# NMAK14026U  Topics on type III von Neumann algebras and ergodic theory

Volume 2014/2015
Education
MSc Programme in Mathematics
MSc Programme in Statistics
MSc Programme in Mathematics-Economics
Content

The course will focus on learning basic examples of type III factors
and its connection to ergodic theory.

In the first part we study basic structure theory of type III factors
and (non-singular) ergodic theory, such as

(1) Basic facts about Tomita-Takesaki theory (mostly for faithful
normal states)
(2) S and T-invariant of Connes, ratio set for ergodic group actions
(which is in fact the same as the S-invariant).
(3) Takesaki duality and Flow of weights. Krieger factors coming from
ergodic flows.
(4) Central sequence algebras and full/McDuff property.
(5) Refined version of (1): Sd, Tau-invariant and almost periodic
states. This is important for determining whether a given type III_1

Since the full treatment of modular theory requires a lot of work (left
Hilbert algebra, etc), we do not intend to cover the whole theory and
would rather focus on examples.

In the second part we learn various examples of type III factors and
compute their invariants.
The possible lists are:
Some structural properties of hyperfinite/Araki-Woods type III factors.
Its realizations as group measure space constructions.
Connes' construction of full factors with prescribed tau or
Sd-invariant.
Free product type III factors. They are generically full factors.
Shlyakhtenko's Free Araki-Woods factors. This family gives examples of
type III factors without Cartan subalgebra (Houdayer-Ricard).

If possible we could also go on to type III factors unique Cartan
subalgebra (Houdayer-Vaes' recent result) coming from non-singular
ergodic action of free groups.

Learning Outcome

After completing the course the students are expected:

Knowledge: To know basic properties of modular flows, Connes
classification of type III factors, and several examples.

Skills: To be able to construct examples of type III factors from
ergodic actions and compute S or T-invariants of them (at least for
particular Z-actions).

Competence: To master fundamental concepts for studying type III
factors.

4 hours of lectures and 3 hours of tutorials each week for 9 weeks
Credit
7,5 ECTS
Type of assessment
Continuous assessment
Overall evaluation based on the student's presentation of the proofs during the exercise classes.
Marking scale
passed/not passed
Censorship form
No external censorship
One internal examiner
Re-exam
Oral exam, 30 minutes with 30 minutes preparation time during which all written aids are allowed.
##### Criteria for exam assesment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.

• Category
• Hours
• Lectures
• 36
• Exercises
• 27
• Preparation
• 143
• Total
• 206