C*-algebra seminar - Page 2

Time and place: May 19, 2016 2:15 PM - 4:00 PM, NHA B735

Abstract: This talk addresses some of the fundamental barriers in the theory of computations. Many computational problems can be solved as follows: a sequence of approximations is created by an algorithm, and the solution to the problem is the limit of this sequence (think about computing eigenvalues of a matrix for example). However, as we demonstrate, for several basic problems in computations such as computing spectra of operators, solutions to inverse problems, roots of polynomials using rational maps, solutions to convex optimization problems, imaging problems etc. such a procedure based on one limit is impossible. Yet, one can compute solutions to these problems, but only by using several limits. This may come as a surprise, however, this touches onto the boundaries of computational mathematics. To analyze this phenomenon we use the Solvability Complexity Index (SCI). The SCI is the smallest number of limits needed in order to compute a desired quantity. The SCI phenomenon is independent of the axiomatic setup and hence any theory aiming at establishing the foundations of computational mathematics will have to include the so called SCI Hierarchy. We will specifically discuss the vast amount of classification problems in this non-collapsing complexity/computability hierarchy that occur in inverse problems, compressed sensing problems, l1 and TV optimization problems, spectral problems, PDEs and computational mathematics in general.

Time and place: May 4, 2016 2:15 PM - 4:00 PM, NHA B735

Abstract:  In a recent work with R. Conti (La Sapienza Univ., Rome), we have introduced a notion of positive definiteness for certain functions associated to a (unital, discrete) C*-dynamical system. We will sketch the proof of a Gelfand-Raikov type theorem for such functions and use it to construct complete positive maps on the full and the reduced C*-crossed products of the system. We will also explain how a natural definition of amenability for C*-dynamical systems emerges from our work. 

Time and place: Apr. 20, 2016 2:15 PM - 3:15 PM, NHA B735

Abstract: The talk will be on positive linear maps of the n x n matrices into itself, a topic which has become quite popular in quantum information theory.  The maps closest to physics are the completely positive ones. I´ll discuss an approximation by a completely positive map to a positive map via the trace , called the “structural physical approximation”, the SPA of the map. Much of the talk will circle around a counter example  to a conjecture on the SPA.

Time and place: Apr. 13, 2016 2:15 PM - 4:00 PM, NHA B735

Abstract: In the classification program for C*-algebras some of the usual assumptions put on the algebras are that they are simple or have at most have finitely many ideals. We often also want algebras that have real rank 0. In this talk we will discuss how to classify certain graph algebras with uncountably many ideals and without real rank 0. There will be examples and applications. Joint work with S. Eilers, G. Restorff, and E. Ruiz

Time and place: Apr. 6, 2016 2:15 PM - 4:00 PM, NHA B735

Abstract: There are many interesting examples of groups acting on trees, arising in various fields (e.g. combinatorial group theory, number theory, geometry). When a group acts on a tree, it necessarily also acts on the boundary of the tree, a (totally disconnected) compact Hausdorff space.  The C*-algebras obtained from the crossed product construction include many fundamental examples.  I will describe methods for analyzing such crossed products, developed in joint work with Nathan Brownlowe, Alex Mundey, David Pask and Anne Thomas.

Time and place: Mar. 16, 2016 2:15 PM - 4:00 PM, NHA B735

Abstract: In this follow-up talk, I shall outline how the boundary quotient diagram may be useful for K-theoretic considerations. We start with the diagram within the context of integral dynamics, and then speculate about potentially promising directions of generalizations.

Time and place: Mar. 9, 2016 2:15 PM - 4:00 PM, NHA B735

Abstract: In this follow-up talk, we shall review the results on the structure of KMS states from the case studies of - the ax+b semigroup over the natural numbers (Laca-Raeburn and Brownlowe-an Huef-Laca-Raeburn), - integer dilation matrices (Laca-Raeburn-Ramagge), - self-similar actions (Laca-Raeburn-Ramagge-Whittaker), and - Baumslag-Solitair monoids (Clark-an Huef-Raeburn) from the perspective of the boundary quotient diagram for the respective right LCM semigroups. We will also discuss (to some extent) similarities and differences of the proofs among these cases.

Time and place: Mar. 2, 2016 2:15 PM - 4:00 PM, NHA B735

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.

Time and place: Feb. 2, 2016 2:15 PM - 4:00 PM, NHA B638

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.

Time and place: Jan. 27, 2016 2:15 PM - 4:00 PM, NHA B735

 

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.

Time and place: Jan. 20, 2016 2:15 PM - Jan. 13, 2016 4:00 PM, B735

Bas Jordans will continue his talk from last week.

Time and place: Jan. 13, 2016 2:15 PM - 4:00 PM, B735

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.

 

Time and place: Nov. 25, 2015 2:15 PM - 4:00 PM, NHA 735

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

Time and place: Nov. 24, 2015 2:00 PM - 4:00 PM, NHA B735

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.

Time and place: Oct. 21, 2015 2:15 PM - 4:00 PM, B738

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.

Time and place: Sep. 30, 2015 2:15 PM - 4:00 PM, B738

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.

 

Time and place: Sep. 25, 2015 10:15 AM - 12:00 PM, B 735

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. 

Time and place: Sep. 2, 2015 2:15 PM - 4:00 PM, NHA 738

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.

Time and place: June 24, 2015 10:15 AM - 12:00 PM, NHA B735

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

Time and place: June 17, 2015 2:15 PM - 4:00 PM, NHA B735

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.

 

 

Time and place: June 9, 2015 2:15 PM - 4:00 PM, NHA Building B738

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.

 

Time and place: May 27, 2015 2:15 PM - 4:00 PM, NHA 738

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.

Time and place: May 26, 2015 2:00 PM - 4:00 PM, NHA B735

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.

Time and place: May 12, 2015 11:00 AM - 12:00 PM, B735

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

Time and place: May 6, 2015 2:15 PM - 4:00 PM, B735

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.