s Freie Wahrscheinlichkeit Lehre

## Von Neumann algebras, subfactors, and planar algebras

(Sommersemester 2016)

### Vorlesung

Mo und Do, 12 - 14, im SR 6, Geb. E2 4

Operator algebras are generalizations of matrix algebras to the infinite dimensional setting; their theory, however,
becomes much more involved and combines linear algebra and analysis. There are two main classes of operator
algebras: C*-algebras and von Neumann algebras. 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.
Von Neumann algebras themselves are already very intriguing, but the theory becomes even more interesting if one
tries to understand subfactors, i.e., the question how one von Neumann algebra can be embedded into another one.
Vaughan Jones addressed this question in the 1980's and found an amazing link to knot theory. In the end this resulted
in a new invariant for knots, the Jones polynomial, and earned Jones the Fields Medal.
Whereas the first investigations of the subfactor problem were quite analytical, Jones introduced, motivated by the
relation with knots, in the 1990's a more combinatorial and diagrammatical description, which goes under the name of
planar algebras. This can on one side be seen as a special example of the more general theory of operads, but has
on the other side also a very planar, i.e., non-crossing structure, which makes it resemble the combinatorics of free
probability. There has been some interesting consequences coming out of this apparent connection, but the final word
on this has not yet been spoken.

In the lecture, I will try to give an introduction into this circle of ideas and I will hopefully also convey some of the
excitement of the subject. Formally, the lecture is a continuation of Funktionalanalysis 2 by M. Weber from the
Winter term 15/16. I will recall (but not prove) all the needed definitions and facts on von Neumann algebras and
then present in detail the analytical and diagrammatic theory of subfactors and planar algebras. So it would be good
to have some prior knowledge on operator algebras, and perhaps von Neumann algebras, but no prerequisites on
subfactors or planar algebras are assumed.

Ankündigung

Eintrag im KVV

Fragen zur Vorlesung können gerne an Tobias Mai gerichtet werden.

### Übungen

Die Übungen finden statt:    Di, 16:00 - 17:30 Uhr, SR 6

Übungsblatt 1
Übungsblatt 2
Übungsblatt 3
Übungsblatt 4
Übungsblatt 5
Übungsblatt 6

### Scheinvergabe

Durch regelmäßige und aktive Teilnahme an der Vorlesung und an den Übungen
wird die Zulassung zur Prüfung erworben. Das Bestehen der Prüfung ist die
Voraussetzung für den Schein und die Grundlage der Note.

### Literatur

Operator Algebras: Theory of C*-Algebras and von Neumann Algebras, Springer

V. Jones
Index for subfactors, Invent. Math., Vol. 72 (1983), No. 1, 1-25

V. Jones, D. Shlyakhtenko, and K. Walker
An orthogonal approach to the subfactor of a planar algebra, ArXiv Preprint (2008)
appeared in Pacific Journal of Mathematics, Vol. 246 (2010), No. 1, 187-197

A. Guionnet, V. Jones, D. Shlyakhtenko
Random matrices, free probability, planar algebras and subfactors, ArXiv Preprint (2007)
appeared in Quanta of maths, 201-239, Clay Math. Proc., 11, Amer. Math. Soc., Providence, RI (2010)

 Aktualisiert am: 29. Juli 2016  Tobias Mai Impressum