THEMATIC PROGRAMS

September 23, 2014

Thematic Program on Operator Algebras July-December, 2007

Graduate Courses held at the Fields Institute

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Introduction to Operator Algebras (Sep. 11 - Dec. 4, Tuesdays 1 p.m., Fields 230)
Instructors: Man-Duen Choi (Toronto), Ken Davidson (Waterloo)

Structure of C*-Algebras (Sep. 11 - Dec. 18, Tuesdays 10 a.m., Fields 230)
Instructors: George Elliott (Toronto), Chris Phillips (Oregon), Mikael Rørdam (Odense)

Free Probability (Sep. 13 - Dec. 20, Thursdays 10 a.m., Fields 230)
Instructors: Roland Speicher (Queen’s), Jamie Mingo (Queen's)

Functional Analysis
(Jul. 3-Aug. 30, Tuesdays 10 a.m., Fields Library)
*To register, please contact Andrew Toms (atoms@mathstat.yorku.ca)


Introduction to Operator Algebras
Audio and Slides

Instructor: Man-Duen Choi (Toronto), Ken Davidson (Waterloo)

Fields Institute
Tuesdays, 1:00--3:00

This course is an introduction to abstract operator algebras. It has roots in the theory of completely positive maps and dilation theory, and Arveson's approach to studying nonself-adjoint algebras via the minimal enveloping C*-algebra, the C*-envelope. We will take much of the material from Vern Paulsen's book ``Completely bounded maps and operator algebras'', Cambridge Studies in Advanced Mathematics vol. 78, Cambridge University Press, 2002.

As the course is divided into ten two hour lectures (Sept.11 through Dec.4, not including Sept. 18 and Nov.13), each lecture will be an overview of the week's topic; but serious students will need to read the omitted proofs and related material on their own.

Students who wish to receive credit should obtain permission from their home university to count the course there, and we can provide a letter stating your participation and level of performance. If you take it for credit, then you will be asked to hand in a written report including exercise solutions and possibly a write-up of some topic related to the course. There will be no exams.

1. Background on C*-algebras
2. CP maps, Stinespring's Theorem.
3. Sz.Nagy and Ando dilation theorems, Commutant lifting theorem.
4. Arveson extension and dilation theorems.
5. The injective and C*-envelopes
6. CB maps, Wittstock, etc.
7. CB homomorphisms, Paulsen's Thm., Haagerup, Christensen.
8. Polynomially bounded operators, Pisier's counterexample.
9. Abstract operator algebras, Blecher--Ruan--Sinclair Thm.
10. Universal operator algebras, factorization, Pisier's similarity degree.

 

Schedule (tentative):(Sep. 11 - Dec. 4, Tuesdays 1 p.m., Fields 230)

Tuesday 11 Sep. 1:00-3:00 p.m. room 230
Tuesday 25 Sep. 1:00-3:00 p.m. room 230
Tuesday 2 Oct. 1:00-3:00 p.m. room 230
Tuesday 9 Oct. 1:00-3:00 p.m. room 230
Tuesday 16 Oct. 1:00-3:00 p.m. room 230
Tuesday 23 Oct. 1:00-3:00 p.m. room 230
Tuesday 6 Nov. 1:00-3:00 p.m. room 230
Tuesday 20 Nov. 1:00-3:00 p.m. room 230
Tuesday 27 Nov. 1:00-3:00 p.m. room 230
Tuesday 4 Dec. 1:00-3:00 p.m. room 230

Structure of C*-Algebras
Audio and Slides

Instructors: George Elliott (Toronto), Chris Phillips (Oregon), Mikael Rørdam (Odense)

Schedule (tentative):(Sep. 11 - Dec. 18, Tuesdays 10 a.m., Fields 230)

Tuesday 11 Sep. 10:00-12:00 room 230
Tuesday 25 Sep. 10:00-12:00 room 230
Tuesday 2 Oct. 10:00-12:00 room 230
Tuesday 9 Oct. 10:00-12:00 room 230
Tuesday 16 Oct. 10:00-12:00 room 230
Tuesday 23 Oct. 10:00-12:00 room 230
Tuesday 6 Nov. 10:00-12:00 room 230
Tuesday 20 Nov. 10:00-12:00 room 230
Tuesday 27 Nov. 10:00-12:00 room 230
Tuesday 4 Dec. 10:00-12:00 room 230
Tuesday 18 Dec. 10:00-12:00 room 230

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Free Probability
Audio and Slides

Instructors: Roland Speicher (Queen’s), Jamie Mingo (Queen's)

The goal of the course is to bring students to the frontier of research in free probability and its relation to random matrices in ten lectures. The lectures will present the basic facts of free probability and show how free probability can simplify and conceptualize the study of random matrices through a careful exploration of basic but instructive examples.

The course will consist of ten lectures and five exercise sets for those taking the course for credit.

Lecture 1

o the distribution of a Gaussian random variable
o moments and cumulants
o multivariate moments and cumulants
o Wick's formula for the moments of a multivariate Gaussian random variable
o the genus expansion for the moments of the GUE
o asymptotic freeness of independent GUE's
o free random variables and free groups

Course Outline

Lecture 1 Excercise1 Problem Set 2
Lecture 2 Excercise 3 Problem Set 4
Lecture 3 Excercise 5  
Lecture 4    
Lecture 5    
Lecture 6    
Lecture 8    
Lecture 10
Thursday 13 Sep. 10:00-12:00
Thursday 27 Sep. 10:00-12:00
Thursday 4 Oct. 10:00-12:00
Thursday 11 Oct. 10:00-12:00
Thursday 18 Oct. 10:00-12:00

Thursday 25 Oct. 10:00-12:00
Applications of Free Probability to von Neumann algebras
In 1990 D. Voiculescu showed that if one takes the group von Neumann algebra of the group F_n and compresses it by a projection of trace 1/k, then one obtains the group von Neumann algebra of F_m where

m - 1 2
_____ = k
n - 1
This week J. Mingo will show how this result is obtained using the techniques of free probability.

Thursday 22 Nov. 10:00-12:00
Free entropy and large deviations
Free entropy is the analogue of the classical notion of entropy in free probability theory. The hope is that eventually it should lead to invariants for von Neumann algebras.
In this lecture, free entropy will be motivated from the point of view of large deviations from Wigner's semicircle law. A moderate introduction to the general idea of the theory of large deviations will also be provided. This lecture will be given by R. Speicher.

Thursday 29 Nov. 10:00-12:00
The second last lecture in the graduate course on free probability will be given by Uffe Haagerup

UFFE HAAGERUP
Brown's spectral distribution measure for operators in finite factor.

Brown's spectral distribution measure was introduced by L. G. Brown in 1983 based on the Fuglede-Kadison determinant in a finite factor and the theory of subharmonic functions in two real variables. The Brown measure has been computed for a number of (non-normal) operators that occur naturally in free probability: Voiculescu's circular operator and more generally R-diagonal operators and elliptic operators. We will discuss the various methods, that have been used for such computations, and also briefly discuss the applications of the Brown measure to construction of invariant subspaces.

Thursday 6 Dec.
Free Entropy and Operator Algebras
This will be the last lecture of the course. Free entropy will be defined for tuples of non-commuting random variables and it will be explained how this concept can be used to prove the absence of property Gamma or of Cartan subalgebras, or the primeness of von Neumann algebras. This lecture will be given by R. Speicher.

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