Graduate Course: Introduction to Operator Algebras
Description
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.
Schedule
| 13:00 to 15:00 |
Background on C*-algebras
Man-Duen Choi, University of Toronto, Kenneth Davidson, University of Waterloo |
| 13:00 to 15:00 |
CP maps, Stinespring's Theorem
Man-Duen Choi, University of Toronto, Kenneth Davidson, University of Waterloo |
| 13:00 to 15:00 |
Sz.Nagy and Ando dilation theorems, Commutant lifting theorem
Man-Duen Choi, University of Toronto, Kenneth Davidson, University of Waterloo |
| 13:00 to 15:00 |
Arveson extension and dilation theorems
Man-Duen Choi, University of Toronto, Kenneth Davidson, University of Waterloo |
| 13:00 to 15:00 |
The injective and C*-envelopes
Man-Duen Choi, University of Toronto, Kenneth Davidson, University of Waterloo |
| 13:00 to 15:00 |
CB maps, Wittstock, etc.
Man-Duen Choi, University of Toronto, Kenneth Davidson, University of Waterloo |
| 13:00 to 15:00 |
CB homomorphisms, Paulsen's Thm., Haagerup, Christensen
Man-Duen Choi, University of Toronto, Kenneth Davidson, University of Waterloo |
| 13:00 to 15:00 |
Polynomially bounded operators, Pisier's counterexample
Kenneth Davidson, University of Waterloo, Man-Duen Choi, University of Toronto |
| 13:00 to 15:00 |
Abstract operator algebras, Blecher--Ruan--Sinclair Thm.
Man-Duen Choi, University of Toronto, Kenneth Davidson, University of Waterloo |
| 13:00 to 15:00 |
Universal operator algebras, factorization, Pisier's similarity degree
Man-Duen Choi, University of Toronto, Kenneth Davidson, University of Waterloo |
