What is a free group factor

Prof. Dr. Roland memory

Jun.-Prof. Dr. Moritz Weber

Seminar / advanced seminar
to operator algebras and free probability theory

(Summer semester 2015)

Current

Another room change: The seminar is now taking place in SR 2!

time and place

Mon 10-12, SR 2 (E2 5)

  • Monday, April 20 --- no meeting
  • Monday, April 27 Pascal
    Lecture Gelfandt theory
  • Monday, 4.5. Simon
    Lecture C * -Algebras
  • Monday, May 11th --- no meeting
  • Monday, May 18 Meeting to discuss the first reading
    Gelfandt theory and C * algebras
  • Monday, May 25 --- no meeting
  • Monday, 1.6. Felix
    Lecture Von Neumann Algebras
  • Monday, 8th June --- no meeting
  • Monday, June 15 Meeting to discuss the second reading
    Von Neumann algebras
  • Monday, June 22nd Marvin
    Lecture Free Probability Theory
  • Monday, June 29th --- no meeting
  • Monday, 6.7. --- Meeting to discuss the third reading
    Free probability theory
  • Monday, July 13th --- no meeting
  • Monday, July 20 --- Meeting to discuss the fourth reading
    Free probability theory
  • Monday, July 27 --- no meeting

content

In order to study operators on the Hilbert space it is often helpful not only to consider individual operators,
but the algebras they generate, provided with a topological closure. The study of such
Operator algebras provide e.g. the functional calculus, one of the most powerful tools in functional analysis.
In addition, some methods and thought patterns from algebra can also be developed for operators on Hilbert spaces
become. In a way it is the study of algebras of (complex valued) functions that do not commute,
so where fg = gffor functions fand Gis no longer fulfilled. Such mathematics is used, among other things, for quantum mechanics
needed, but also for some novel concepts of non-commutative geometry or non-commutative analysis.

Following on from the functional analysis, we will develop the basics of Gelfandt theory and GNS construction,
C *- Define and Von Neumann algebras and study their basic properties and finally in a
current and very active research area immersing free probability theory. The latter is historically off
a still (for more than 70 years) open problem about Von Neumann algebras arose (the free
Group factor problem), but has now become an independent mathematical subject with relationships
developed for functional analysis, combinatorics, function theory, random matrices and much more. We will
Choose operator-algebraic approach and treat the basic concepts of freeness and non-commutative distributions.

Bachelor or master theses can be awarded after the seminar.

Announcement of the seminar

literature

Jacques Dixmier, Les C * -algebres et leurs representations, 1969.
Gerard Murphy, C * algebras and operator theory, 1990.
Bruce Blackadar, Operator algebras. Theory of C * -algebras and von Neumann algebras, 2006.
Kenneth Davidson, C * algebras by example, 1996.
Alexandru Nica and Roland Speicher, Lectures on the Combinatorics of Free Probability, 2006 (call number Bibl. Inf + Math: LMS 335).

If you have any further questions, please contact Moritz Webermelden!