
THE ABEL SYMPOSIUM 2004 

Operator Algebras  Titles and abstractsThis page will also contain the unrefereed contributions to the proceedings of the symposium. The refereed versions of these contributions will appear as volume 1 in a series of books published by Springer Verlag in collaboration with the Norwegian Mathematical Society. For details regarding submitting contributions, see here. Renormalization and motivic symmetry We describe our recent work with M. Marcolli on the classification of equisingular flat connections. We show that the divergencies of quantum field theory exactly provide the data allowing to define an action of a specific ``motivic symmetry group" U* on the set of physical theories. The renormalization group appears as a canonical one parameter subgroup of U*. Our result allows to interpret renormalization as a RiemannHilbert correspondence thus clarifying the role of the Birkhoff decomposition appearing in my joint work with Kreimer. It also allows to build a universal singular frame in which the divergencies disappear. Connections between the algebraic KTheory of C*algebras and the topological Ktheory of locally convex algebras We introduce a new version of topological Ktheory, Khomology and bivariant Ktheory in a nearly completely algebraic manner. These theories can be applied and computed for many new types of algebras. A prominent example is the Weyl algebra. The determination of the coefficients of the bivariant theory which is crucial for explicit computations, relies on some arguments from algebraic Ktheory for C*algebras. Hyperinvariant subspaces for some Bcircular operators Some basic facts and questions about hyperinvariant subspaces and von Neumann algebras will be reviewed. We will introduce the Bcircular operators, which are a spcial case of Speichers BGaussian operators. A new technique of Foias, Jung, Ko and Pearcy will be used to construct hyperinvariant subspaces for a large class of Bcircular operators. Automorphisms on nonsimple purely infinite C*algebras Most C*classification results are established in such a way that one with little or no
extra effort can prove that any isomorphism between a pair of invariants may be lifted
to a *isomorphism. Rørdam's classification result concerning purely infinite
C*algebras with one nontrivial ideal forms a notable exception to this rule.
Modular invariants, subfactors and twisted Ktheory
Orbit equivalence of minimal actions on the Cantor set Both in the measurable and in the Borel case, hyperfinite actions are well understood and classified up to orbit equivalence. In particular any free Borel action of Z^{2} is (orbit equivalent to) hyperfinite. In the case of (minimal) topological actions of Z^{2} on the Cantor set, the similar question is still open. In this talk, I will present recent developments in the study of this problem. These developments come from a work in progress with I. Putnam (University of Victoria, Victoria) and C. Skau (NTNU, Trondheim). Nonseparable AFalgebras The theory of classification of C*algebras started with the classification of separable AFalgebras by George Elliott after the works by Glimm, Dixmier and Bratteli. Since then, many properties have been studied for separable AFalgebras. In my talk, I will give two phenomena of nonseparable AFalgebras which never occur in the separable world. One phenomenon is that a same Bratteli diagram can give different AFalgebras. The other is that there exists a prime but not primitive AFalgebra. Properties of algebras defined by central sequences PDF: Abstract Flows on a separable C*algebra Qlattices: quantum statistical mechanics and Galois theory The talk is based on joint work with Alain Connes. The noncommutative space of 2dimensional Q lattices up to scaling modulo the equivalence relation of commensurability generalizes the BostConnes system (which corresponds to the analogous space of 1dimensional Qlattices). The system has phase transitions and spontaneous symmetry breaking. The action of the symmetry group of the system on the KMS states at zero temperature can be expressed in terms of the Galois theory of the modular field. On the assembly map and the BaumConnes conjecture for quantum groups The life and works of GKP Gert Kjærgård Pedersen (GKP) was born in April 1940 and died in March 2004. At this point, he had been an active mathematical researcher for more than 40 years, a piano player for more than 50 years and a singer for more than 60 years. The talk will try to describe some aspects of his life and works  the latter amounting to more than 100 publications. C*algebras that absorb the JiangSu algebra The JiangSu algebra Z is a unital, simple, quasidiagonal, separable, nuclear, infinite dimensional C*algebra with the same Elliott invariant as the complex numbers. It absorbs itself tensorially, and any C*algebra with weakly unperforated K_{0}group which falls into Elliott classification scheme will absorb Z. In this talk we show that if A is a Zabsorbing C*algebra, then the Cuntz semigroup W(A) of equivalence classes of positive elements in matrix algebras over A is almost unperforated; if A is exact, then A is purely infinite if and only if A is traceless; if A is separable and nuclear, then A absorbs the Cuntz algebra O_{∞} if and only if A is traceless; and if A is simple and unital, then A is of stable rank one if and only if A is finite. One can also describe when A is of real rank zero. Computations of free entropy dimension and L^{2} cohomology We discuss connections between free probability theory and L^{2} cohomology. This connection leads to estimates and computations of free entropy dimension. For example (joint work with I. Mineyev) we compute the nonmicrostates free entropy dimension of an arbitrary finitelygenerated group in terms of its L^{2} Betti numbers. Pentagonal transformations and Kac cohomology (joint work with S. Baaj and S. Vaes) Outer actions of a group on a factor Reduced HNN extensions of von Neumann algebras There are two fundamental constructions in combinatorial or geometric
group theory, which are those of free products with amalgamations and
of HNN (G.Higman, B.H.Neumann and H.Neumann) extensions. In this talk,
I will introduce reduced HNN extensions of von Neumann algebras as well
as those of C On quantum ax+b group One of the unexpected features of the known constructions of the quantum ax+b group is the fact that they work only for a small set of distinguished values of the deformation parameters h. Up to now it is not clear, whether this phenomenon is a result of the particular method of construction, or reflects some more fundamental properties of locally compact quantum groups. We shall present a new approach to the problem. 