Seminars - Page 4

Abstract: For the ax+b semigroup over the natural numbers, which is known to be part of a quasi-lattice ordered group, Laca and Raeburn considered its Nica-Toeplitz algebra and its Cuntz-Nica-Pimsner algebra, with a special appeal to nice presentations by generators and relations as well as the structure of KMS states for a natural dynamics. Shortly thereafter, Brownlowe-an Huef-Laca-Raeburn showed that there are two intermediate quotients between the Nica-Toeplitz algebra and the Cuntz-Nica-Pimsner algebra that exhibit interesting structural properties, especially with regards to KMS states. Since then, analogous quotients have been considered (partly in disguise) in a growing list of case studies on the KMS state structure, e.g. for dilation matrices, self-similar actions, and Baumslag-Solitair monoids. Somewhat surprisingly, all these case studies can be viewed from the perspective of semigroup C*-algebras of right LCM semigroups, and in this talk, I shall describe a unifying perspective on such boundary quotient diagrams. Thereby several questions concerning the general structure of right LCM semigroups are raised.

Abstract: As has been observed by many authors, the Drinfeld double of the q-deformation of a compact Lie group can be regarded as a quantization of the complexification of the original Lie group. Using this point of view, I will discuss irreducible unitary representations of these Drinfeld doubles.

 

Abstract: We discuss a way of constructing noncommutative projective manifolds as inductive and projective limits, generalizing the so-called Berezin quantization for ordinary compact Kähler manifolds. We first review the physical motivation for Berezin quantization and then discuss how the restriction to commutative manifolds limits the use of this quantization. We will also outline how our more general construction appears naturally in the study of the long-time limit of open quantum systems.

Bas Jordans will continue his talk from last week.

Bas Jordans will give a talk with title " Random walks on discrete quantum groups: convergence to the boundary"

Abstract:

For classical random walks there exist two boundaries: the Poisson boundary and the Martin boundary. The relation between these two boundaries is described by the so-called "convergence to the boundary". For random walks on discrete quantum groups both the Poisson boundary and Martin boundary are defined and a non-commutative analogue of convergence to the boundary can be formulated. However, no proof is known for a such a theorem. In the first part of the talk we will discuss the classical and quantum version of convergence to the boundary, explain how these are related and give an overview of what is known in general for the quantum case. In the second part we will discuss the boundary convergence for SUq(2) and for monoidally equivalent quantum groups.

 

Piotr Soltan (University of Warsaw) will give a talk with title: Subgroups of quantum groups, the center and inner automorphisms

Arnaud Brothier, Vanderbilt University (USA) will give a talk with title: Analytic properties for subfactors

Abstract:

I will discuss analytic properties for groups and their generalizations to subfactors, standard invariants, and certain tensor categories.

I will present a class of subfactor planar algebras that are constructed with a group acting on a bipartite graph.

I will show that if the group satisfies a given approximation property (such as amenability, Haagerup property, or weak amenability), then the subfactor planar algebra satisfies it as well.

I will exhibit an infinite family of subfactor planar algebras with non-integer index that are non-amenable, have the Haagerup property, and have the complete metric approximation property.

Alfons van Daele (Leuven) will give a talk with title: The Haar measure on quantum groups

Abstract: At this moment, there is no theory of locally compact quantum groups with axioms from which one can prove the existence of the Haar weights. In general, the existence is part of the axioms. There are however a few cases where the existence can be proven. This is true for compact quantum groups and for discrete quantum groups. We will discuss some aspects of these proofs and see which of them are useful in the general case.

Judith Packer, University of Colorado (Boulder), USA, will give a talk with title: Wavelets associated to representations of higher-rank graph C*-algebras

Abstract: Let $\Lambda$ denote a finite $k$-graph in the sense of A. Kumjian and D. Pask that is strongly connected, and let $\Lambda^{\infty}$ denote its infinite path space. I discuss some recent joint work with C. Farsi, E. Gillaspy, and S. Kang, where we construct a system of functions that we call ``wavelets" on a Hilbert space of square-integrable functions on $\Lambda^{\infty}.$ In so doing, we generalize work of M. Marcolli and A. Paolucci for finite directed graphs to the higher rank case. The key tool is the construction of a representation of the graph $C^*$-algebra $C^{\ast}(\Lambda)$ on $L^2(\Lambda^{\infty},M)$ for the appropriate measure $M.$ When the finite $k$-graph $\Lambda$ in question is strongly connected and aperiodic, the representation of $C^{\ast}(\Lambda)$ that we obtain is faithful.

