Moritz Weber Lehre

Functional analysis 2b (Operator algebras)

(Summer term 2021)

News

Sadly, Prof. Eschmeier passed away recently.

This year's functional analysis 2 lecture is split into two parts:

Functional analysis 2a (Applied Functional Analysis), Prof. Groves, Tue, 10-12, 4.5 CP

Functional analysis 2b (Operator algebras), Prof. Weber, Thu, 10-12, 4.5 CP

You are welcome to attend any of these lectures!

Please register via CMS. You will find a link to MS Teams there.

Lecture

Thursday, 10-12, Teams room "Functional analysis 2b (Operator algebras)"

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), we will focus on the operator algebraic aspects of functional analysis.

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 the theory of C*-algebras covering amongst others the GNS construction, representation theory, and universal C*-algebras. We might briefly mention von Neumann algebras in the end (such as factors and their classification, the hyperfinite factor, and group factors).

Script

We will follow closely Chapters 1-7 of the following script:
ISem24 Lecture Notes

Exercises

The exercise sessions will be held by Marcel Scherer on Fridays, 10-12 via Teams.

How to obtain the credit points

In order to obtain the credit points for this course, you must actively take part at the exercise sessions
(not missing them more than twice) and obtain 50% of the total of all points on the exercise sheets.
You will then be permitted to take part at the oral exams at the end of the term which are the basis
for your grade.

References

Books on operator algebras/C*-algebras:

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

Books and lecture notes on von Neumann algebras:

Claire Anantharaman and Sorin Popa, An introduction to II₁ factors, preliminary version.
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

Updated: 12 April 2021 Moritz Weber

Prof. Dr. Moritz Weber

Alexander Mang