Past seminars in 2006.

Alfons van Daele, University of Leuven:

Abstract: In this talk, we plan to explain how the dual $(hat M, hatDelta)$ of a locally compact quantum group $(M,Delta)$ is constructed (within the von Neumann algebra setting). We will explain how the dual Haar weights are constructed (using the theory of left Hilbert algebras) and indicate how it is shown that they are invariant. Finally, we will explain why the dual of $(hat M,hatDelta)$ is again $(M,Delta)$. We will also show how various objects of the original pair $(M,Delta)$ and of the dual pair $(hat M,hatDelta)$ are related with each other through a collection of nice formulas (thus illustrating the beauty of the Kustermans-Vaes theory). 

Ulrich Kramer, Berlin:

Sergey Neshveyev, University of Oslo:

Abstract: An account is given of the first three papers here: http://front.math.ucdavis.edu/search?a=Sorin+Popa&t=&q=&c=&n=40&s=Listings

Olav Gravir Imenes, University of Oslo:

Abstract: How can the theory of moduli spaces be used to model physical systems? Every physical system can be described by certain parameters (position, momentum, etc.). The idea of Laudal is that the geometry (necessarily non-commutative) of the moduli-space (i.e. the space consisting of all such possibilities, or possible "states" of the system) describes how these parameters change. We start by considering an algebra Ph(A) parameterising position and momenta. A given representation of Ph(A) into an algebra of operators together with a force-law are the initial conditions of the system. The geometry of the moduli-space of the representations then governs how the system evolves. Our first example will be of the system consisting of a classical observer observing a classical object. Later we consider quantum-mechanical systems, by considering representations of Ph(A) on both finite and infinite dimensional vector spaces.

Olav Gravir Imenes, University of Oslo:

Abstract: This is a continuation of the seminar talk 15 February

Michael Mueger, Radboud University Nijmegen::

Abstract: The talk is based on the paper with the same title in http://front.math.ucdavis.edu/math.QA/0403322

We show that a quantum field theory A living on the line and having a group G of inner symmetries gives rise to a category GLoc A of twisted representations. This category is a braided crossed G-category in the sense of Turaev. Its degree zero subcategory is braided and equivalent to the usual representation category Rep A. Combining this with [29], where Rep A was proven to be modular for a nice class of rational conformal models, and with the construction of invariants of G-manifolds in [60], we obtain an equivariant version of the following chain of constructions: Rational CFT -> modular category -> 3-manifold invariant.

We then study the relation between GLoc A and the braided (in the usual sense) representation category Rep A^G of the orbifold theory A^G. We prove the equivalence Rep A^G = (GLoc A)^G, which is a rigorous implementation of the insight that one needs to take the twisted representations of A into account in order to determine Rep A^G. In the opposite direction we have GLoc A = Rep A^G times S, where S subset Rep A^G is the full subcategory of representations of A^G contained in the vacuum representation of A, and times refers to the Galois extensions of braided tensor categories of [44,48].

If A is completely rational and G is finite we prove that A has g-twisted representations for every g in G. In the holomorphic case (where Rep A = Vect_C) this allows to classify the possible categories GLoc A and to clarify the role of the twisted quantum doubles D^omega(G) in this context, as will be done in a sequel. We conclude with some remarks on non-holomorphic orbifolds and surprising counterexamples concerning permutation orbifolds.

Kjetil Røysland, University of Oslo:

Dorin Dutkay, Rutgers University:

Abstract: We show how the multiresolution technique from wavelet theory can be implemented in the setting of affine fractals. The essential link is the self-similarity, present both in scaling functions and in fractals. The rich multiresolution structure enables us to construct Fourier bases on fractal measures. The construction involves certain covariant representations and dilations of endomorphisms to automorphisms. The talk is mainly based on the following joint paper with Palle Jorgensen: Iterated function systems, Ruelle operators, and invariant projective measures. See http://front.math.ucdavis.edu/math.DS/0501077

Matilde Marcolli, Max Planck Institute, Bonn::

Abstract: I will talk about joint work with Alain Connes where we describe the Riemann-Hilbert correspondence underlying the Connes-Kreimer theory of perturbative renormalization. The resulting Galois group is universal with respect to physical theories and can be identified with a motivic Galois group.

