MAGIC040: Operator Algebras

Course details

A specialist MAGIC course


Autumn 2022
Monday, October 3rd to Friday, December 9th


Live lecture hours
Recorded lecture hours
Total advised study hours


10:05 - 10:55 (UK)


I. C*-algebras (3 lectures) 
  • 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 


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. 


See description.


  • MD

    Dr Michael Dritschel

    University of Newcastle


No bibliography has been specified for this course.


The assessment for this course will be released on Monday 9th January 2023 at 00:00 and is due in before Sunday 22nd January 2023 at 23:59.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.


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