 

Antoine Julien, NTNU, will give a talk with title: Links between cut-and-project tilings and Diophantine approximation

Abstract: Cut-and-project tilings are obtained by cutting a slice of a higher dimensional lattice and projecting it on a lower dimensional space. The result is a point set which is regular enough (since it originates from a lattice), but is not periodic, provided the direction of the slice is irrational in a suitable sense. In one dimension, typical examples of this construction are Sturmian subshifts. It is known that some of their dynamical properties depend on the arithmetic properties of a certain parameter. In this talk, I will recall some known results by Hedlund and Morse on Sturmian subshifts. Then, I will describe how, even in higher dimensions, the repetition properties of some cut-and-project sets can be linked to problems of simultaneous Diophantine approximation. This is joint work with A. Haynes, H. Koivusalo and J. Walton. 

Nicolai Stammeier (Münster) will give a talk with title "Aiming for accuracy - boundary quotients of right LCM semigroups revisited "

Abstract: I will recall the notions of foundation sets and the boundary quotient for right LCM semigroups. This C*-algebra is obtained by modding out products of defect projections over foundation sets in the full semigroup C*-algebra of the right LCM semigroup. Observing that this is in stark contrast to the standard presentations of C*-algebras in the spirit of Cuntz algebras, where a summation relation gets used, we will discuss the possibility of replacing the product relation by a summation relation and arrive at the accurate refinement property. This feature turns out to be quite common among right LCM semigroups. In fact, we are yet to see an example of a right LCM semigroup that has an insufficient supply of accurate foundation sets. Time permitting, we will leave the realm of right LCM semigroups for the sake of finding semigroups without the accurate refinement property.

Klaus Thomsen, Aarhus Universitet, Denmark, will give a talk with title "KMS states and ground states for generalized gauge actions on graph C*-algebras"

John Quigg, Arizona State University at Tempe, USA, will give a talk with title: Landstad duality and a theorem of Pedersen

Abstract:

In joint work with Steve Kaliszewski and Tron Omland, we show how a theorem of Pedersen characterizing exterior equivalent actions on a C*-algebra can be parlayed into an equivalence between two equivariant categories of C*-algebras. In one category, isomorphisms correspond to outer conjugacies of actions, while isomorphisms in the other category are equivariant isomorphisms of the crossed products that respect the generalized fixed point algebras. This category equivalence is a variation of Landstad's original characterization of actions up to equivariant isomorphism, where we now allow more morphisms. Time permitting, we will compare our "outer duality" with Landstad duality and also with Imai-Takai crossed-product duality.

 

 

Alfons van Daele (University of Leuven, Belgium) will give a talk with title:  Constructing locally compact quantum groups from pairs of *-algebras

Abstract: Let (A,\Delta)  be a finite-dimensional Hopf *-algebra. The dual B of A is again a finite-dimensional Hopf *-algebra. The entire structure of these two Hopf *-algebras is encoded in the *-algebras A and B and the pairing between the two. We will explain how this works. This is in fact true in many more, and more general situations.  In particular, we give an example constructed from a pair of subroups H,K of a group G with the property that the map (h,k)-> hk is a bijection from HxK to G. This method is used in a joined paper with Magnus Landstad where we construct a pair of locally compact quantum groups from such pairs of subgroups of a locally compact group G.

 

Adam Sørensen (UiO) will give a talk with title "Leavitt path algebras - a connection between pure algebra and operator algebras"

Abstract: We will discuss Leavitt path algebras, the purely algebraic cousins of the analytic graph C*-algebras. We will discuss similarities and differences, in particular recent work with Brownlowe on a purely algebraic version of Kirchberg's theorem that all exact C*-algebras embed into O_2.