Sindre Duedal, University of Oslo:

Abstract: The talk is based on the following paper by Robertson and Sims: http://front.math.ucdavis.edu/math.OA/0602120

Maxim Nazarov, York University, UK:

Abstract: The group S of all finite permutations of the infinite sequence 1,2,... is called the infinite symmetric group. This is the inductive limit of the sequence of finite symmetric groups S(1),S(2),... where each S(N-1) is the subgroup of S(N) preserving the number N, when S(N) acts by permutations of the numbers 1,...N. In 1964 Thoma gave a classification of the II(1) factor representations of the infinite symmetric group S.

The group S has a non-trivial central extension T, by the group of 2 elements. Hence the group S has projective representations that cannot be reduced to linear ones. It turns out that the classification of Thoma can be extended to all II(1) factor representations of the group T. Moreover, each of these representations admits an explicit realization. This talk reviews the classical results of Thoma about S, and new results about T. 

Maxim Nazarov, York University, UK:

Abstract: The degenerate affine Hecke algebra H(N) is a deformation of the crossed product of the symmetric group S(N) with the ring of polynomials in n variables. This algebra appears naturally when considering the action of the general linear group GL(n) on the tensor product of N copies of the n-dimensional vector space. According to the classical theorem of Schur and Weyl, the actions of S(N) and GL(n) then generate mutual commutants.

The aim of this talk is to describe the algebra which plays the role of H(N) when the the group GL(n) is replaced by any other classical group, namely by the orthogonal group O(n), or by the symplectic group Sp(n). This algebra was introduced by myself and is called the degenerate BMW algebra. Its definition involves the "basic construction" of Jones.

Rashid elHarti, University Hassan I, Settat, Morocco::

Abstract: A pro-C*-algebra is a (projective) limit of C*-algebras in the category of topological *-algebras. From the perspective of non-commutative geometry, pro-C*-algebras can be seen as non-commutative k-spaces. An element of a pro-C*-algebra is bounded if there is a uniform bound for the norm of its images under any continuous ?-homomorphism into a C*-algebra.

The *-subalgebra consisting of the bounded elements turns out to be a C*-algebra. In this paper, we investigate pro-C*-algebras from a categorical point of view. We study the functor ( )_b that assigns to a pro-C*-algebra the C*-algebra of its bounded elements, which is the dual of the Stone-Cech-compactification. We show that ( )_b is a coreflector, and it preserves exact sequences. A generalization of the Gelfand duality for commutative unital pro-C*-algebras is also presented.

Juliana Erlijman, University of Regina, Canada:

Abstract: Review of certain construction of subfactors in the context of braided tensor categories and of the computation of important invariants.

Roberto Conti, Chulalongkorn University, Bangkok, Thailand:

Abstract: We wish to extend the Gelfand theorem for abelian C*-algebras to a categorical framework. To this end, we'll introduce some relevant tools and present partial results. Some connections to the literature are also pointed out.

Kjetil Røysland, University of Oslo:

Anilesh Mohari, Calcutta:

Abstract: Though our primary motivation is to study properties of the ground states of a Hamiltonian for quantum spin chain, in this exposition we propose a general methodology valid for a translation invariant state. To that end we study the associated Popescu systems [Po,BJKW] representing the translation invariant state and find an useful criteria for the state to be pure. As a simple consequence of this criteria we prove a surprising result which says that there exists no translation invariant and SU(2)-gauge invariant pure state for half-odd integer spin chain. As a consequence of this result we prove that ground state for half-odd integer spin chain is not unique for any SU(2)-gauge invariant Hamiltonian. In other words this exhibits that there is a phase transition in the ground state for anti-ferromagnetic isotropic Heisenberg half-odd integer spin chain.

Anilesh Mohari, Calcutta:

Abstract: This is a continuation of the talk given 1.June, based on the paper http://front.math.ucdavis.edu/math-ph/0509049

Adam Rennie, University of Copenhagen:

Abstract: I will describe, mostly by examples, how one can bootstrap the general semifinite index theorem to study algebras without trace. This leads to an index theorem in twisted cyclic cohomology, and a new K-type group. I will outline some of the many outstanding questions left by our constructions. 

John Quigg, Arizona State University, USA:

Abstract: Our goal is to derive all known imprimitivity theorems involving groups, using Yamagami's symmetric imprimitivity theorem for Fell bundles over groupoids. (Joint with S. Kaliszewski and P. Muhly.) 

