General
Description
I. C*-algebras (3 lectures)
II. Completely bounded and completely positive maps (3 lectures)
IV. Operator Spaces and Algebras (4 lectures)
Semester
- Definitions
- Abstract vs concrete algebras
- Linear functionals, states and representations
- The GNS construction and the Gel'fand and Gel'fand-Naimark theorems, characterizing abstract C*-algebras
- Ideals and approximate units
- Multipliers
- Tensor products
II. Completely bounded and completely positive maps (3 lectures)
- Positivity/boundedness and complete positivity/boundedness
- The Stinespring representation theorem and Arveson extension theorem
- The Wittstock decomposition theorem for completely bounded maps, and the Haagerup-Paulsen-Wittstock theorem
IV. Operator Spaces and Algebras (4 lectures)
- Abstract vs concrete operator spaces, systems and algebras
- The Effros-Ruan theorem, characterizing abstract operator systems
- Ruan's theorem, characterizing abstract operator spaces
- The Blecher-Ruan-Sinclair theorem, characterizing abstract operator algebras
Autumn 2011 (Monday, October 10 to Friday, December 16)
Timetable- Thu 13:05 - 13:55
Lecturer
Michael Dritschel
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Students
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Abdulsatar Al-Juburie
(Newcastle) |
Alex Bailey
(Southampton) |
Matthew Gadsden
(Sheffield) |
Paul Jones
(Loughborough) |
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Umberto Lupo
(York) |
Joshua Tattersall
(Leeds) |
Bryan Williams
(Liverpool) |
Yiwei Zhang
(Exeter) |
Prerequisites
A working knowledge of functional analysis and operator theory, as well as some topology, as provided in, for example, MAGIC061. We lightly skirt over some of this material in the first couple of lectures.
Syllabus
No syllabus information is available yet.
Bibliography
| C*-algebras and operator theory | Murphy |
| C*-algebras by example | Davidson |
| C*-algebras | Dixmier |
| Fundamentals of the Theory of Operator Algebras: Elementary theory | Kadison and Ringrose |
| Fundamentals of the Theory of Operator Algebras: Advanced theory | Kadison and Ringrose |
| Completely bounded maps and operator algebras | Paulsen |
| Hilbert C*-modules: a toolkit for operator algebraists | Lance |
| Hilbert C*-modules | Manuĭlov and Troit︠s︡kiĭ |
| Operator algebras and their modules: an operator space approach | Blecher and Merdy |
| Operator algebras: theory of C*-algebras and von Neumann algebras | Blackadar |
| What are operator spaces? | G. Wittstock, et al. |
Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.)
Assessment
Throughout the lectures you will find ∗s in the margins. These indicate material that has been stated without justification. You are asked to fill in the details on as many of these as you can. The material will be collected at the end of the semester, and you are asked to provide your work in electronic form as a pdf file, preferably produced from a TeX source file.
Assignments
MAGIC040 Course Work
| Deadline: | Friday 16 December 2011 (551.2 days ago) | ||
| Instructions: | Throughout the lectures you will find ∗s in the margins. These indicate material that has been stated without justification. You are asked to fill in the details on as many of these as you can. The material will be collected at the end of the semester, and you are asked to provide your work in electronic form as a pdf file, preferably produced from a TeX source file. The due date is 16 December. | ||
Files
Files marked L are intended to be displayed on the main screen during lectures.
| Week(s) | File | |
|---|---|---|
| 1-10 | lecture-1.pdf | L |
| 1-10 | lecture-10.pdf | L |
| 1-10 | lecture-2.pdf | L |
| 1-10 | lecture-3.pdf | L |
| 1-10 | lecture-4.pdf | L |
| 1-10 | lecture-5.pdf | L |
| 1-10 | lecture-6.pdf | L |
| 1-10 | lecture-7.pdf | L |
| 1-10 | lecture-8.pdf | L |
| 1-10 | lecture-9.pdf | L |
| 1-10 | Topology_&_FA_lecture_notes.pdf | L |