This school is divided in three parts:
Part A: K-theory for operator algebras. Classification of C*-algebras.
Part B: Modular theory for von Neumann algebras and applications to quantumfield theory
Part C: Amenability, hyperbolic groups and operator algebrasPart C provides connections between part A (C*-algebras) and part B (von Neumann algebras).
The theory of operator algebras introduced in the thirties byJ. von Neumann was developed in close relationship with fundamental aspectsof functional analysis, ergodic theory, harmonic analysis and quantum physics.More recently this field has shown many further fruitful interrelations with otherareas of mathematics and mathematical physics.
This school addresses two fundamental aspects of the theory of operator algebraswhich are interesting in themselves and important in applications: K-Theory forC*-algebras and Modular Theory for von Neumann algebras.
This school aims to train PHD students and young researcherswith interdisciplinary interests in mathematics or mathematical physics in somefundamental aspects of operator algebras.Background: Rudiments of functional analysis,
algebra or quantum theory.
Bibliography (some background text books; more specific references will be distributedduring the course)
[1] H. Baumgaertel, Operatoralgebraic methods in quantum field theory, AkademieVerlag, 1995.
[2] B. Blackadar, K-Theory for Operator Algebras, Spinger Verlag, 1986
[3] A. Connes (et al. eds),"Noncommutative Geometry, Springer Verlag, 2004
[4] M. Rordam, Classifications of nuclear C*-algebras, Springer Verlag 2002.
[5] M. Rà¸rdam, F. Larsen, N. Laustsen, An introduction to K-theory for C*-algebras.London Mathematical Society Student Texts, 49. Cambridge University Press,Cambridge, 2000.
[6] S. Stratila, Modular Theory in Operator Algebras, Abacus Press, 1981.
[7] V.S. Sunder, An Invitation to von Neumann algebras, Springer Verlag, 1987.
[8] M. Takesaki, Theory of operator algebras. Vols I,II and III, Springer-Verlag,2002
[9] N.E. Wegge-Olsen, K-Theory and C*-Algebras. An Introduction, Oxford UniversityPress, 1993
Colaborador: Real Sociedad Matemática Española
ProgramaDirector/aPere Ara BertránUniversitat Autà²noma de BarcelonaSecretarios/asFrancesc PereraUniversitat Autà²noma de BarcelonaFernando Lledó MacauUniversidad Carlos III de MadridRWTH-Aachen UniversityLunes 21
10:00 h.Informal overview of C*-algebras and von Neumann algebras
Fernando Lledó MacauUniversidad Carlos III de MadridRWTH-Aachen University 11:30 h.Basics on C*-algebras: positive elements, functional calculus, K-Theory. Part A
Andrew S. TomsYork University, Toronto 15:30 h.Von Neumann algebras and examples. Part B
Fernando Lledó MacauUniversidad Carlos III de MadridRWTH-Aachen University 17:30 h.The Cuntz semigroup: historical origin, essential technical devices. Part A
Francesc PereraUniversitat Autà²noma de Barcelona Martes 22
09:30 h.Hilbert modules and hereditary subalgebras: Kasparov´s theorem. Part A
Pere Ara BertránUniversitat Autà²noma de Barcelona 11:30 h.Modular theory: definition and main results. Part B
Fernando Lledó MacauUniversidad Carlos III de MadridRWTH-Aachen University 15:30 h.The category Cu and its order structure. Part A
Andrew S. TomsYork University, Toronto 17:30 h.Free quantum fields and local quantum theories. Part B
Daniele GuidoUniversità degli studi di Roma "Tor Vergata", Roma Miércoles 23
09:30 h.Coward-Elliott-Ivanescu´s picture of the Cuntz semigroup in terms of the category Cu. Part A
Francesc PereraUniversitat Autà²noma de Barcelona 11:30 h.Examples of modular objects. Part B
Fernando Lledó MacauUniversidad Carlos III de MadridRWTH-Aachen University 13:00 h.From C*-algebras to von Neumann algebras, and back. Part C
Nathanial P. BrownPenn State University Jueves 24
09:30 h.Functorial properties of the Cuntz semigroup: continuity and half exactness. Part A
Andrew S. TomsYork University, Toronto 10:15 h.Amenability for operator algebras. Part C
Nathanial P. BrownPenn State University 11:30 h.Bisognano-Wichmann relations: proofs and interpretations. Part B
Daniele GuidoUniversità degli studi di Roma "Tor Vergata", Roma 15:30 h.Amenability for group actions and exactness. Part C
Nathanial P. BrownPenn State University 17:30 h.Geometric modular action and modular localization. Part B
Daniele GuidoUniversità degli studi di Roma "Tor Vergata", Roma Viernes 25
09:30 h.Classification of C*-algebras, or why do we care about the Cuntz semigroup? Part A
Francesc PereraUniversitat Autà²noma de Barcelona 11:30 h.Hyperbolic groups and their von Neumann algebras. Part C
Nathanial P. BrownPenn State University 13:15 h.Clousure: open problems
