Lecturers: Roland Speicher and Moritz Weber
Assistant: Luca Junk
- The language of the course is English by default, unless all participants speak German.
- In this lecture, which is formally a continuation of the lecture Functional Analysis (Funktionalanalysis) held in the previous semester, we will focus on the operator algebraic aspects of functional analysis.
- We will mainly follow the manuscript Functional Analysis II by Tobias Mai and Moritz Weber.
- Problem Set 1 (due April 18)
- Problem Set 2 (due April 25)
- Problem Set 3 (due May 2)
- Problem Set 4 (due May 9)
- Problem Set 5 (due May 16)
- Problem Set 6 (due May 23)
- Problem Set 7 (due May 30)
- Problem Set 8 (due June 6)
- Problem Set 9 (due June 13)
- Problem Set 10 (due June 20)
- Problem Set 11 (due June 27)
- Problem Set 12 (due July 4)
- Problem Set 13 (due July 11)
- Problem Set 14 (optional)
Time and Place
The lectures and tutorials will be held in person.
Mondays 14-16 and Wednesdays 10-12 in HS IV, E 2.4
Wednesdays 12 -14 in SR 5
Operator algebras are generalizations of matrix algebras to the infinite dimensional setting; they are given as subalgebras of the algebra of all bounded linear operators on some Hilbert space that are invariant under taking adjoints and closed with respect to some specific topology. Roughly speaking, operator algebras are used to study by algebraic means the analytic properties of several operators simultaneously; their theory thus combines in a fascinating way linear algebra and analysis.
The most prominent examples of such operator algebras are C*-algebras and von Neumann algebras, which show a very rich structure and have various applications both in mathematics and physics, especially in quantum mechanics.
Whereas the former have a more topological flavour (and their theory is thus often addressed as non-commutative topology), the latter has more measure theoretic and probabilistic sides and gives rise to non-commutative measure theory and non-commutative probability theory. We give an introduction to both the basics and some more specialized topics of the theory of C*-algebras (such as the GNS construction, their representation theory, and universal C*-algebras) and von Neumann algebras (such as factors and their classification, the hyperfinite factor, and group factors).
- Claire Anantharaman and Sorin Popa, An introduction to II1 factors, preliminary version.
- Bruce Blackadar, Operator algebras. Theory of C*-algebras and von Neumann algebras, 2006.
- Kenneth Davidson, C*-algebras by example, 1996.
- Jacques Dixmier, Les C*-algebres et leurs representations, 1969.
- Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras.
Volume I-IV, 1997.
- Gerard Murphy, C*-algebras and operator theory, 1990.
- Masamichi Takesaki, Theory of Operator Algebras I-III, 2002/2003
- Script by Vaughan Jones, Berkeley, 2009.
- Lecture Notes Von Neumann algebras and ergodic theory of group actions, IHP 2011.
- Script by Jesse Petersen, Vanderbilt, 2013.
- Script by Sven Raum, Münster, 2015/2016.
- Lecture Notes Von Neumann Algebras, Subfactors, Knots and Braids, and Planar Algebras
by Roland Speicher, Saarbrücken, 2016.
- Lecture Notes by Cyril Houdayer, Paris
- Lecture Notes of the 24th Internet Seminar on C*-algebras and dynamics by Xin Li, Christian Voigt and Moritz Weber
Department of Mathematics
Postfach 15 11 50
Campus building E 2 4