Math 9101 - Seminar in Analysis: Operator Algebras and Acylindrically Hyperbolic Groups - Vanderbilt University - Spring 2021

Lecture notes (updated March 24th, 2021)
Jesse Peterson's homepage

All lectures will be held MWF 3:00-3:50pm over Zoom. Meeting ID: 941 4756 2219. Email the instructor for the password.

Key: [BO] Brown, Ozawa: C*-algebras and finite-dimensional approximations, (2008), American Mathematical Society, Providence, Rhode Island.
[AP] Anantharaman, Popa: An introduction to II_1 factors, preprint.
[P] Paulsen: Completely bounded maps and operator algebras, (2002), Cambridge University Press, Cambridge, UK.
[Os] Osin: Acylindrically hyperbolic groups, Transactions of the American Mathematical Society (2016), 368 (2): 851-888.
[Ba] Balasubramanya: Acylindrical group actions on quasi-trees, Algebr. Geom. Topol. Volume 17, Number 4 (2017), 2145-2176.
[BBF] Bestvina, Bromberg and Fujiwara: Bounded cohomology with coefficients in uniformly convex Banach spaces, Commentarii Mathematici Helvetici 91(2) (2016, 203-218.
[Oz] Ozawa: Weak amenability of hyperbolic groups, Groups Geom. Dyn. 2 (2008), 271-280.
[ER] Effros, Ruan: Multivariable multipliers for groups and their operator algebras, Proceedings of Symposia in Pure Mathematics, Volume 51 (1990), part 1, 197-218.

Schedule: (Subject to change)
Date Material covered Reference Video
January 25 Introduction/Invariant measure and Vitali sets Lecture 1
January 27 Banach Tarski Paradox/Amenable groups, paradoxical decomposition 2.6 in [BO] Lecture 2
January 29 Amenable groups II/Folner sequences, almost invariant vectors. 2.6 in [BO] Lecture 3
February 1 Operator algebras associated to amenable groups, Relative property (T) 12.1 in [BO] Lecture 4
February 3 Relative property (T) for (Z^2, SL(2, Z) \ltimes Z^2) 12.1 in [BO] Lecture 5
February 5 Positive definite functions/cocycles and Schoenberg's theorem Appendix D in [BO] Lecture 6
February 8 Characterizations of relative property (T) 12.1 in [BO] Lecture 7
February 10 Property (T) and reduced cohomology 12.1 in [BO] Lecture 8
February 12 Property (T) for SL(3, Z) 12.1 in [BO] Lecture 9
February 15 Haagerup's property for free groups 12.2 in [BO] Lecture 10
February 17 Haagerup's property and wreath products 12.2 in [BO] Lecture 11
February 19 Finite von Neumann algebras Review of selected sections in [AP] Lecture 12
February 22 Completely positive maps/Correspondences 13.1 in [AP] Lecture 13
February 24 Applications to II_1 factors 14.4 in [AP] Lecture 14
February 26 Property (Gamma) and inner-amenability 15.4 in [AP]
February 26 Hyperbolic graphs/groups 5.3 in [BO]
February 26 Hyperbolic graphs/groups II 5.3 in [BO]
February 26 Amenable actions 4.3 in [BO]
February 26 Exact groups 5.1 in [BO]
March 1 Biexact groups 15.2 in [BO]
March 3 Wreath products and Haagerup's property 12.2 in [BO]
March 5 Wreath products and biexactness 15.3 in [BO]
March 8 Acylindrically hyperbolic groups [Os]
March 10 Acylindrical actions on Quasi-trees [Ba]
March 12 Quasi-cocycles from quasi-trees [BFF]
March 15 Herz-Schur multipliers Appendix D in [BO]
March 17 Herz-Schur multipliers II Appendix D in [BO]
March 19 Weak amenability 12.3 in [BO]
March 22 Weak amenability for free groups 12.3 in [BO]
March 24 Weak amenability for hyperbolic groups [Oz]
March 26 Non-weak amenability for wreath products 12.3 in [BO]
March 29 Operator systems and completely positive maps Chapter 3 in [P]
March 31 Arveson's Extension Theorem Chapter 7 in [P]
April 5 Operator spaces and completely bounded maps Chapter 8 in [P]
April 7 Wittstock's Extension Theorem Chapter 8 in [P]
April 9 The Haagerup tensor product Chapter 17 in [P]
April 12 Completely bounded multilinear maps Chapter 17 in [P]
April 14 Multivariable Herz-Schur multipliers [ER]
April 16 Multivariable Herz-Schur multipliers II [ER]
April 19 Applications to von Neumann algebras I
April 21 Applications to von Neumann algebras II
April 23
April 26
April 28
April 28