MAGIC040: Operator Algebras

Course details

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

Course forum

Visit the MAGIC040 forum

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

Coming soon

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Files

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Week File
1-10 lecture-1.pdf Lecture
1- rl.pdf Lecture
1-10 lecture-2.pdf Lecture

Recorded Lectures

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