Table Of ContentABEL SYMPOSIA
Edited by the Norwegian Mathematical Society
First row from left: George Elliott, Dorte Olesen, Masamichi Takesaki, Joachim Cuntz, Matilde
Marcolli, Alain Connes, Erling Størmer, Ola Bratteli.
Second row from left: Trond Digernes, Ken Dykema, Sergey Neshveyev, Nadia Larsen, Thierry
Giordano, Heidi Dahl, Takeshi Katsura, Mikael Rørdam, Søren Eilers, Erik Alfsen, David Evans,
Magnus Landstad, Christian Skau, John Rognes.
Last row from left: Anders Hansen, Akitaka Kishimoto, Yosimichi Ueda, Sindre Duedahl, Dimitri
Shlyakhtenko, Kjetil Røysland, Stanisław Lech Woronowicz, Georges Skandalis, Toke Meier Carlsen,
Eberhard Kirchberg, Erik Bedos, Ragnar Winter, Ryszard Nest.
Alain Connes with predecessors Niels Henrik Abel and Evariste Galois.
Photo credits: Samfoto/Svein Erik Dahl
.
Ola Bratteli Sergey Neshveyev
Christian Skau
Editors
Operator Algebras
The Abel Symposium 2004
Proceedings of the First Abel Symposium, Oslo,
September 3- 5, 2004
ABC
Editors
Ola Bratteli Sergey Neshveyev
Department of Mathematics Department of Mathematics
University of Oslo University of Oslo
PB 1053 - Blindern PB 1053 - Blindern
0316 Oslo 0316Oslo
Norway Norway
e-mail:[email protected] e-mail:[email protected]
Christian Skau
Department of Mathematical Sciences
Norwegian University of Science and Technology
7491 Trondheim
Norway
e-mail:[email protected]
LibraryofCongressControlNumber:2006926217
MathematicsSubjectClassification(2000):37A20, 37A60, 46L35, 46L40, 46L54, 46L55,
46L60, 46L80, 46L85, 46L87
ISBN-10 3-540-34196-XSpringerBerlinHeidelbergNewYork
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Preface to the Series
TheNielsHenrikAbelMemorialFundwasestablishedbytheNorwegiangov-
ernmentonJanuary1.2002.ThemainobjectiveistohonorthegreatNorwe-
gian mathematician Niels Henrik Abel by awarding an international prize for
outstanding scientific work in the field of mathematics. The prize shall con-
tributetowardsraisingthestatusofmathematicsinsocietyandstimulatethe
interest for science among school children and students. In keeping with this
objective the board of the Abel fund has decided to finance one or two Abel
Symposia each year. The topic may be selected broadly in the area of pure
and applied mathematics. The Symposia should be at the highest interna-
tionallevel,andservetobuildbridgesbetweenthenationalandinternational
researchcommunities.TheNorwegianMathematicalSocietyisresponsiblefor
the events. It has also been decided that the contributions from these Sym-
posia should be presented in a series of proceedings, and Springer Verlag has
enthusiastically agreed to publish the series. The board of the Niels Henrik
AbelMemorialFundisconfidentthattheserieswillbeavaluablecontribution
to the mathematical literature.
Ragnar Winther
Chairman of the board of the Niels Henrik Abel Memorial Fund
Preface
The theme of this symposium was operator algebras in a wide sense. In the
last 40 years operator algebras has developed from a rather special disci-
pline within functional analysis to become a central field in mathematics
often described as “non-commutative geometry” (see for example the book
“Non-Commutative Geometry” by the Fields medalist Alain Connes). It has
branched out in several sub-disciplines and made contact with other subjects
like for example mathematical physics, algebraic topology, geometry, dynam-
ical systems, knot theory, ergodic theory, wavelets, representations of groups
and quantum groups. Norway has a relatively strong group of researchers in
the subject, which contributed to the award of the first symposium in the
series of Abel Symposia to this group. The contributions to this volume give
a state-of-the-art account of some of these sub-disciplines and the variety of
topicsreflecttosomeextenthowthesubjecthasbranchedout.Wearehappy
that some of the top researchers in the field were willing to contribute.
