Pure Math 822 - Dilation Theory - Winter 2013
Meeting Time: Tuesday and Thursday 9:00 -- 10:30
Room: MC 5046
Course outline
This course is an introduction to dilation theory and 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.
- Background on C*-algebras and operator theory
- Sz.Nagy and Ando dilation theorems, Commutant lifting theorem.
- Completely positive maps, Stinespring's Theorem.
- Arveson extension and dilation theorems.
- C*-envelope (Dritschel--McCullough, Arveson approach).
- Completely bounded maps, Wittstock's theorem.
- Completely bounded homomorphisms (Paulsen, Haagerup, Christensen)
- Polynomially bounded operators, Pisier's counterexample.
Time permitting:
- Universal operator algebras, factorization, Pisier's similarity degree.
- Dilation theory and semicrossed products.
Required Background
The student needs a good course in functional analysis. Some further
exposure to operator theory or operator algebras would be an asset.
Reference Texts
The main text for the course will be:
Other useful references are:
- B. Sz. Nagy and C. Foias (with H. Bercovici and L. Kerchy),
Harmonic analysis of operators on Hilbert space, 2nd ed., Springer 2010.
Online version available through UW Library.
- G. Pisier, Completely bounded maps and similarity problems, 2nd ed., Springer, 1996.
Online version available through UW Library.
- K.R. Davidson and E. Katsoulis, Dilation theory, commutant lifting and semicrossed products,
Documenta Math. 16 (2011) 781-868.
Grading
| 1. Problem Sets | 70 |
| 2. Talk and paper | 30 |
Topics for student seminars
Assignments
Solutions
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