Iain Raeburn, University of Newcastle, Australia:

Masaki Izumi, Kyoto University, Japan:

Talk given as part of Felleskollokviet.

Masaki Izumi, Kyoto University, Japan:

This talk is a continuation of the talk given on Thursday, August 31.

Erik Bedos, Oslo:

Abstract: In a recent paper, B. Abadie and K. Dykema introduced a notion of unique ergodicity relative to the fixed point algebra for a single automorphism of a unital C*-algebra. We will discuss some thoughts about a corresponding notion for an action of a discrete group on a unital C*-algebra, relative to an invariant C*-subalgebra.

Nadia S. Larsen, Oslo:

Abstract: I plan to go through the material of the preprint "Hecke algebras of semidirect products and the finite part of the Connes-Marcolli C*-algebra", joint with M. Laca and S. Neshveyev.

Nadia S. Larsen, Oslo:

Abstract: After a brief review of the talk given on September 26, I will continue towards proving the theorem on phase-transition.

Kjetil Røysland, Oslo:

Toke M. Carlsen, University of Newcastle, Australia:

Abstract: Shift spaces come in two flavours: one-sided and two-sided. The main difference between these two flavours is that the shift map of a two-sided shift space is a homeomorphism, whereas the shift map of a one-sided shift space is not injective (unless the shift space is finite). This difference is reflected in the way we associate C*-algebras to one- and two-sided shift spaces; to a two-sided shift space we associate the crossed product of the continuous functions of the shift space by the integer, and to a one-sided shift space we associate a Cuntz-Krieger like algebra (it is a Cuntz-Krieger algebra if the shift space is sofic).

One can in a natural way associate a one-sided shift space to every two-sided shift space (one just restrict every two-sided sequence to a one-sided sequence). In general the C*-algebra of the two-sided shift space and the C*-algebra of the corresponding one-sided shift space are quite different, but for a certain subclass of shift spaces which includes shift spaces of primitive substitutions and of Sturmian and non-regular Toeplitz sequences, the former is a quotient of the latter. 

Kamran Reihani, Oslo:

Abstract: Following our joint work with G. Cornelissen, M. Marcolli and A. Vdovina, "Noncommutative Geometry on Trees and Buildings", we discuss construction of spectral triples for Kleinian Schottky groups and for actions of groups on trees. The Cuntz-Krieger algebras appear as the crossed product C*-algebra associated with these actions. We review the theta summable case given by Consani and Marcolli. Then we improve it to finitely summable construction, upon passing to the stabilization of the corresponding Cuntz-Krieger algebra.

Kamran Reihani, Oslo:

Abstract: This talk is a continuation of the talk given on October 25.

Erik Alfsen, Oslo:

Abstract: In this talk, I will sketch a proof which involves some background material on Jordan algebras. In fact, I shall need the structure theory of the Jordan algebras called JBW-algebras. It resembles the structure theory of von Neumann algebras, but it has a much richer supply of "factors" (among those the "spin factors", which will play an important role in my proof). 

Erik Alfsen, Oslo:

Abstract: This is a continuation of the talk from November 8.

Toke M. Carlsen, University of Newcastle, Australia:

Abstract: Higher rank graphs and their C*-algebras were introduced in 2000 by Kumjian and Pask in order to generalise a recent construction by Robertson and Steger and graph C*-algebras. The advantage of C*-algebras of higher rank graphs over ordinary graph C*-algebra is that we get more C*-algebras from higher rank graphs than from ordinary graphs. Pask, Raeburn, Rørdam and Sims have for instance recently shown how to construct AT-algebras, such as for example Bunce-Deddens algebras, as C*-algebras of higher rank graphs.

Kumjian and Pask's original construction of C*-algebras of higher rank graphs only worked for row-finite higher rank graphs. The construction was later extended by Raeburn, Sims and Yeend to so-called finitely aligned higher rank graphs.

I will in this talk show how to extend the definition of the C*-algebra of a higher rank graph to the whole class of higher rank graphs in a natural way involving inverse semigroups and Boolean rings. (This is joint work with Gwion Evans.)

Roberto Conti, Rome:

Abstract: We will present a general introduction to the various notions of independence that have appeared in operator algebras and quantum physics, and discuss some examples.