The basic field of operator algebras is classified within mathematics as
part of functional analysis. Functional analysis treats analysis on infinite di-
mensional spaces by using topological concepts. A linear map between two
such spaces is called an operator. Examples are differential and integral op-
erators. An important feature is that the composition of two operators is a
non-commutative operation. It is often convenient not just to consider a sin-
gleoperator,butawholeclassofoperatorswhichformanalgebraandsatisfy
sometechnicalconditions.ThebasictheoryofOperatoralgebrasencompasses
C∗-algebras and von Neumann algebras. The study of C∗-algebras could be
called non-commutative topology and the study of von Neumann algebras
non-commutative measure theory, since this study reduces to the study of
topology and measure theory in the case that the algebras are abelian.
The symposium took place in Oslo, September 3–5, 2004, and was orga-
nized by
• Ola Bratteli, University of Oslo
• Alain Connes, Collège de France, Paris
VIII Preface to the Series
• Joachim Cuntz, Westfälische Wilhelms-Universität Münster
• Sergei Neshveyev, University of Oslo
• Christian Skau, Norwegian University of Science and Technology,
Trondheim
• Erling Størmer, University of Oslo
ThesymposiumwasdedicatedtothememoryofGertKjærgaardPedersen,
the pater familias of the operator algebraists in Denmark, who was invited
to give a talk, but died March 15, 2004. One of his last contributions to
mathematics is published in these proceedings.
The following senior researchers from abroad participated and all gave talks:
• Alain Connes, Paris • Matilde Marcolli, Bonn
• Joachim Cuntz, Münster • Ryszard Nest, Copenhagen
• Ken Dykema, Texas A&M • Dorte Olesen, Copenhagen
• Søren Eilers, Copenhagen • Mikael Rørdam, Odense
• George Elliott, Toronto • Dimitri Shlyakhtenko, UCLA
• David Evans, Cardiff • Georges Skandalis, Paris
• Thierry Giordano, Ottawa • Masamichi Takesaki, UCLA
• Takeshi Katsura, Sapporo • Yoshimichi Ueda, Kyushu
• Eberhard Kirchberg, Berlin • Stanisław Lech Woronowicz,
• Akitaka Kishimoto, Sapporo Warsaw
Senior researchers and postdocs from Oslo and Trondheim who participated:
• Erik Alfsen • Magnus Landstad
• Erik Bedos • Nadia Larsen
• Ola Bratteli • Sergey Neshveyev
• Toke Meier Carlsen • Christian Skau
• Trond Digernes • Erling Størmer
Doctoral students from Oslo and Trondheim who participated:
• Sindre Duedahl
• Kjetil Røysland
• Heidi Dahl
More information about the symposium may be found at this web page:
http://abelsymposium.no/2004
Oslo and Trondheim 27 March 2006
Ola Bratteli Sergei Neshveyev Christian Skau
Contents
Interpolation by Projections in C∗-Algebras
Lawrence G. Brown, Gert K. Pedersen.............................. 1
KMS States and Complex Multiplication (Part II)
Alain Connes, Matilde Marcolli, Niranjan Ramachandran ............. 15
An Algebraic Description of Boundary Maps Used
in Index Theory
Joachim Cuntz .................................................. 61
On Rørdam’s Classification of Certain C∗-Algebras with One
Non-Trivial Ideal
Søren Eilers, Gunnar Restorff ..................................... 87
Perturbation of Hausdorff Moment Sequences,
and an Application to the Theory
of C∗-Algebras of Real Rank Zero
George A. Elliott, Mikael Rørdam .................................. 97
Twisted K-Theory and Modular Invariants: I Quantum
Doubles of Finite Groups
David E Evans ..................................................117
The Orbit Structure of Cantor Minimal Z2-Systems
Thierry Giordano, Ian F. Putnam, Christian F. Skau.................145
Outer Actions of a Group on a Factor
Yoshikazu Katayama, Masamichi Takesaki ..........................161
Non-Separable AF-Algebras
Takeshi Katsura .................................................165
X Contents
Central Sequences in C∗-Algebras and Strongly Purely
Infinite Algebras
Eberhard Kirchberg...............................................175
Lifting of an Asymptotically Inner Flow for a Separable
C∗-Algebra
Akitaka Kishimoto ...............................................233
Remarks on Free Entropy Dimension
Dimitri Shlyakhtenko .............................................249
Notes on Treeability and Costs for Discrete Groupoids
in Operator Algebra Framework
Yoshimichi Ueda.................................................259
