### EuroSummer School: Instructional Conference on
Operator Algebras and Operator Spaces
*5-14 April*Principal
Organisers: Chris Lance (Leeds) and Allan Sinclair (Edinburgh)
**Scientific Committee: ** Vaughan Jones (UC Berkeley), G K
Pedersen (Copenhagen), Gilles Pisier (Paris), Sorin Popa (UCLA) and
Georges Skandalis (Paris)
Supported by: European
Commission (Framework V) and the London Mathematical Society
The latest
in a series of instructional meetings in analysis, the aim of the meeting was
to introduce this active field of mathematics to younger scientists and to
provide an opportunity for specialists to exchange ideas. The scientific
programme consisted of eight series of three lectures and seven tutorials.
Several themes connected the different lecture series with cross-references
common amongst the talks. Exact C*-algebras was one theme running through the
first week’s lectures. Free products and free probability linked two of
the series of lectures and subfactors were the final linking idea. In addition
to their three hour-long talks, each lecturer also had the opportunity to take
some extra time in the tutorial periods: some used this, some did not. The
first three lectures – by Anantharaman-Delaroche, Pisier and Kirchberg
– set the tone for the conference with very high standards of presentation
and exposition. Open problems were mentioned in all the talks. Proofs, either
full or sketched, were given to many results in all lecture courses and these
were amplified as necessary in the tutorials. A summary of each lecturer’s
talks follow.
Claire Anantharaman-Delaroche: ‘Groups, dynamical
systems and their C*algebras: a survey on amenability, exactness and uniform
embeddings.’ These lectures covered the basic theory of nuclear
C*-algebras, amenable groups, amenable dynamical systems, and uniform
embeddings in Hilbert space. The maximum and minimum tensor of C*-algebras, and
the full and reduced crossed products were introduced during the course of the
lectures. During one tutorial Claire gave a brief introduction to the Novikov
conjecture in response to a request for a description of it. The level of these
lectures went from introductory to advanced; they were excellently presented.
Gilles Pisier: ‘Local theory of operator spaces’: These
demanding and well-presented lectures were a study of the asymptotic properties
of the metric space of all n-dimensional operator spaces as n tends to
infinity: the relevant background and some applications were discussed.
Operator spaces, Ruan’s Theorem, dual operator spaces, quotients, tensor
products and the usual n-dimensional operator spaces formed from rows, columns,
generators of the free group algebra (both reduced and full) were discussed in
some detail. The n-dimensional self-dual operator Hilbert space and infinite
dimensional one, OH, were defined and their crucial properties mentioned. WEP
and QWEP were defined for C*-algebras and relationships with exact operator
spaces were studied. The compactness of the space of all n-dimensional Banach
spaces was contrasted with the strong lack of compactness of the space of all
n-dimensional operator spaces for n greater than 2.
Eberhard Kirchberg:
‘Exact C*-algebras and groups’: This intense course of lectures
defined and studied exact C*-algebras and groups listing several equivalent
conditions on an algebra, or group, for it to be exact. Technical conditions
involving tensor products were introduced and studied. Kirchberg had himself
discovered many of the important properties of C*-algebras discussed.
Uffe Haagerup: ‘Free probability, random matrices and exact
C*-algebras’: These lectures defined and studied free probability and
random matrices, and used them to sketch a proof of the key theorem of the
speaker and Steen Thorbjornsen. Voiculescu’s circular and semi-circular
systems were introduced carefully and the analogies between free and ordinary
probability were drawn. The applications of these free methods to the states on
the K-zero group of an exact C*-algebra was given.
Ken Dykema:
‘Free products of C*-algebras’: The free product of C*-algebras with
fixed states were defined and studied thoroughly with careful calculations of
several examples to illustrate how these products behave. Full proofs were
given of several important results including Haagerup’s estimates on the
norm of linear combinations of words of length k in the reduce C*-algebra of a
free group. Stable rank one was defined and the cases of when free products had
this property were studied.
Dietmar Bisch: ‘Introduction to
subfactors’ and ‘Combinatorial aspects of subfactors’: The first
lecture was a rapid development of the basic theory of von Neumann algebras
with a brief discussion of classification, the coupling constant and Jones
index for subfactors of a type II1 factor. The second lecture covered the
‘basic construction’, the tower of II1 factors and the proof of
Jones’ theorem on the values of the index. His final lecture covered the
Popa axiomatization of the tower of II1 factors associated with a subfactor of
finite index, the standard lambda lattice invariant and bimodules. These
lectures were very clearly presented and an excellent foundation to the most
recent results in subfactors.
