Issue No 10


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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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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