after Houdayer and Vaes).

Resources

Topic 2: Classification of II₁ Factors

Survey Articles

A. Ioana, "Classification and rigidity for von Neumann algebras", Proceedings of the European Congress of Mathematicians (Krakow, 2012).

S. Popa, "Deformation and rigidity for group actions and von Neumann algebras", Proceedings of the ICM (Madrid, 2006).

S. Vaes, "Rigidity for von Neumann algebras and their invariants", Proceedings of the ICM (Hyderabad, 2010).

S. Vaes, "Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa)", Seminaire Bourbaki, exp. 961, Asterisque 311 (2007), 237-294.

Articles and Lecture Notes

J. Peterson, "Lecture Notes in Ergodic Theory".

S. Popa, "Correspondences".

N. Brown, "The Symbiosis of C*- and W*-Algebras".

N. Ozawa, "A Kurosh type theorem for type II₁ factors," Int. Math. Res. Not. (2006), article id 97560.

N. Ozawa and S. Popa, "On a class of II₁ factors with at most one Cartan subalgebra," Ann. Math. 172 (2010), 713-749.

I. Chifan and T. Sinclair, "On the structural theory of II₁ factors of negatively curved groups," Ann. Sci. Ec. Norm. Sup. 46 (2013), 1-33.

Books

N. P. Brown and N. Ozawa, "C*-Algebras and Finite-Dimensional Approximations", Graduate Studies in Mathematics 88, AMS (2008).

M. Takesaki, "Theory of Operator Algebras, II, III", Springer.

Topic 1: Basic Theory of C*- and W*-Algebras (Fall 2012)

Books

J. B. Conway, "A Course in Functional Analysis", 2nd ed., Grad. Texts in Math. 96, Springer, New York (1990).

K. R. Davidson, "C*-Algebras by Example", Fields Inst. Monographs, AMS, Providence (1996).

J. Dixmier, "C*-Algebras", revised ed., North Holland Math. Library 15, North Holland, Amsterdam (1982)

R. G. Douglas, "Banach Algebra Techniques in Operator Theory", 2nd ed., Grad. Texts in Math. 179, Springer, New York (1998).

R. Kadison and J. R. Ringrose, "Fundamentals of the Theory of Operator Algebras, Volume II", reprint, Graduate Studies in Mathematics 16, AMS (1997).

G. J. Murphy, "C*-Algebras and Operator Theory", reprint, Academic Press (1990).

M. Takesaki, "Theory of Operator Algebras, I", Springer (1979).