MAGIC040: Operator Algebras

Course details

A specialist MAGIC course

Semester

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

Hours

Live lecture hours
10
Recorded lecture hours
0
Total advised study hours
40

Timetable

Thursdays
10:05 - 10:55 (UK)

Description

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 

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

See description.

Lecturer

  • MD

    Dr Michael Dritschel

    University
    University of Newcastle

Bibliography

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Assessment

The assessment for this course will be released on Monday 11th January 2021 at 00:00 and is due in before Sunday 24th January 2021 at 23:59.

The assessment is by means of a final written examination consisting of seven questions.  Answering at least four of these in a substantially correct manner will suffice to pass the course.

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

Files

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Lectures

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