Marc Rieffel: ‘Non-commutative
metric spaces’: Lip-norms on the space of continuous functions on a
compact Hausdorff space were introduced and used to motivate the corresponding
non-commutative ideas. The Gromov-Hausdorff distance between two metric spaces
was revised before the non-commutative Gromov-Hausdorff distance between two
unital C*-algebras was defined and studied. These ideas were related to
non-commutative tori and the action of compact Lie groups on C*-algebras.
Vaughan Jones: ‘Consequences of annular invariance for
subfactors’: Planar algebras were introduced and the basic axioms
carefully stated. The theorem showing that a standard lambda lattice is
isomorphic to a planar algebra with positivity was stated and discussed. The
principal graph of an inclusion of subfactors and its relation to planar
algebras was discussed as a route to showing that there are no subfactors with
principal graphs Dn, with n odd, or E7. The ideas in these lectures were
illustrated through out by diagrams of tangles, their shading and the * points.
Tutorials were chaired by Chris Lance and Rob Archbold. All
participants were invited to submit questions, which were then edited by the
chairmen. Major questions were presented to the lecturers well before the
tutorial so that they had time to prepare their answer [e.g. ‘What is the
Novikov conjecture?’]. The remaining questions were read out by the
chairman in the tutorial. Questions were also invited at several stages during
the afternoon from the floor and were forthcoming from both novices and
experts. Having the chairman act as speaker elicited a good flow of questions
from students and younger mathematicians. The lecturers brought up problems
that had been discussed informally over lunch. All the lecturers made a great
effort to deal with the questions thoroughly. The tutorials were one of the
great successes of the conference.
The meeting was held in the James
Clerk Maxwell Building of the University of Edinburgh. In all, there were 65
participants, including the 21 younger scientists from Belgium, Denmark,
France, Germany, Poland, Portugal, Romania, Slovenia and the UK funded by the
EC award.
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**Participants: **
Anantharaman-Delaroche, Claire,
Universite de Orleans Archbold, Robert, University of Aberdeen Beggs,
Edwin J, University of Wales Swansea Belinschi, Serban Teodor, The Romanian
Academy Bisch, Dietmar, UC Santa Barbara Boulton, Lyonell, King’s
College London Catterall, Stephen, University of Glasgow Collins,
Benoit, Ecole Normale Superieure de Paris Curtis, Robyn, University of
Geneva Deicke, Klaus, University of Paderborn Desmedt, Pieter,
University of Leuven Drivaliaris, Dimos, University of Edinburgh
Drnovsek, Roman, University of Ljubljana Dykema, Ken J., Texas A&M
University Garling, David (Ben), London Mathematical Society Grasselli,
Matheus da R, King’s College London Haagerup, Uffe, Odense
University Halanay, Andrei, Politechnica University of Bucharest
Harnisch, Hergen, Humboldt Universitaet, Berlin Haworth, Paul,
Lancaster University Hazel, Graham, Edinburgh University Hjelmborg,
Jacob, Odense University Iordache, Adrian, The Romanian Academy Jones,
Vaughan, University of California at Berkeley Katavolos, Aristides,
University of Athens Kirchberg, Eberhard, Humboldt Universitat Klein,
Silvius, Romanian Academy Kleveland, Rune, University of Oslo Kramar,
Marjeta, University of Ljubljana Lance, Chris, Leeds University
Laustsen, Niels Jakob, University of Leeds Leobacher, Gunther,
University of Salzburg Lioudaki, Vasiliki, University of Edinburgh
Louvet, Nicholas, Universite de Neuchate Lykova, Zinaida, Unviersity of
Newcastle Martinez, Antonio, Uinversity of Oviedo Martins, Nuno,
Instituto Superior Tecnico, Lisbon Mason, Colin, Kings College London
Ng, Chi-Keung, Queens University, Belafast Ozawa, Narutaka, Tokyo
University Pardo, Enrique, Unviersity of Cadiz Parnovski, Leonid,
University of Sussex Perera, Francesc, Universitat Autonoma de Barcelona
Pinto, Paulo, University of Wales, Cardiff Pisier, Gilles, Univ Paris
VI (currently) Pop, Ciprian, The Romanian Academy Power, Stephen,
Lancaster University Pushnitski, Alexander, Loughborough
University Rieffel, Marc, University of California, Berkeley Sinclair,
Allan, University of Edinburgh Smith, Martin, University of Leeds
Smith, Roger, Texas A&M University Sniady, Piotr, Heidelberg
University Srivastava, Sachi, University of Oxford Todorov, Ivan,
University of the Aegean Turowska, Lyudmila, Chalmers University of
Technology Twomey, Brian, University College Cork Vassout, Stephane,
Universite Pierre et Marie Curie Verrill, Robert, University of Cambridge
Wassermann, Simon, University of Glasgow Wills, Stephen, University of
Nottingham Youngson, Martin, Heriot-Watt University Zsak, Andras,
University of Cambridge
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