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General


Description

I. C*-algebras (3 lectures)
  1. Definitions
  2. Abstract vs concrete algebras
  3. Linear functionals, states and representations
  4. The GNS construction and the Gel'fand and Gel'fand-Naimark theorems, characterizing abstract C*-algebras
  5. Ideals and approximate units
  6. Multipliers
  7. Tensor products

II. Completely bounded and completely positive maps (3 lectures)
  1. Positivity/boundedness and complete positivity/boundedness
  2. The Stinespring representation theorem and Arveson extension theorem
  3. The Wittstock decomposition theorem for completely bounded maps, and the Haagerup-Paulsen-Wittstock theorem

IV. Operator Spaces and Algebras (4 lectures)
  1. Abstract vs concrete operator spaces, systems and algebras
  2. The Effros-Ruan theorem, characterizing abstract operator systems
  3. Ruan's theorem, characterizing abstract operator spaces
  4. The Blecher-Ruan-Sinclair theorem, characterizing abstract operator algebras

Semester

Spring 2017 (Monday, January 23 to Friday, March 31)

Timetable

  • Thu 11:05 - 11:55

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

I. C*-algebras (3 lectures)
  1. Definitions
  2. Abstract vs concrete algebras
  3. Linear functionals, states and representations
  4. The GNS construction and the Gel'fand and Gel'fand-Naimark theorems, characterizing abstract C*-algebras
  5. Ideals and approximate units
  6. Multipliers
  7. Tensor products

II. Completely bounded and completely positive maps (3 lectures)
  1. Positivity/boundedness and complete positivity/boundedness
  2. The Stinespring representation theorem and Arveson extension theorem
  3. The Wittstock decomposition theorem for completely bounded maps, and the Haagerup-Paulsen-Wittstock theorem

IV. Operator Spaces and Algebras (4 lectures)
  1. Abstract vs concrete operator spaces, systems and algebras
  2. The Effros-Ruan theorem, characterizing abstract operator systems
  3. Ruan's theorem, characterizing abstract operator spaces
  4. The Blecher-Ruan-Sinclair theorem, characterizing abstract operator algebras

Lecturer


Michael Dritschel
Email m.a.dritschel@ncl.ac.uk
Phone (0191) 2227229
Interests operator theory, operator algebras, function theory
vcard
Photo of Michael Dritschel


Students


Photo of Amjad Alghamdi
Amjad Alghamdi
(Leeds)
Photo of Paul Druce
Paul Druce
(Nottingham)
Photo of Neil Hansford
Neil Hansford
(Sheffield)
Photo of Dale Hodgson
Dale Hodgson
(Leeds)
Photo of Tanmay Inamdar
Tanmay Inamdar
(East Anglia)
Photo of Ben Jones
Ben Jones
(Cardiff)
Photo of James Kulmer
James Kulmer
(York)
Photo of Rosario Mennuni
Rosario Mennuni
(Leeds)
Photo of Francesco Parente
Francesco Parente
(East Anglia)
Photo of Joakim Stromvall
Joakim Stromvall
(Surrey)


Bibliography


C*-algebras and operator theoryMurphy
C*-algebras by exampleDavidson
C*-algebrasDixmier
Fundamentals of the Theory of Operator Algebras: Elementary theoryKadison and Ringrose
Fundamentals of the Theory of Operator Algebras: Advanced theoryKadison and Ringrose
Completely bounded maps and operator algebrasPaulsen
Hilbert C*-modules: a toolkit for operator algebraistsLance
Hilbert C*-modulesManuĭlov and Troit︠s︡kiĭ
Operator algebras and their modules: an operator space approachBlecher and Merdy
Operator algebras: theory of C*-algebras and von Neumann algebrasBlackadar
What are operator spaces?G. Wittstock, et al.


Note:

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



The course will be assessed via a final exam consisting of seven questions. Answering at least four correctly will constitute a pass. The exam will be made available in April. Students will be given two weeks to complete this and submit it on-line.

MAGIC040 final exam

Files:Exam paper
Released: Monday 24 April 2017 (5.0 days ago)
Deadline: Sunday 7 May 2017 (9.0 days to go)
Instructions:

Students are encouraged to attempt all questions. Successfully completeing four of the seven will constitute a pass.



Files


Files marked L are intended to be displayed on the main screen during lectures.

Week(s)File
1-10lecture-1.pdfL
1-10lecture-2.pdfL
1-10lecture-3.pdfL
1-10lecture-4.pdfL
1-10lecture-5.pdfL
1-10lecture-6.pdfL
1-10lecture-7.pdfL
1-10lecture-8.pdfL
1-10lecture-9.pdfL
1-10reading_list.pdf


Recorded Lectures


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