MAGIC040: Operator Algebras

Course details


Autumn 2020
Monday, October 5th to Friday, December 11th


Live lecture hours
Recorded lecture hours
Total advised study hours


10:05 - 10:55

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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


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