Educational Week on Noncommutative Integration

June 9 - June 13, 2008

Organized by the research school Thomas Stieltjes Institute for Mathematics.
Co-sponsored by the research cluster for Geometry and Quantum Theory GQT, the research school MRI and the Netherlands Organisation for Scientific Research.

Venue: Snellius building, Leiden University.
Organizers: Marcel de Jeu, Erik Koelink, Ben de Pagter and Fedor Sukochev.
Main lecturers: Ben de Pagter and Fedor Sukochev.
Guest speakers: Thierry Fack, Bruno Iochum and Christian Le Merdy.
Contact: noncomint@math.leidenuniv.nl.

Description

In this Educational Week some aspects of noncommutative integration theory associated with von Neumann algebras will be discussed, as well as some of its applications. After a review of the fundamentals of the theory of von Neumann algebras, the noncommutative L^p-spaces will be introduced for semi-finite von Neumann algebras, as well as the (Haagerup) L^p-spaces in the type III setting. The duality theory for these spaces (including e.g. the noncommutative Hölder inequality) will be discussed. Moreover, some attention will be given to the geometric properties of these L^p-spaces (exhibiting the specific differences with the classical (commutative) L^p-spaces).
After these introductory lectures a selection of the following topics will be treated:
- operator functions and perturbation theory in noncommutative spaces;
- Dixmier traces and their applications;
- Connes trace formulas;
- operator zeta-function formulas;
- noncommutative heat semigroups;
- spectral triples and spectral flows;
- noncommutative Chern character and Hochschild cohomology.
During the week there will be ample time for informal discussions and to become acquainted with the material by working on problem sets. There will be lecture notes available.

Lecture notes

There are already some notes available on the following topics:
Review of operator theory and operator algebras;
Noncommutative Banach function spaces;
Noncommutative L^p-spaces and their generalizations;
Matricial structure of the Haagerup L^p-spaces;
Dixmier traces.
Some of these notes well exceed what will/can be covered about these topics during the week and it is not necessary to study them beforehand, but people interested may use them to their advantage.
Exercises are to be found here.
Slides of presentations used by Fedor Sukochev during the week (see below for the slides of the guest speakers):
Noncommutative L^p-spaces and their generalizations;
Matricial structure of the Haagerup L^p-spaces.

Prerequisites

A short reminder of the basic theory of C^*-algebras and von Neumann algebras theory will be presented, but participants will benefit most of this week by studying beforehand the following material in Takesaki's "Theory of Operator Algebras I": Chapter I, Sections 1-6; Chapter II, Section 3; Chapter V, Sections 1 and 2.

Audience

This week is intended for advanced students, PhD-students, postdocs and other people interested.

Staff

The lectures will be given by Ben de Pagter (Delft) and Fedor Sukochev (Adelaide).
Assistance at the problem sessions will be given by Marcel de Jeu and Erik Koelink.

Dates

Start: Monday June 9, 2008, 9.30-10.00: arrival of participants at the Snellius building.
End: Friday June 13, 2008, 16.00: closing reception with drinks.

Schedule and guest speakers

On Monday through Thursday we have the following day schedule:

On Friday the schedule is different as there will be invited lectures by the guest speakers:

Thierry Fack

Singular traces in noncommutative geometry

Spectral action in noncommutative geometry

Christian Le Merdy

Rademacher averages on noncommutative symmetric spaces

Abstracts:

Thierry Fack: Singular traces in noncommutative geometry .
We give some motivations to introduce singular traces in noncommutative geometry, including the explicit computation of the Dixmier trace in the case of Riemannian spin manifolds. We also explain how to use Dixmier traces in the computation of the index map associated with a spectral triple. Finally, we give a characterization of finite sums of commutators in Banach function spaces, which is used to discuss the existence and spectrality of singular traces on ideals of operators.
Slides.

Bruno Iochum: Spectral action in noncommutative geometry.
Some motivations from physics are given to justify the notion of spectral action for a given spectral triple. Its explicit computation deals with the heat kernel approach, Dixmier traces and zeta functions. Different examples are considered: Riemannian spin manifolds, the noncommutative torus and the quantum groups related to SU(2).
Slides.

Christian Le Merdy: Rademacher averages on noncommutative symmetric spaces
Let E be a separable (or the dual of a separable) symmetric function space, let M be a semifinite von Neumann algebra and let E(M) be the associated noncommutative function space. Our aim is to discuss Rademacher averages and Khintchine type inequalities on this space. We will actually review various classical results regarding these averages, starting with the scalar Khintchine inequalities, and including the the noncommutative Khintchine inequalities on noncommutative L^p spaces.
Slides.

Fee

Participation is free. Lunch and drinks will be provided by the organization.

Support

We are trying to obtain funds for limited partial support for participants from outside the Netherlands. Those who wish to be eligible for such partial support should apply before Monday April 21 by sending an email to the organizers, mentioning the amount applied for and also including an estimate of the total expenditure for travel and accommodation, specifying each. The organization will then decide on the support in the first week of May.
Important: in view of the limited expected amount of available funds, we urgently ask that you only apply for support by the organization as a last resort.

Registration and accommodation

To register, send an email to the organizers, mentioning your affiliation (university or employer), situation (student + year, PhD-student, postdoc, professor,...) and field.
There is accommodation available at hotel Het Haagsche Schouw at walking distance (1500 m) from the institute. Standard hotel accommodation costs approximately 85 euro (single occupancy without breakfast) or 42.50 euro (double occupancy without breakfast) per person per night, taxes included. Breakfast is 12.50 euro per person. When registering, please indicate clearly whether you need a hotel and, if so, for which nights and whether you prefer single or double occupancy. The hotel has agreed to guarantee accommodation for names we pass on by April 30; after that it will be on availability.

Participants

Frederic Albert, Metz
Martijn Caspers, Nijmegen
Gilles Castro, Orleans
Sonja Cox, Delft
Malgorzata Czerwinska, Memphis
Steven Deprez, Leuven
Sjoerd Dirksen, Delft
Sebastien Falguieres, Leuven
Tobias Fritz, Bonn
Stanislaw Goldstein, Lodz
Wolter Groenevelt, Delft
Nikolai Ivankov, Bonn
Evgenios Kakariadis, Athens
Krysztof Kaniowski, Lodz
Damian Kubiak, Memphis
Anna Kula, Krakow
Katarzyna Lubnauer, Lodz
Jan Maas, Delft
Tom Melotte, Leuven
Pieter Naaijkens, Nijmegen
Jan van Neerven, Delft
Mihai Popa, Bloomington
Maarten van Pruijssen, Nijmegen
Marie Sabbe, Leuven
Richard Sanders, Nijmegen
Piotr Soltan, Warsaw
Christian Svensson, Leiden/Lund
Reiji Tomatsu, Leuven
Stefan Wagner, Darmstadt
Marten Wortel, Leiden
Janusz Wysoczanski, Wroclaw
Chen Yang, Durham
Ioannis Zarakas, Athens
Dapeng Zhang, Bonn
Bartek Zielinski, Lodz

Here is a picture, taken on Friday when the guest speakers were present but some of the participants had already had to leave.

Further information

For further information, contact Marcel de Jeu or Erik Koelink. There is general information on Stieltjes Educational Weeks available and on that page you will also find links to the pages of previous weeks.

Maintained by Marcel de Jeu
Last modified: June 16, 2008