All seminars during 2005.

Kjetil Røysland, University of Oslo:

Iain Raeburn, University of Newcastle:

Arnoud Jacobs, University of Leuwen:

Abstract: We start where we left off in the first seminar (17 Nov. 2004) and go further into the world of multiplicative unitaries and Haar weights. Thanks to the results obtained in Seminar I, we can now use quantum group techniques to obtain some interesting new results. In particular, we will strenghten the main result announced in Seminar I.

Stefaan Vaes, Paris:

Abstract: We construct new examples of ergodic coactions of compact quantum groups, in which the multiplicity of an irreducible corepresentation is strictly larger than the dimension of the latter. These examples are obtained using a bijective correspondence between certain ergodic coactions on C*-algebras and unitary fiber functors on the representation category of a compact quantum group. Nevertheless we do not make use of the abstract language of categories and so, the talk will be essentially elementary.

Nadia S. Larsen, University of Oslo:

Abstract: We propose a generalisation of Exel's crossed product by a single endomorphism and a transfer operator to the case of actions of abelian semigroups of endomorphisms and associated transfer operators. The motivating example for our definition yields new crossed products, not obviously covered by familiar theory. Our technical machinery builds on Fowler's theory of Toeplitz and Cuntz-Pimsner algebras of discrete product systems of Hilbert bimodules, which we need to expand to cover a natural notion of relative Cuntz-Pimsner algebras of product systems.

Ulrich Krahmer, Berlin:

Abstract: We calculate the twisted Hochschild and cyclic homology (in the sense of Kustermans, Murphy and Tuset) of the coordinate algebra of the quantum SL(2) group relative to twisting automorphisms acting by rescaling the standard generators a,b,c,d. We discover a family of automorphisms for which the "twisted" Hochschild dimension coincides with the classical dimension of SL(2, C), thus avoiding the "dimension drop" in Hochschild homology seen for many quantum deformations. Strikingly, the simplest such automorphism is the canonical modular automorphism arising from the Haar functional. In addition, we identify the twisted cyclic cohomology classes corresponding to the three covariant differential calculi over quantum SU(2) discovered by Woronowicz. Se http://front.math.ucdavis.edu/math.QA/0405249

David Kerr, University of Tokyo:

Abstract: We introduce a version of Voiculescu-Brown approximation entropy for isometric automorphisms of Banach spaces and develop within this framework the connection between dynamics and the local theory of Banach spaces discovered by Glasner and Weiss. Our fundamental result concerning this contractive approximation entropy, or CA entropy, characterizes the occurrence of positive values both geometrically and topologically. This leads to various applications; for example, we obtain a geometric description of the topological Pinsker factor and show that a C*-algebra is type I if and only if every multiplier inner *-automorphism has zero CA entropy. We also examine the behaviour of CA entropy under various product constructions and determine its value in many examples, including isometric automorphisms of l_p spaces and noncommutative tensor product shifts. Se http://front.math.ucdavis.edu/math.FA/0407386

Kamran Reihani, University of Oslo:

Abstract: We present some results on classification of simple quotients of the group C*-algebra of a discrete cocompact subgroup of a 6-dimensional nilpotent Lie group. Then we present some calculations for the K-groups of certain transformation group C*-algebras of the tori, including an explicit computation of the ranks of the K-groups of the crossed products generated by Furstenberg transformations.

Kjetil Røysland, University of Oslo:

Kamran Reihani, University of Oslo:

Abstract: We present some calculations for the K-groups of certain transformation group C*-algebras of the n-dimensional torus including the crossed products generated by Furstenberg transformations. In general, It is proved that the ranks of the K-groups of the crossed products associated with the transformations on the n-torus whose corresponding integer matrices over $Bbb{Z}^n$ are unipotent of order n are the same number $a_n$. These computations lead to combinatorial generating functions and an asymptotic formula for the sequence ${a_n}$.

Roberto Conti, Chulalongkorn:

Sergei Silvestrov, Lund Institute of Technology:

Marcelo Laca, University of Victoria:

Abstract: I will discuss addition and multiplication on the rational numbers, and how they lead to a dynamical system related to the Hecke C*-algebra of Bost and Connes. I will then use a theorem of Williams to describe in detail the primitive ideal space of the associated crossed product. This is joint work with Iain Raeburn.

Erling Størmer:

Sergei Neshveyev:

Nadia S. Larsen:

Abstract: This is joint work with Iain Raeburn.

Alfons van Daele, Leuven:

Abstract: The operator algebra approach to quantum groups resulted in a nice theory of locally compact quantum groups. It seems natural to formulate such a theory in the context of C*-algebras because the theory of C*-algebras can be viewed as the non-commutative version of the theory of locally compact spaces. However, for many other, more practical reasons, it turns out to be more appropriate to develop locally compact quantum groups in a von Neumann algebraic framework. In this talk, we will explain why this is so. We will also show that it is rather easy to pass from one setting to the other and in particular, we will see how the two approaches eventually yield the same objects and the same results.

The results have been known for some time, but our approach is somewhat new.

Ola Bratteli:

Abstract: This is a lecture given in Singapore 5 August 2004. The starting point is Baggett and Merrills construction of scaling functions and wavelet functions from the action of dilations and translations on a preferred scaling subspace. The method can be extended to construct wavelet frames on more non-traditional structures by replacing the circle and the map z ---> z^N by more general dynamical systems as, say, a Julia set and the action of the corresponding rational function on the set. On the way we construct solutions of operator relations of the form UT = P(T)U where U is unitary, T is normal and P is a rational or more general function.

Kjetil Røysland:

Kjetil Røysland:

Sindre Duedahl:

Louis Labuchagne, Pretoria:

Abstract: We show that almost all of the famous circle of theorems characterizing the generalization of classical H^infty known as `weak* Dirichlet algebras', can be extended to Arveson's very general noncommutative setting of maximal subdiagonal subalgebras of finite von Neumann algebras. This solves the long-standing open question of the equivalence of principles such as Szegö's theorem, the weak* density of A + A*, and so on, within the noncommutative setting. Included in this circle of equivalent conditions is a description of the A-invariant subspaces of L^2(M, au), and some general Beurling-Nevanlinna factorization formulae for functions in noncommutative L^2. These results should be useful in the future development of noncommutative H^p theory. The theory presented is based on joint work with David Blecher.

Louis Labuchagne, Pretoria:

Abstract: We give an account of A.E.F. Djemai´s paper with this title in International Journal of Theoretical Physics, 34 (1995).

Kamran Reihani:

Abstract: We review the concept of spectral triples for C*-algebras by some examples. Then we introduce a method to construct spectral triples for crossed products.

Erik Bedos:

Abstract: We will report on a joint work with Roberto Conti and discuss summation processes à la Fejer/ Abel-Poisson for Fourier series of elements in the reduced twisted group C*-algebra of a discrete group.

Erik Bedos:

Abstract: We will continue the report on the joint work with Roberto Conti and discuss summation processes à la Fejer/ Abel-Poisson for Fourier series of elements in the reduced twisted group C*-algebra of a discrete group.