Marco Matassa (UiO) will give a talk with title The Dolbeault-Dirac operator on quantized projective spaces, revisited

Abstract: In this talk I will present a new construction for the Dolbeault-Dirac operator on quantum projective spaces, the main result being the computation of its square. This clarifies and generalizes some results of D'Andrea-Dąbrowski. Moreover it gives a class of explicit examples of the general construction of Krähmer-Tucker Simmons, which deals with such operators on irreducible generalized flag manifolds.

Roberto Conti (La Sapienza, Rome) will give a talk with title "C*-algebras and Fourier theory"

Robert Yuncken (Univ. Clermont-Ferrand II, France) will give a talk with title: A groupoid approach to pseudodifferential operators

Abstract: Connes introduced the "tangent groupoid" of a manifold as a geometric device for linking a classical pseudodifferential operator to its symbol, yielding a novel proof of the Atiyah-Singer index theorem.  Since then, numerous variations on the tangent groupoid have been produced, each adapted to a different class of pseudodifferential operators.  In this talk we will consider the reverse problem: associating to a given tangent groupoid a pseudodifferential calculus.  We shall show that the kernels of classical pseudodifferential operators are precisely the essentially homogeneous fibrewise distributions on Connes' tangent groupoid.  This leads to a natural pseudodifferential calculus of subelliptic type on a manifold with a filtration on its Lie algebra of vector fields.

Réamonn Ó Buachalla (IMPAN) will give a talk with title: Noncommutative Kähler structures on quantum homogeneous spaces

Abstract:

Building on the definition of a noncommutative complex structure for a general algebra A, I will introduce the notion of a noncommutative Kähler structure for A. In the special case where A is a quantum homogeneous space, I show that many of the fundamental results of classical Kähler geometry follow from the existence of such a structure: Hodge decomposition, Serre duality, the Hard Lefschetz theorem, the Kähler identities, and collapse of the Frölicher spectral sequence at the first page. We then apply these results to Heckenberger and Kolb's differential calculus for quantum projective space, and show that they have cohomology groups of at least classical dimension. Time permitting, I will also discuss the relationship of this work to Connes proposal to study positive Hochschild cocycles as a starting point for noncommutative complex geometry, and Fröchlich, Grandjean, and Recknagel's definition of a Kähler spectral tuple.

Eduard Ortega, NTNU, will give a talk with title: Cuntz-Krieger uniqueness theorems

Abstract: I will make a little survey about Cuntz-Krieger uniqueness theorems and how they help to the study of the ideal structure of the rings to which one can apply them. In certain classes of (C*-)algebras this is described as topologically freeness or condition (L). However they are important classes of rings for which are not known Cuntz-Krieger type theorems. I will present a class of rings, that generalize Leavitt path algebras and Passman crossed products, for which I can totally characterize the Cuntz-Krieger uniqueness theorem.

Erik Bédos will give a talk with title: On the Fourier-Stieltjes algebra of a C*-dynamical system

Abstract: When G is a discrete group, its Fourier-Stieltjes algebra B(G) may be described as the set of coefficient functions associated with unitary representations of G on Hilbert spaces. In a similar way, if Sigma=(A, G, alpha, sigma) is a unital discrete twisted C*-dynamical system, one may let the Fourier-Stieltjes algebra B(Sigma) consist of the functions from G x A into A that arise as coefficient functions of equivariant representations of Sigma on Hilbert A-modules. We will explain how B(Sigma) may be organized as an algebra with conjugation, and show that it may be represented as completely bounded multipliers on the full crossed product C*(Sigma). (This is also known to be true for the reduced crossed product). This is part of an ongoing project with Roberto Conti (Rome).

These are the fifth and and the sixth lectures given by Anders Hansen (Cambridge Univ. and UiO) on Compressed sensing - Theory and Applications.

These are the third and fourth lectures given by Anders Hansen (Cambridge Univ. and UiO) on Compressed sensing - Theory and Applications.

These are the first two of six 45-min. lectures given by Anders Hansen (Cambridge Univ. and UiO) on Compressed sensing - Theory and Applications.