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Operator Algebras - A Tool Kit, fall 2014


Kay Schwieger
Stefan Wagner


5 sp.


Advanced studies


The course is intended for advanced undergraduate students of mathematics and physics. Prior courses in topology or functional analysis are recommended but not required, and all the necessary theory will be developed during the course.


The theory of operator algebras emerges as a central tool in many branches of mathematics. Among them are combinatorics, geometry, group theory, logic, and stochastics. For instance the classical notions of points and sets (e. g. in analysis or topology) can be replaced by operator algebraic notions often leading to an alternative point of view on problems. Moreover, operator algebras fruitfully provide input for the foundations of new branches of mathematics (e. g. non-commutative geometry). But also outside of mathematics operator algebras have many applications for instance in physics and computer science. In this context quantum mechanics is the most prominent example. In fact, the heart of quantum mechanics is its mathematical formalism in terms of operator algebras.


In this course we will introduce the fundamentals of operator algebras and provide a standard tool kit for the theory. We will also discuss leading examples, of which some are still subject to state-of-the-art research.


Weeks 36-42 and 44-50, Tuesday 14-16 in room C122. Two hours of exercise classes per week.




  • J. W. Conway, “A Course in Functional Analysis”, Graduate Texts in Mathematics 96, Springer, 2007.
  • R. Kadison and R. Ringrose, “Fundamentals of the Theory of Operator Algebras Volume I, II”, Academic Press, 1983, 1986.
  • G. J. Murphy, “C*-Algebras and Operator Theory”, Academic Press, 1990.
  • G. K. Pedersen, “Analysis Now”, Graduate Texts in Mathematics 118, Springer, 1989.


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Exercise class

1.Monday 14-16 B321  

Exercise Sheets

Exercise 11

Exercise 10

Exercise 9

Exercise 8

Exercise 7 

Exercise 6

Exercise 5 

Exercise 4 

Exercise 3

Exercise 2

Exercise 1, Solution for Exercise 3

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  1. Unknown User (

    The first exercise sheet can be found here. The general idea is to discuss the exercises during the exercise class (maybe in small groups). So it is not required to solve them in advance, but of course you are free to do that.

  2. Unknown User (

    Here you find a possible solution for Exercise 3 on Sheet 1.

  3. Unknown User (

    The second exercise sheet can be found here.

  4. Unknown User (

    The third exercise sheet can be found here.

  5. The 4th exercise sheet can be found in the new section "Exercise Sheets".